STARTED— JULY 8, 2017

FINISHED—august 6, 2017 11:30pm

8/19/17; 9/1/17;11/1/17

*****MATHEMATICS*****

DEDICATED TO

***** MERYEM HEBERT

***** CATHY

RIP:

C.P. RAMANUJAM

RENATO CACCIOPPOLI

TAIRA HONDA

FELIX HAUSDORFF

YUTAKA TANIYAMA

ALAN TURING

JULIA ROBINSON’S FATHER

CONTENTS

1. DEFINITION OF MATHEMATICS

1. MATHEMATICAL QUOTES

2. HISTORY OF MATHEMATICS

3. MATHEMATICIANS

4. FIELDS IN MATHEMATICS:

A. ARITHMETRIC

B. ALGEBRA

C. GEOMETRY

D. TRIGONOMETRY

E. PRECALCULUS

F. CALCULUS, DIFFERENTIAL EQUATIONS, ANALYSIS

G. REST MATH FIELDS

1. NUMBER THEORY

2. TOPOLOGY

3. APPLIED

4. MISCELLANEOUS FIELDS

5. ALL THE REST

5. FORMULAS:

A. PRACTICAL FORMULAS

B. BEAUTIFUL FORMULAS

1. Pythagorean theorem

2. Euler’s formula

3. Laplace equation

4. Poisson equation

5. Riemann zeta function

6. Navier-stokes equation

7. Pythagorean triples

8. Euler’s identity

9. De moivre’s formula

10. Mini-max method

11. Euler’s line

12. Baye’s theorem

13. Hamilton quanternion formula

14. Euler-lagrange equations

15. 3 interesting math items

6. THEOREMS

7. ODDS AND ENDS:

A. ARITHMETIC

B. ALGEBRA

C. GEOMETRY

D. TRIGONOMETRY

E. LINEAR ALGEBRA

F. CALCULUS

G. DIFFERENTIAL EQUATIONS

H. ANALYSIS

I. LOGIC

J. SET THEORY

K. TOPOLOGY

L. STATISTICS

M. PROBABILITY

N. REST OF MATH

8. PARADOXES

9. SOLVED PROBLEMS

10. UNSOLVED PROBLEMS

A. MILLENNIAL PROBLEMS

B. HILBERT’S PROBLEMS

C. SMALE’S PROBLEM

D. LANDAU’S PROBLEMS

E. ALGEBRA

F. ALGEBRAIC GEOMETRY

G. ANALYSIS

H. COMBINATORICS

I. DIFFERENTIAL GEOMETRY

J. DISCRETE GEOMETRY

K. EUCLIDEAN GEOMETRY

L. DYNAMIC SYSTEMS

M. GRAPH THEORY

N. GROUP THEORY

O. MODEL THEORY

P. NUMBER THEORY

Q. FEW OTHER UNSOLVED PROBLEMS

11. MATH PROBLEMS IMPOSSIBLE TO SOLVE

12. AWARDS

13. FACTS

DEFINITION OF MATHEMATICS

Mathematics comes from the greek word meaning ‘inclined to learn.’

it is the abstract science of number, quantity, and space. It can be studied in its own right (pure mathematics), or it can be applied to other disciplines such as physics or engineering (applied mathematics).

QUOTES ABOUT MATHEMATICS

1. ‘The laws of nature are written in the language of mathematics.’-Galileo

2. ‘Mathematics knows no race or geography; for mathematics, the cultural world is one country.’-David Hilbert

3. ‘God used beautiful mathematics in creating the world.’-Paul Dirac

4. ‘Pure mathematics is, in a way, the poetry of logical ideas.’-Albert Einstein

5. ‘Mathematics is not only for solving numbers. It is also for dividing sorrows, subtracting sadness, adding happiness, and multiplying love and forgiveness.’

6. ‘Mathematics is the door and key to the sciences’.-roger bacon

7. ‘Mathematics is the language which God wrote the universe.’-Galileo

8. mathematics is like love. It is a simple idea, but it gets complicated.

9. ‘mathematics is the queen of the sciences.’-gauss

10. ‘Pure mathematicians just love to try unsolved problems-they love a challenge.’-andrew wiles

11. ‘If I have seen further than others, it is by standing upon the shoulders of giants.’-isaac newton

12.’I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself now in and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.’-isaac newton

13. ‘Mathematics-the unshakeable foundation of sciences, and the plentiful fountain of advantage in human affairs.’-isaac barrows

14. Mathematics is a great motivator for all humans. Because its career starts with zero and never ends (infinity).

15. ‘Mathematics is written for mathematicians.’-copernicus

16. ‘Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.’-p. Dirac

17. ‘Mathematics is the cheapest science. Unlike physics or chemistry, it does not need expensive equipment. All one needs for mathematics is a pencil and paper.’

18. ‘Mathematics is an independent world created out of pure intelligence.’-william woods worth

19. ‘Mathematics is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.’-descartes

20. ‘Mathematics is the science of definiteness, the necessary vocabulary of those who know it.’-w.h. white

21. ‘Mathematics is pure language-the language of science. It is unique among languages in its ability to provide precise expressions for every thought or concept that can be formulated in its terms.’-a. Adler

22. ‘in the eyes of some, there is no finer beauty than that found in mathematics.’

HISTORY OF MATHEMATICS

Ca. 70,000 BC- south african ochre rocks adorned with scratches of geometric patterns.

C. 20,000 BC-1st reference to prime numbers.

2500 bc-multiplication tables

C. 2,000 BC-1st known approximation of the value of pi.

C. 1,800 BC-berlin papyrus 6619 contains the quadratic equation and its solution.

1,650 BC- attempt to square a circle and solutions to linear equations, 1st use of

cotangent, solving linear equations

ANCIENT BABYLON-practical arithmetic, and geometry, use both duodecimal and

decimal systems, composite and prime numbers, arithmetic, geometric, and

harmonic means, sieve of eratosthenes, perfect number theory (number 6), linear

equations, arithmetic and geometric series, solving 2nd order equations

ANCIENT EGYPT- arithmetic and cadastral surveys, use of decimal system

C. 2000-1800 bc-pythagorean triples and pythagorean theorem

1,046 BC-256 BC-arithmetic and geometric algorithms and proofs.

C. 8th century BC-earliest concept of infinity.

800 BC-use of quadratic equation

C. 600 BC-calculation of pythagorean triples, quadratic equations

CLASSICAL GREECE-

key people: thales, pythagoras, euclid

main accomplishments: geometry, logical reasoning

thales-pure geometry, Pythagoras-deductive geometry, special

numbers and ratios, eudoxus-method of exhaustion,

Mathematical rigor, conics, spherical geometry summation of

infinite series

C. 530 BC- discovery of irrational numbers

C. 400 BC-recognition of 5 kinds of infinity

300 BC-earliest information on combinations, fundamental theorem of arithmetic

HELLENISTIC-

Key people: archimedes, apollonius, diophantus

Key accomplishments: algebra

euclid- systematization of deductive geometry, archimedes-applied

math, apollonius-conics, hero-algebraic method, archimedes-rational approximation

of irrational numbers, Diophantus-algebraic notation

300 BC-proof of infinitely many prime numbers; proof of the fundamental theorem of

arithmetic, 1st use of fibonacci numbers and pascal’s triangle

300 BC-abacus

C. 300 BC-1st use of fibonacci numbers and pascal’s triangle

150 BC-knowledge of the theory of numbers, arithmetic operations, geometry,

operations with fractions, simple equations, cubic equations, quartic equations,

permutations, and combinations

150 BC-gaussian elimination, horner’s method, negative numbers, negative numbers

190-120 BC- development of trigonometry

1st century-earliest reference to the square root of a negative number

70-140-menelaus of alexandria on spherical trigonometry

250-350- diophantus- diophantine analysis, diophantine approximations, solutions of

indeterminate equations

179- china- mathematical formula for gaussian elimination

300-earliest use of zero

300-500-chinese remainder theorem

3rd century- cavalier’s principle to find volume of sphere

C. 400-understanding of logarithms to base 2 and computes square roots of numbers

as large as a million correct to 11 decimal places

C. 340-hexagon theorem and centroid theorem

MIDDLE AGES-

key people: khwarizmi

Key accomplishments: algebra

Khwarizmi- development and name of algebra

4th century-hexagon theorem and centroid theorem

500-aryabhata introduces the trigonometry of functions and how to calculate them

628-brahagupta sums series, Brahmagupta identity, brahamguptl theorem

8th century-virasena gives explicit rules for the fibonacci sequence, derivation of the

volume of a frustum using an infinite procedure, knows the laws for base 2

logarithm; shridhara-volume of sphere and formula for solving quadratic equations

9th century-govindsvamin-newton-gauss interpolation formula

C. 850—mahavira rules for expressing a fraction as sum of unit fractions; alkindi-

Frequency analysis

895-thabit Ibn qurra-solution and properties of cubic equations, and theorem of how

amicable numbers can be found

953-al-karaji-algebra replaces geometric operations with arithmetic operations, and

discovered the binomial theorem

975-al-batani-inverse of trigonometric functions

895-solutions and properties of cubic equations

10th to 15 centuries-

Key people: fibonacci

Key accomplishment: trigonometry

rediscovery of ancient knowledge, arabic-hindu numerals to west,

fibonacci-rediscovery of arabic (greek) mathematics

9th century- algebra of irrational numbers with square roots of 4th roots as solutions and

coefficients to quadratic equations, solving 3 nonlinear equations with 3 unknowns

C. 1000-abu-mahmud al-khujandi-special case of fermat’s last theorem, law of sines,al-

karaji-1st known proof by induction and proof of binomial theorem, pascal’s

triangle, and sum of integral cubes, and was 1st to introduce integral calculus

C. 1000-solving equations higher than degree 2

1000-1st proof by mathematical induction and used to prove binomial theorem, pascal’s

triangle, and the sum of integral cubes

1020-adul waft-formula for sin(a+b) and quadrature of parabola and volume of

paraboloid

1100-omar khayyam-general geometric solutions to cubic equations using intersecting

conic sections and laid foundations for development of analytic geometry and non-

euclidean geometry

12th century- bhaskara recognizes that a positive number has 2 square roots,

conceives differential calculus, develops rolle’s theorem, pell’s equation, a proof of

the pythagorean theorem, proves division by zero is infinity, and calculated time

which the earth orbits sun to 9 decimal places

1135-sharafeddin tush-uses algebra to study curves by means of equations-beginning

of analytic geometry

C. 1250-nasir al-din-tusi-attempt to develop non-euclidean geometry

1280-1303-china-solution to simultaneous higher order algebraic equations using

method similar to horner’s method

1303-zhu shijie-method of arranging binomial coefficients in a triangle

14th century-madhava-father of mathematical analysis, worked on power series for pi,

sine, and cosine, and founded important concepts of calculus, parameshvara-

series form of sine function equivalent to Taylor series expansion, and states mean

value theorem of differential calculus, bhaskara-2-mathematical objects equivalent

or approximately equivalent to infinitesimals, derivatives, mean value theorem, and

derivative of sine

1400-madhava-series expansion of inverse-tangent function, infinite series for arctan

and sin

15th century-symbolic algebra and for mathematics in general; nilakantha somayaji-

infinite series expansion

1424-ghiyath al-kashi-computes pi to 16 decimal places using inscribed and

circumscribed polygons

16th centuries-

Key people: tartaglia, cardano, napier, kepler

Key accomplishment: logarithms, infinitesimal, solving cubic and quartic equations

Tartaglia, cardano, bombelli-solution of cubic equations, vieta-

algebraic problems solved trigonometrically, Napier- logarithms, briggs-log tables

1639-cardono solves cubic equations

1540-ferrari solves quartic equations

1545-cardano conceives complex numbers

1572-bombelli uses imaginary numbers to solve cubic equations

17th century-

Key people: newton, Leibniz, Jakob and john Bernoulli, Descartes, fermat

Key accomplishments: calculus, analytic geometry, probability, number

theory, codification algebra development of calculus, kepler-conics as modified

circles, Wallis-mathematics of infinitesimal, limits, Newton and Leibniz-calculus,

Bernoulli’s Jacques and jean-application of calculus, Descartes-coordinate

geometry, desargues- projective geometry, pascal and format-statistical probability

1618- napier uses e in logarithmic work

1619-descartes and fermat- analytic geometry

1629-fermat- rudimentary differential calculus

1637-fermat claims to have proven fermat’s last theorem, 1st use of imaginary numbers

1654-pascal and fermat create theory of probability

1665-newton-fundamental theorem of calculus and infinitesimal calculus

1668-mercator and brouncker discover infinite series for the logarithm while attempting

to calculate area under hyperbolic segment

1671-gregory develops series expansion for inverse-tangent function

1673-leibniz-infintesimal calculus

1675-newton-computation of functional roots

1680s-leibniz-symbolic logic

1683-seki-resultant, determinant, elimination theory

1691-leibniz-separation of variables for ordinary differential equations

1696-l’hospital-rule for computing certain limits

1696-jakob and Johann bernoulli solve brachistochrone problem, the first result in

calculus of variations

18th century-

Key people: euler, Laplace, Lagrange, de moivre, Daniel bernoulli

Key accomplishment: analysis, topology, mathematical physics, complex numbers,

Metric system

cotes,de moivre, maupertius-complex numbers and trigonometry, Euler-

development of analysis, and foundation of topography, Lagrange-calculus of

variations, laplace-mathematical physics

1706-john machin develops a quickly converging inverse-tangent series for pi and computes pi to 100 decimal places

1708-seki-bernoulli numbers

1712-taylor series

1722-de moire formula stated connecting trigonometric functions and complex numbers

1733-de moire introduces normal distribution to approximate binomial distribution in probability

1734-euler’s integrating factor technique to solve 1st-order ordinary differential

equations

1735-euler solves Basel problem relating an infinite series to pi.

1736-euler-7 bridges of konigsberg, graph theory

1739-euler-solves general homogeneous linear ordinary differential equations with

constant coefficients

1742-goldbach’s conjecture

1761-proof of bake’s theorem, lambert-pi irrational

1762-lagrange-divergence theorem

1796-gauss-regular 17-gon constructed with compass and straightedge

1796-legrendre-prime number theorem conjectured

1797-wessel-associates vectors with complex numbers and studies complex numbers

in geometric terms

1799-gauss-fundamental theorem of algebra; ruffini partial proof (abel-ruffini theorem)

quintic or higher equation cannot be solved by general formula

19th century-

Key people- legendre, gauss, Riemann, Hamilton, abel, galois, Cauchy, Jacobi,

dirichlet, labochevski, weierstrauss, lie, klein, Hausdorff, Fourier

Key accomplishments- non-euclidean geometry, vector calculus, matrices, set

theory, group theory, boolean algebra, n-dimestions, theory of infinite

gauss-number theory and method of least squares, fundamental theorem of

algebra, quadratic law of reciprocity, bolyais (fracas and janos), lobachevski,

Riemann-noneuclidean geometry, dedikind-dedikind cuts and

number theory, cantor-theory of infinite sets, mobiles-topology, babage-

calculatikng engine, hamilton-quanternions, grassmann-vector analysis,

Boole-boolean algebra, cayley-matrices, Venn-graphic solutions of set theory

19th century-hermann grassmann-vector spaces, galois-group theory, cantor-set theory

1805-legendre-methods of least squares for fitting a curve to given set of observations

1806-argand diagram

1807-fourier-trigonometric decomposition of functions

1811-gauss-integrals with complex limits

1815-poisson-integration along paths of complex plane

1817-bolzano-intermediate value theorem

1822-cauchy-cauchy integral theorem-integrate around boundary of rectangle in

complex plane

1823-sophie germain theorem

1824-abel partial proof abel-ruffini theorem

1825-cauchy integral theorem for general integration paths, and theory of residues in

complex analysis; dirichlet and legendre-proof Fermat last theorem n=5, ampere-

stoke’s theorem

1828-proof green’s theorem

1829-bolyai, gauss, labochevsky-hyperbolic non-euclidean geometry

Early 1800s-elliptical geometry

1831-ostrogradsky-proof divergence theorem and founding group and galois theory

1832-dirichlet-proof n=14 format’s last theorem, galois general condition for solving of

algebraic equations, thereby founding group theory and galois theory

1835-proof dirichlet theorem about primes in arithmetic progressions

1837-wantzel-proof impossibility double cube and trisect angle with compass and

straight edge, and constructability of regular polygon; dirchlet-analytic number

theory

1841-1843-weierstrauss and laurent-laurent expansion theorem

1843-hamilton-quanternions

1847-boole-symbolic logic/boolean algebra

1850-stoke’s theorem proved

1854-riemann geometry

1854-cayley-quanternions in 4-dimensional space

1858-mobius strip

1859-riemann hypothesis

1870-klein-analytic geometry for labochevskian geometry

1872-dedikind cut

1873-hermite- e transcendental; frobenius-series solutions to linear differential

equations with regular singular points

1874-cantor-real numbers uncountably infinite, but all algebraic numbers countably

infinite

1882-lindemann-pi transcendental and circle cannot be squared with compass and

straightedge; klein bottle

1895-cantor-set theory containing arithmetic of infinite cardinal numbers and continuum

hypothesis

1896-hadamard and vallée poussin prove prime number theorem

1899-cantor-contadiction in set theory

20th century-

Key people: poincare, von neuman, godel, Russell, hilbert, lebesgue, manelbrot

Key accomplishments: game theory, mathematical logic, chaos theory, statistics

einstein-theories of relativities, von neuman-game theory, Russell-math

logic, model- yodel’s theorem, measure theory, qualitative study of dynamical

systems, axiomatization of probability theory, development of functional analysis,

distribution theory, fixed point theory, knot theory, ergodic theory, singularity theory,

catastrophe theory, lie theory, non-standard analysis, information theory, control

theory

1900-hilbert problems

1901-lebesgue integration

1903-runge-fast fourier transform algorithm

1903-landau-much simpler proof prime number theorem

1908-zermelo-axiometization of set theory avoiding cantor’s condradiction; plemelj-

solves reman problem about existence of differential equation with given

monochromic group and uses sokhotsky-plemelj formula

1912-brouwer fixed-point theorem

1915-noether-proves her symmetry theorem which shows symmetry in physics has a corresponding conservation law

1916-ramanujan conjecture

1919-brun’s constant b2 for twin primes

1921-noether-commutative ring

1930-church-lambda calculus

1931-godel imcompleteness theorem

1931-de rham-theorems in cohomology and characteristic classes

1933-boruk-ulam antipodal-point; kolmogorov-axiomatization of probability based on

measure theory

1940-gode-neither continuum hypothesis nor axiom of choice can be disproven from standard axioms of set theory

1942-fast fourier transform algorithm

1945-mac lane and eilenbery-category theory

1949-shannon-information theory

1950-ulam and von neumann-cellular automata dynamical systems

1957-ito calculus

1958-grothendieck-riemann-roch theorem

1959-iwasawa theory

1961-qr algorithm to calculate eigenvalues and eigenvectors of a matrix; smale-proof

dimensions >=5 for poincare conjecture

1963-cohen-neither continuum hypothesis nor axiom choice can be proven with

standard axioms of set theory; butterfly effect

1965-zadeh-fuzzy mathematics

1966-robinson-non-standard analysis

1967-langlands program of conjectures relating number theory and representation

theory

1968-proof of atiyah-singer-index theorem about index of elliptical operators

1975-mandelbrot-fractal

1976-proof 4-color theorem

1973-zadeh-fuzzy logic

1983-falting proves morsel conjecture-only infinite many whole number solutions for

each exponent fermat last theorem

1985-branges proves biererbach conjecture

1986-proof rib’s theorem

1991-connes and lott-non-commutative geometry

1994-shor’s algorithm, a quantum algorithm for

integer factorization

1995-wiles-proof format’s last theorem

1995-baily-borwein-plouffe formula to find nth binary digit of pi

1998-hales-almost certainly proof kepler conjecture

1999- proof taniyama-shimura conjecture

2000-clay mathematics institute millenium prize problems

21st century—

2002-mihailescu proof Catalan’s conjecture

2003-perelman-proof poincare conjecture

2009-ngo baa-proof fundamental lemma (langlands program)

2005-green and tao prove the green-tao theorem

2009-fundamental lemma (langlands program) proved by ego ban Chau

2014-complete kepler’s conjecture proof, calculation of pi to 13.3 trillion digits

2015-tao-solves erdos discrepancy problem, lassie babai found that a quasi polynomial
complexity algorithm would solve the graph isomorphism problem

MATHEMATICIAN BIOGRAPHIES

PYTHAGORAS- c. 570-495 BC. He is known for the Pythagorean theorem, credited to

him, was known to the babylonians and Indians before him. It states that the squares of each side of a right triangle add together is equal to the square of the hypotenuse of the triangle.

ZENO- c. 490-c. 430 bc. He is best known for his paradoxes, particularly zeno’s paradox.

THEAETETUS- c. 417-c. 368 BC. His principle contributions to math were on irrational lengths, which were included in Euclid’s work, and in proving that there are precisely 5 regular convex polyhedra.

EUCLID- mid-4th century BC-mid-3rd century BC. (fl. 300 BC). Greek mathematician often referred to as the father of geometry. His work, the elements, is one of the most influential works in the history of mathematics. He deduced the principles of what is now called Euclidean geometry. He wrote about perspective, conic sections, spherical geometry, number theory, and rigor. Although best known for its geometrical results, the elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid-Euler theorem), the infinitude of prime numbers, Euclid’s lemma on factorization, which leads to the fundamental theorem of arithmetic on the uniqueness of prime factorizations, and the Euclidean algorithm for finding the greatest common divisor of 2 numbers. Euclid’s geometry was considered the only geometry possible, until non-euclidean geometry was discovered in the 19th century.

EUDOXUS- 355 bc. Developed the method of exhaustion, a precursor to integral calculus. He was very familiar with geometry and number theory. He used the theory of proportions to allow the possibility of irrational numbers.

ARCHIMEDES- c. 287-c. 212 BC. Regarded as one of the leading scientists in classical antiquity. He anticipated modern calculus and analysis by applying concepts of infinitesimal and the method of exhaustion, to derive and rigorously prove a range of geometric theorems, including the area of a circle, the surface area and volume of a sphere, and the area under a parabola. Other mathematical achievements include deriving an accurate approximation of pi, defining and investigating the Archimedean spiral, and creating a system using exponentiation for expressing very large numbers. He also applied the method of exhaustion similar to integral calculus to answer problems involving finding areas. He concluded that the number of grains of sand needed to fill the universe would be 8×10^63 grains.

APOLLONIUS- late 3rd century-early 2nd century. Known for his theories on the topic of conic sections.

HIPPARCHUS- c. 190-c. 120 BC. He was the first mathematician to possess trigonometric tables used to compute the eccentricity of the orbits of the Sun and Moon.

HERON- C. 10-C. 70. He described how to compute square roots iteratively, but is best known for hero’s formula, which gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin. The formula is—

area=sqrt(s(s-a)x(a-b)x(s-c)), where sides are lengths a, b, and c, and s=(a+b+c)/2.

NICOMACHUS- c. 60-c. 120. He was more interested in the mystical properties of numbers than their mathematical properties. He put significance in prime and perfect numbers.

DIOPHANTUS- 201/215-285/299. Sometimes called the father of algebra. He was the first greek mathematician to recognize fractions as numbers. Diophantine equations are named after him for usually algebraic equations with integer coefficients in which integer solutions are sought.

PAPPUS- c. 290-c. 350. One of the last great Greek mathematicians. He is known for his Pappus’s hexagon theorem in projective geometry.

HYPATIA- c. 350-415. She is known for her intense interest and reverence of mathematics.

ARYABHATA- 476-550. He is known for the sinusoidal functions, solution of the single variable quadratic equation, calculating pi to 4 decimal places, and measuring the circumference of the Earth with 99.8% accuracy.

BRAHMAGUPTA- c. 598-after 685. First to gives rules to compute zero. He solved ax^2+bx=c and came up with x=(sqrt(4ac)/2a and x=((sqrt(ac+b^2/4)-b/2)/a. He also calculated a/c+b/d*a/c=a(d+b)/cd, and a/c-b/d*a/c=a(d-b)/cd. He provided a formula for generating pythagorean triples.the formula is-a=mx is one leg of the right triangle. b=m+d is the other leg of the right triangle. M and x are rational numbers d=mx/(x+2),

And c=m(1+x)-d. You can generate pythagorean triples with the previous formulas and arrive at pythagorean triples of a^2+b^2=c^2. He also went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the 2nd degree as nx^2+1=y^2, called pell’s equation. Brahmagupta most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths for the sides of any cyclic quadrilateral, he gave an approximate and exact formula for the figure’s area.the formula is let the lengths be: p,q,r and s. The approximate area is (p+r)/2, q+s)/2, and t=(p+q+r+s)/2, and the exact formula is sqrt((t-p)(t-q)(t-r)(t-s)). Heron’s formula is a special case of this formula and it is derived by setting one of the sides equal to zero.he also provided a way to determine pi with very little margin of error. He presented a sine table using names of objects to represent the digits of place-value numerals. In 665, he devised and used a special case of the Newton-Sterling interpolation formula of the second order to interpolate new values of the sine function and other values.

BHASKARA 1- c. 600-c. 680. He was the first to write numbers in the Hindu decimal system with a circle for the zero, and he gave a remarkable and unique rational approximation of the sine function.

AL-KHWARIZMI- c. 780-c. 850. He is credited with the first systematic solution of linear and quadratic equations in Arabic. He is considered one of the fathers of algebra. He also did innovative work in trigonometry.

AL-KARAJI- c. 953-c. 1029. He was influenced by Diophantus, and be began the freeing of algebra from geometry. He systematically studied the algebra of exponent and was the first to realize that the sequence x, x^2, x^3…, and its reciprocal can be extended indefinitely. He wrote on the binomial theorem and Pascal’s triangle.

ABU A-HAYTHAM- c. 965-c. 1040. He worked on the beginning of the link between algebra and geometry. He explored Euclid’s parallel postulate, using the concept of motion into geometry using a proof by contradiction. He worked on perfect numbers, where 2^(n-1)x(2^n-1) is a perfect number where 2^n-1 is prime. He found the volume of a paraboloid.

OMAR KHAYYAM-

May 18, 1048-December 4, 1131. He worked on the theory of parallels, geometric algebra, and the binomial theorem and extraction of roots. He developed general methods for solving cubic equations and equations of higher order.

BHASKARA 2- 1114-1185. He proved the Pythagorean theorem by calculating the same area in 2 different ways and then cancelling out terms to arrive at a^2+b^2=c^2. He solved quadratic equations with more than one unknown, and found negative and irrational solutions. He arrived at preliminary concepts of mathematical analysis and infinitesimal calculus, conceived differential calculus, after discovering an approximation of the derivative and differential coefficient, stated Rolle’s theorem, developed spherical trigonometry, and found methods of solving Pell’s equation.

FIBONACCI- c. 1175-c. 1250. Considered the most talented mathematician of the Middle Ages. He popularized the Hindu-Arabic numeral system in the west. in his book, liber abaci, he introduced the sequence of fibonacci numbers and also discussed irrational and prime numbers. The golden ratio is the limit of the ratio of consecutive numbers in the fibonacci sequence.

NASIR AL-DIN AL-TUSI- February 18, 1201-June 26, 1274. He was the first to list the six distinct cases of a right triangle in spherical trigonometry. He stated the law of sines for plane and spherical triangles and the law of tangents for spherical triangles along with proof for all of these.

QIN JIUSHAO- 1202-1261. He produced indeterminate equations and numerical solutions of certain polynomial equations up to the 10th order, a general form of the Chinese remainder theorem, and a formula for finding the area of a triangle from the given lengths of 3 sides, also known as Heron’s formula proved by Heron about 60 BC.

He also found ways for finding sums of arithmetic series.

JORDANUS SE NEMORE- fl. 13th century. He wrote wonderful treatise on practical arithmetic, pure arithmetic, algebra, and geometry.

NOCOLE ORESME- c. 1320-July 11, 1382. he is known for the proof of the divergence of the harmonic series.

REGIOMONTANUS- June 6, 1436-July 6, 1476. He did work on arithmetic and symbolic algebra.

DEL FERRO- February 6, 1465-November 5, 1526. He provided a solution of the depressed cubic equation.

MICHAEL STIFEL- 1487-April 19, 1567. He was the first to use the term ‘exponent’ and give a few of the exponential laws. He was the first who used a standard model to solve quadratic equations.

NICCOLO FONTANA TARTAGLIA- 1499/1500-December 13, 1557. He is best known for his conflicts with Gerolamo Cardano, where Cardano cajoled Tartaglia into revealing his solution to the cubic equation by promising not to publish them. Cardano saw an unpublished pre-Tartaglian solution to the cubic equation by del ferro who independently came up with the same solution as Tartaglia, so cardamon broke his promise to Tartaglia and published Tartaglia’s solution. Both Tartaglia and cardano are credited with solving the cubic equation, which is now known as the Cardano-Tartaglia formula. He is also known for giving the volume of a tetrahedron.

GEROLAMO CARDANO- September 24, 1501-September 21, 1576. One of the most influential mathematicians of the renaissance, and was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficient and the binomial theorem to the western world. He made the first systematic use of negative numbers, published solutions from other mathematicians for the cubic and quartic equations, acknowledged the existence of imaginary numbers, and made the first systematic treatment of probability.

LODOVICO FERRAI- February 2, 1522-October 5, 1565. Was mainly responsible for solving quartic equations. He proved the intermediate value theorem for polynomials, using the procedure of conquer and divide by subdividing the interval into equal parts. His decimals were an inspiration to Newton.

RAPHAEL BOMBELLI- January 20, 1526-1572. He authored a treatise on algebra and is a central figure in the understanding of imaginary numbers. He gave a comprehensive account of the algebra known at the time, and was the first European to write down the way of performing computations with negative numbers. He made monumental contributions to complex numbers. He saw that imaginary numbers were crucial and necessary to solving quartic and cubic equations.

FRANCOIS VIETE- 1540-February 23, 1603. He is known for the first notation of new algebra (symbolic notation).

SIMON STEVIN- 1548-1620. Brought to the western world for the first time the general solution to the quadratic equation.

JOHN NAPIER- February 1, 1550-April 4, 1617. Best known for discovering logarithms.

HENRY BRIGGS- February 1561-January 26, 1630. He changed the original logarithms invented by Napier into common base 10 logarithms.

GALILEO GALILEI-

February 15, 1564-January 8, 1642. He made original contributions to the science of motion through an innovative combination of experiment and mathematics. He was one of the first modern thinkers to clearly state that the laws of nature are mathematical. Galileo showed a modern appreciation of the proper relationship between mathematics, theoretical physics, and experimental physics. He understood the parabola, both in terms of conic sections, and in terms of the ordinate (y) varying as the square of the abscissa (x). He asserted that a parabola was the theoretical ideal trajectory of a uniformly accelerated projectile in the absence of air resistance.

MARIN MERSENNE- September 8, 1588-September 1, 1648. He is known for Mersenne primes of the form M(n)=2^n-1.

GIRARD DESARGUES- February 21, 1591-September 1661. He is one of the founders of projective geometry, and known for Desargues theorem and Desargues graph.

Albert Girard- 1595-December 8, 1632. He had early thoughts on the fundamental theorem of algebra and gave the inductive definition for the Fibonacci numbers. He showed that the area of a spherical triangle depends on the interior angles, a result called Gerard’s theorem.

RENE DESCARTES-

March 31, 1596-February 11, 1650. He is mainly known in mathematics for his cartesian coordinate system.

BONAVENTURA. CAVALIERI- 1598-November 30, 1647. He is known for his work on indivisibles, the precursor to infinitesimal calculus, and the introduction of logarithms into Italy. Cavalieri’s principle in geometry partially anticipated integral calculus.

GILLES PERSONNE DE ROBERVAL- August 10, 1602-October 27, 1675. Just before the invention of calculus, he worked with problems which are soluble by some method involving limits and infinitesimals. He coined the term trochoid.

PIERRE DE FERMAT-

October 31 to December 6, 1607-January 12, 1665. He is given credit for the early developments that led to infinitesimal calculus and did work analogous to that of differential calculus, then unknown. He also did research into number theory,, and made notable contributions to analytic geometry and probability. He is known for Fermat’s last theorem, proved in 1994, and Fermat’s little theorem.

JOHN PELL- March 1, 1611-December 12, 1685. He is known for the Pell number and the Pell’s equation.

JOHN WALLIS- December 3, 1616-November 8, 1703. he was given partial credit for the development of infinitesimal calculus, and is also credited with the introduction of the infinity symbol. He made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. He introduced the term continued fraction. He published a treatise on conic sections which were defined analytically. He is also known for the Wallis product.

BLAISE PASCAL- June 19, 1623-August 19, 1662. Helped create 2 major new areas of research: projective geometry at t6 and probability theory with Fermat. He a provided convenient tabular presentation of binomial coefficients, now called pascal’s triangle. Pascal used a probabilistic argument, pascal’s wager, to justify belief in God and a virtuous life.

ISAAC BARROWS- October 1630-May 4, 1677. Generally credited for the early development of infinitesimal calculus, especially the fundamental theorem of calculus. His work was on tangents, and he was the first to calculate the tangents of kappa curves. Newton was his student who went on to develop calculus in its modern form.

JAMES GREGORY- November 1638-October 1675. He discovered infinite series representations for several trigonometric functions and he formulated the Taylor’s series. He was influential with trigonometric series.

ISAAC NEWTON-

December 25, 1642-march 20, 1727. Known for his book Principia Mathematica, invention of infinitesimal calculus, binomial series.

SEKI TAKAKAZU- 1642-December 5, 1708. He has been described as japan’s newton. He did work on infinitesimal calculus and Diophantine equations independently of newton. He is credited with the discovery of Bernoulli numbers, and the resultant and determinant are attributed to him.

GOTTFRIED LEIBNIZ- July 1, 1646-November 14, 1716. Co-inventor of calculus.

MICHEL ROLLE- April 21, 1652-November 8, 1719. he is best known for Rolle’s theorem, which states that any real-valued differentiable function that attains equal values at 2 distinct points must have a stationary point somewhere between them.

THE BERNOULLI FAMILY OF MATHEMATICIANS- Daniel, Jacob, Jacob(2), Johann, Johann(2), Johann(3), Nicolaus, Nicolaus(2)

JACOB BERNOULLI- December 27, 1654-August 16, 1705. Sided with Leibniz against newton in the calculus inventor controversy. He made numerous contributions to calculus, including being one of the inventors of the calculus of variations. He discovered the mathematical constant ‘e’. He has numerous mathematical discoveries to his name.

DE L’HOSPITAL- 1661-February 2, 1704. He is associated with L’Hospital’s rule, a way of calculating limits involving indeterminate forms 0/0 and infinity/infinity. This rule did not originate with him.

GIOVANNI SACCHERI- September 5, 1667-October 25, 1733. His geometric work resulted in the basis of elliptic geometry and theorems of hyperbolic geometry.

ABRAHAM DE MOIVRE- May 26, 1667-November 27, 1754. He is known for de Moivre’s formula that links complex numbers and trigonometry, for his work on the normal distribution and probability theory, and for the theorem of de Moivre-Laplace..

JOHANN BERNOULLI- August 6, 1667-January 1, 1748. One of the many prominent mathematicians in the Bernoulli family, he made many contributions to infinitesimal calculus, and educated Euler in his youth.he is known for the brachistochrone problem, which is a curve which has the fastest descent.

JACOPO RICCATI- May 28, 1676-April 14, 1754. He introduced the hyperbolic functions, and studied the Riccati equation, which is a first-order differential equation that is used to refer to matrix equations with an analogous quadratic term, which occurs in both continuous-time and discrete-time linear quadratic-gaussian control. Introduced hyperbolic functions.

ROGER COTES- July 10, 1682-June 5, 1716. Invented the quadrature formulas and introduced the Euler formula.

TAYLOR BROOK- August 18, 1685-December 1731. Best known for Taylor series and the Taylor’s theorem.

NICOLAUS BERNOULLI- October 21, 1687-November 29, 1759. He is one of the many prominent mathematicians of the Bernoulli family. He did work in probability, differential equations, and geometry.

JAMES STERLING- May 1692-December 5, 1770. He. Is known for sterling numbers, sterling permutations, sterling approximations, and he proved the correctness of newton’s classification of cubics.

CHRISTIAN GOLDBACH- March 18, 1690-November 20, 1764. Known for Goldbach’s conjecture.

COLIN MACLAURIN- February 1, 1698-June 14, 1746. The Maclaurin series, a special case of the Taylor series, is named after him. He was influential in trigonometric series.

DANIEL BERNOULLI- February 8, 1700-March 17, 1782. One of the prominent mathematicians in the Bernoulli family, he applied math to mechanics, especially fluid mechanics, and did pioneering work in probability and statistics. He is known for the Bernoulli principle, which in fluid dynamics, states that an increase in the speed of a fluid occurs simultaneously with an decrease in pressure or a decrease in the fluid’s potential energy. This principle explains why an airplane wing provides lift and enables airplane flight.

THOMAS BAYES- c. 1701-April 7, 1761. Bayes theorem is named after him, dealing with probability theory, and led to Bayesian probability.

GABRIEL CRAMER- July 31, 1704-January 4, 1752. He is known for Cramer’s rule (an explicit formula for the solution of a system of linear equations with as many equations as unknowns and is valid whenever the system has a unique solution), Cramer’s paradox, and Cramer’s theorem for algebraic curves.

LEONARD EULER- April 15, 1707-September 18, 1783. Swiss mathematician Leonard Euler is one of the most eminent mathematicians of the 18th century and it held as one of the greatest in history. He is also one of the very few that were the most prolific ones. He worked in almost all areas of mathematics, from geometry, infinitesimal calculus, trigonometry, algebra, and number theory. The number ‘e’ is named after him, being equal to 2.71828…, as is the Euler-Mascheroni constant (gamma), which is equal to approximately .57721, and it is not know whether this number is rational or irrational. The concept of a function and summations originated from him.He worked in the power series and proved the power series expansions for ‘e’ and the inverse tangent function along with the Basel function (summation of 1/n^2=pi^2/6 in 1735. He introduced the use of the exponential function and logarithms into analytic proofs. He is known for the Euler identity- e^(I*pi)+1=0. It was voted in 1988 the most beautiful formula ever. Euler is responsible for 3 to the top 5 formulas in 1988.De movire’s formula resulted directly from Euler’s formula. He elaborated on the theory of higher transcendental functions by introducing the gamma function. He used new methods to sole quartic equations. He foreshadowed the development of complex analysis and invented the calculus of variations, including its best known result, the Euler-Lagrange equation. He incorporated complex numbers into trigonometry. He did pioneering work in the use of analytic methods to solve number theory problems. He introduced analytic number theory, and created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. He used the divergence of the harmonic series to prove the infinitude of primes, and used analytic methods to gain understanding of the way prime numbers are distributed. His work led to the development of the prime number theorem. He proved that the sum of the reciprocals of primes diverges. Doing this, he discovered the connection between the Riemann zeta function and the primes. This is known as the Euler product formula for the Riemann zeta function. Euler proved newton’s identities, format’s little theorem, Fermat’s theory on sums of 2 squares, and made distinct contributions to lag range’s 4-square theorem. He invented the totient function, using properties of this function to generalize format’s little theory in what is now known as ruler’s theorem. He proved the relationship shown between Mersenne primes and perfect numbers earlier proved by Euclid was 1-to-1, known as Euclid-Euler theory. He conjectured about the law of quadratic reciprocity, which is fundamental to number theory. Gauss used these ideas. In 1772, Euler proved that 2^31-1-2,147,483,647 is a Mersenne prime, and remained so until 1867. In 1735, he proved that the 7 brides of Konigsberg problem in graph theory is not possible. He discovered the formula V-E+F=2 for the number of vertices, edges, and faces of a convex polyhedron, and hence a planar graph. Its constant is known as the Euler characteristic. The study and generalization of this formula, by Cauchy and l’hillier is the origin of topology. He described many applications of the Bernoulli numbers, Fourier series, Euler numbers, and the constants e and pi, continued fractions, and integrals. He integrated Leibniz’s and Newton’s differential calculus. He made great progress in improving numerical approximation of integrals, inventing Euler approximations. Some of these approximation are ruler’s method and the Euler-Maclaurin formula. He facilitated the use of differential equations, in particular introducing the Euler-Mascheroni constant. He calculated with great accuracy the orbits of comets and other celestial bodies, and calculated the parallax of the sun.he did work in set theory and logic.

THOMAS SIMPSON- August 20, 1710-May 14, 1761. He is the inventor of Simpson’s rule, used to approximate definite integrals.

ALEXIS CLAIRAUT- May 13, 1713-May 17, 1765. He is known for Clairaut’s theorem, which helped to establish Newton’s principles and results. He is also credited with Clairaut’s equation and Clairaut’s relation.

JEAN LE ROND D’ALEMBERT- November 17,1717-October 29, 1783. He is known for d’Alember’s principle concerning dynamics as related to newton’s 3rd law of motion, applying calculus to vibrating strings, and his research in integral calculus where he devised relationships of variables by means of rate of change of their numerical value.

MARIA AGNESI- May 16, 1718-January 9, 1799. first woman to write a mathematics handbook on both differential and integral calculus.

JOHANN LAMBERT- August 26, 1728-September 25, 1777. He was the first to prove that pi is an irrational number. He is also known for the Lambert w function, and he introduce hyperbolic functions.

ALEXANDRE-THEOPHILE VANRMONDE (28 February 1735 – 1 January 1796) was a French mathematician, musician and chemist who worked with Bezout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.

JOSEPH LOUIS LAGRANGE- January 25, 1736-April 10, 1813. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. He invented Lagrange multipliers.

GASPARD MONGE- May 9, 1746-July 28, 1818. Inventor of descriptive geometry, the mathematical basis of technical drawing. (father of descriptive geometry.)

PIERRE SIMON DE LAPLACE- March 23, 1749-March 5, 1827. He wrote a 5 volume Mechanique Celeste (1799-1825).it translates the geometric study of celestial mechanics to one based on calculus. In statistics, bayesian interpretation of probability was developed by him. He is known for Laplace’s equation, Laplacian, Laplace transform, block holes, and the nebular hypothesis of the solar system’s formation.

ADRIEN-MARIE LEGENDRE- September 18, 1752-January 10, 1833. He is known for Legendre transformation, Legendre polynomials, Legendre transform, and elliptic functions.

PAOLO RUFFINI- September 22, 1765-May 10, 1822. He is known for an incomplete proof that quintic and higher ordered equations cannot be solved by radicals (Abel-Ruffini theorem), the Ruffini rule, and a fast method for polynomial division. He made contributions to group theory, probability, and the quadrature of a circle.

JOSEPH FOURIER- March 21, 1768-May 16, 1830. Known for initiating the investigation of Fourier series and their application to problems of heat transfer and vibrations. The Fourier transform and Fourier’s law are named in his honor.

SOPHIE GERMAIN- April 1, 1776-June 27, 1831. She did work in differential geometry, number theory, and is known for Sophie Germain prime numbers along with proving format’s last theorem for one of its exponents.

FRIEDRICH GAUSS- April 30, 1777-February 23, 1855. He contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, mechanics, matrix theory, and astronomy.

MARY FAIRFAX SOMERVILLE- December 26, 1780-November 29, 1872. She studied math and astronomy and her writings influenced James Clerk Maxwell. She was a polymath.

SIMON POISSON- June 21, 1781-April 25, 1840. He is known for the Poisson distribution, poisson regression, poisson summation, poisson algebra, and much more.

BERNARD BOLZANO- October 5, 1781-December 18, 1848. He is known for Bolzano’s theory, the first purely analytical proof of the intermediate value theorem.

JEAN-VICTOR PONCELET- July 1, 1788-December 22, 1867. He is considered a reviver of projective geometry and did notable work in this area. He developed the concept of parallel lines meeting at infinity, and aided in the development of complex numbers.

AUGUSTIN LOUIS CAUCHY- August 21, 1789-May 23, 1857. He made pioneering work in analysis and was one of the first to star and prove theorems of calculus rigorously. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra.

AUGUST MOBIUS- November 17, 1790-September 26, 1868. He is known for the Mobius strip, Mobius transform, Mobius function, and more.

NIKOLAI LABACHEVSKY- November 20, 1792-February 12, 1856. He is known primarily for his work on hyperbolic/Labachskian geometry (non-Euclidean).

GEORGE GREEN- July 14, 1793-May 31, 1841. He is known for Green’s theorem (the idea of potential functions as currently used in physics), Green’s functions, Green’s identity, and more.

JULIUS PLUCKER- June 16, 1801-May 22, 1868. He made fundamental contributions to analytic geometry. He vastly extended the study of lame curves. He is known for the plucker formula.

MIKHAIL OSTROGRADSKY- September 24, 1801-January 1, 1862. A disciple of Euler and one of the leading mathematicians in imperial Russia. He gave the first general proof of the divergence theorem, created the Ostrogradsky equation, , and method for integrating rational functions.

JANOS BOLYAI- December 15, 1802-January 27, 1860. He is one of the founders of non-Euclidean geometry.

CARL JACOBI- December 10, 1804-February 18, 1851. He made fundamental contributions to elliptical functions, differential equations, and number theory. He is known for the Jacobean, Jacobi’s elliptical functions, Jacobi ellipsoid, Jacobi transform, Jacobi polynomials, and more.

NIELS ABEL- August 5, 1802-April 6, 1829. His most famous single result is the first complete proof demonstrating the impossibility of solving general quintic equations by radicals. He was also an innovator in the field of elliptic functions, discoverer of abelian functions.

PETER DIRICHLET- February 13, 1805-May 5, 1859. He made deep contributions to number theory, including creating analytic number theory, the theory of Fourier series, and other topics of mathematical analysis. He gave the modern formal definition of a function. He is known for much in math, such as the Dirichlet series, the Dirichlet distribution, the Dirichlet integral, the Dirichlet space, and much more.

WILLIAM HAMILTON- August 4, 1805-September 2, 1865. In pure mathematics, he is best known for the inventor of quanternions.

AUGUSTUS DE MORGAN- June 27, 1806-March 18, 1871. He formulated De Morgan’s law and introduced the term mathematical induction. He is known for De Morgan’s laws, De Moran algebra, relational algebra, and universal algebra.

Johann Benedict Listing (25 July 1808 – 24 December 1882) was a German mathematician.

J. B. Listing was born in Frankfurt and died in Göttingen. He first introduced the term “topology”, in a famous article published in 1847, although he had used the term in correspondence some years earlier. He (independently) discovered the properties of the half-twisted strip at the same time (1858) as August Ferdinand Möbius, and went further in exploring the properties of strips with higher-order twists (paradromic rings). He discovered topological invariants which came to be called Listing numbers.[1]

In ophthalmology, Listing’s law describes an essential element of extraocular eye muscle coordination.

JOSEPH LIOUVILLE- March 24, 1809-September 8, 1882. He is best remembered for Liouville’s theorem, a basic result in complex analysis, and in number theory, he was the first to prove the existence of transcendental numbers by construction using continued fractions (liouville numbers).

HERMANN GRASSMANN- April 15, 1809-September 26, 1877. He is known for multilinear algebra.

ERNST KUMMER- January 29, 1810-May 14, 1893. He is known for Bessel functions, Kummer surfaces, and Kummer theory.

EVARISTE GALOIS- October 25, 1811-May 31, 1832. While still a teen, he was able to determine the necessary and sufficient condition for a polynomial to be solved by radicals, solving a 350 year problem. His work laid the foundations of group theory and Galois theory, two major branches of abstract algebra.

Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician and Military Officer best known as the discoverer of the Laurent series, an expansion of a function into an infinite power series, generalizing the Taylor series expansion.

He was born in Paris, France. Pierre Laurent entered the École Polytechnique in Paris in 1830, Laurent graduated from the École Polytechnique in 1832, being one of the best students in his year, and entered the engineering corps as second lieutenant. He then attended the École d’Application at Metz until he was sent to Algeria.

Laurent returned to France from Algeria around 1840 and spent six years directing operations for the enlargement of the port of Le Havre on the English Channel coast. Rouen had been the main French port up to the nineteenth century but the hydraulic construction projects on which Laurent worked in Le Havre turned it into France’s main seaport. It is clear that Laurent was a good engineer, putting his deep theoretical knowledge to good practical use.

It was while Laurent was working on the construction project at Le Havre that he began to write his first mathematical papers. He submitted a memoir for the Grand Prize of the Académie des Sciences of 1842. His result was contained in a memoir submitted for the Grand Prize of the Académie des Sciences in 1843, but his submission was after the due date, and the paper was not published and never considered for the prize. Laurent died at age 41 in Paris. His work was not published until after his death.

Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specializing in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spaces. The concept of multidimensionality has come to play a pivotal role in physics, and is a common element in science fiction.

JAMES SYLVESTER- September 3, 1814-March 15, 1897. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He is known for the Sylvester constant, the Sylvester sequence, the Sylvester formula, the Sylvester determinant theorem, and more.

KARL WEIERSTRASS- October 31, 1815-February 19, 1897. The father of modern analysis. He formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano-Weierstraus theorem, and used the properties of the latter to study the properties of continuous functions on closed bounded intervals. He brought soundness and rigor to calculus, as did caught earlier. He made significant advancements in the field of calculus of variations.he discovered a function that is continuous, but had no derivative at any point.

GEORGE BOOLE- November 2, 1815-December 8, 1864. He worked in the fields of differential equations and algebraic logic, and for Boolean logic, which laid the foundation for the Information Age.

SIR GEORGE STOKES- August 13, 1819-February 1, 1903. He made seminal contributions to fluid dynamics, including the Navier-Stokes equations. He is known for Stoke’s theorem, and contributed to the theory of asymptotic expansions.

PAFNUTY CHEBYSHEV- May 16, 1821-Decemebr 8, 1894. He worked on probability, statistics, analytical geometry, and number theory. He is known for Chebyshev’s theorem in which there is always a prime number between n and 2n.

ARTHUR CAYLEY- August 16, 1821-January 26, 1895. He helped found the modern British school of mathematics. He is known for algebraic geometry, group theory, the Cayley-Hamilton theorem, the Cayley-Dickerson construction, and Cayley algebra (octonion).

CHARLES HERMITE- December 24, 1822-January 14, 1901. He did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. He is known for the proof the ‘e’ is a transcendental number.

GOTTHOLD EISENSTEIN- April 16, 1823-October 11, 1852. He specialized in number theory and analysis and proved several results in which Gauss could not solve. He provided 2 proofs of the law of quadratic reciprocity, and analogous laws of cubic and quartic reciprocity.

Enrico Betti- October 21, 1823-August 11, 1892. He wrote a paper in topology in 1871 that led to the later naming after him of the Betti numbers. He also worked on the theory of equations, giving early expositions of Galois theory, he also discovered Betti’s theory, a result in the theory of elasticity, and is known for Betti’s theorem.

LEOPOLD KRONECKER- December 7, 1823-December 29, 1891. He is known for the kronecker product, the kronecker theorem, the kronecker lemma, and other discoveries.

BERNARD RIEMANN- September 17, 1826-July 20, 1866. He made contributions to analysis, number theory, and differential geometry. He rigorously formulated the Riemann integral, worked on Fourier series, contributed to complex analysis, notably the introduction of Riemann surfaces, laid the foundation of the mathematics of general relativity, and in his famous 1859 paper on the prime-counting function, contained the Riemann hypothesis, one of the most famous unsolved problems in pure mathematics.

HENRY SMITH- November 2, 1826-February 9, 1883. He is known for the Smith-Minkowski-Siegel mass formula and Smith normal form.

JAMES CLERK MAXWELL- June 13, 1831-November 5, 1879. Did work in mathematical physics,, his most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as manifestations of the same phenomenon in his 4 differential equations (Maxwell’s equations).

RICHARD DEDIKIND- October 6, 1831-February 12, 1916. He made important contributions to abstract algebra, particularly ring theory, algebraic number theory, and the definition of real numbers. He is known for the Dedikind cut, a method of construction of real numbers.

LAZARUS FUCHS- May 5, 1833-April 26, 1902. Known for fuchsian groups and fuchsia’s theorem.

JOHN VENN- August 4, 1834-April 4, 1923. Logician who introduced the Venn diagram, used in set theory, logic, statics, and computer science.

EUGENIO BELTRAMI- November 16, 1835-February 18, 1900. He was the first to prove the consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and the interior of an n-dimensional unit sphere, the so-called Beltrami-Klein model. He also developed singular decomposition for matrices, and used differential calculus for problems of mathematical physics indirectly influencing the development of tensor calculus developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita.

CAMILLE JORDAN- January 5, 1838-January 22, 1922. He is known for his foundational work in group theory, Jordan curve theorem, Jordan matrix, and more.

JOSIAH WILLARD GIBBS- February 11, 1839-April 28, 1903. Made important theoretical contributions to physics, chemistry, and math. Together with James clerk maxwell and Ludwig Boltzmann, he created statistical mechanics explaining the laws of thermodynamics as consequence of the statistical properties of ensembles of a physical system composed of many particles. He invented modern vector calculus independently of Oliver Heaviside.

FRANCOIS LUCAS- April 4, 1842-October 3, 1891. He studied the Fibonacci sequence, and the Lucas sequence and Lucas numbers are named after him. He is known for the cannonball problem, Lucas primes, and Lucas’ theorem. Also, a proof using elliptic functions was found which as relevance to the boson string theory in 26 dimensions.

SOPHIUS LIE- December 17, 1842-february 18, 1899. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.he is known for Lie groups, lie theory, and Lie algebra, among much more.

GREGOR CANTOR- March 3, 1845-January 6, 1918. He invented set theory, which became a fundamental theory in mathematics, established the importance of one-to-one correspondence between two sets, defined infinite and well=ordered sets, and proved that the real numbers are more numerous than the natural numbers. He showed that there are an infinity of infinities. He defined cardinal and ordinal numbers and their arithmetic.

GOSTA MITTG-LEFFLER- March 16, 1846-July 7, 1927. He is known chiefly for his connection with the theory of functions, today known as complex analysis.

FERDINAND GEOG FROBENIUS- October 26, 1849-August 3, 1917. He is best known for his contributions to the theory of elliptical functions, differential equations, and group theory. He is known for his determinants identities, known as Frobenius-Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was the first to introduce the notion of rational approximations of functions, known as Made approximates, and to give a full proof of the Cayley-Hamilton theorem. He is also known for Frobenius manifolds, which are differential-geometric objects.

SOPHIA KOVALEVSKAYA- 1850-1891. Made noteworthy contributions to analysis, partial differential equations, and mechanics. She is known for the Cauchy-Kowalevski theorem.

OLIVER HEAVISIDE- May 18, 1850-February 3, 1925. Adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations, equivalent to Laplace transforms, reformulated Maxwell’s equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. He is known for the Heaviside-step function, among other achievements.

CARL LINDEMANN- April 12, 1852-March 6, 1939. He is noted for his proof of pi being a transcendental number (a number which is not a root of any polynomial with rational coefficients.).

HENRI POINCARE- April 29, 1854-July 17, 1912. A polymath and described the last universalist in mathematics. He is known for the poincare conjecture (which was solved in 2003), the three-body problem, special relativity, Hilbert-Poincare series, chaos theory, coining the term ‘Betti number’, and the power fixed-point theorem.

EMILE PICARD- July 24, 1856-December 11, 1941. He is known for Painieve transcendentals, Picard group, and Picard theorem, which states that an analytic function with an essential singularity takes every value infinitely often, and perhaps one exception, in any neighborhood of the singularity.

KARL PEASON March 27, 1857-April 27, 1938. He is credited with establishing the discipline of mathematical statistics. He is known for the Pearson distribution, phi coefficient, and Pearson’s chi-squared test.

GIUSEPPE PEANO- August 27, 1858-April 20, 1932. He wrote over 200 books and papers, and was a founder of mathematical logic and set theory, to which he contributed much notation.the axiomatization of natural numbers is named the Peano numbers. He also made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction.

Giulio Ascoli (20 January 1843, Trieste – 12 July 1896, Milan) was an Italian mathematician. He made contributions to the theory of functions of a real variable and to Fourier series. For example, Ascoli introduced equicontinuity in 1884, a topic regarded as one of the fundamental concepts in the theory of real functions.[1] In 1889, Italian mathematician Cesare Arzelà generalized Ascoli’s Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.[

DAVID HILBERT- January 23, 1862-February 14, 1943. He is recognized as one of the most influential and universal mathematicians if the 19th and 20th centuries. He discovered and developed ideas in many areas, including invariant theory and the axiomatization of geometry, one of the foundations of functional analysis. He is known as one of the founders of proof theory and mathematical logic. He put forth his famous 23 problems in 1900 to hopefully be solved in the 20th century.

GOTTLOB FREGE- November 8, 1848-July 26, 1925. He is considered a major figure in mathematics, and responsible for the development of modern logic. He is known for predicate calculus, Frege’s theorem, the Frege-Geach problem, and more.

CHRISTIAN FELIX KLEIN- April 25, 1849-June 22, 1925. Known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connection of geometry and group theory. He classified geometries by their underlying symmetry groups. He is known for a Klein bottle. It is a one sided surface which, if traveled upon, could be followed back to the post of origin while flipping the traveler upside down. it is a one sided bottle.

GREGORIO RICCI-CUBASTRO- January 12, 1853-August 6, 1925. He invented tensor calculus, an extension of vector calculus to tensor fields (tensors that may vary over a manifold I.e. spacetime).

THOMAS STIELTJES- December 29, 1856-December 31, 1894. he pioneered in the field of moment problems and contributed to the study of continued fractions.

GRACE CHISHOLM YOUNG- March 15, 1858-March 29, 1944. she worked in calculus and is known for the Denjoy-Young-Saks theorem.

Vito Volterra (3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations,[2][3] being one of the founders of functional analysis. He is known for the lotka-volterra equations.

CHARLES JEAN DE LA VALLEE-POUSSIN- august 14, 1865-march 2, 1962. he is best known for providing a proof of the prime number theorem.

ERIK IVAR FREDHOLM- april 7, 1866-august 17, 1927. Worked on integral equations and operator theory which foreshadowed hillier spaces.

JACQUES HADAMARD- December 8, 1866-October 17, 1963. He made contributions in number theory, complex function theory, differential geometry, and partial differential equations. He proved the prime number theorem, and is known for the Hadamard product and Hadamard matrices.

FELIX HAUSDORFF- November 8, 1868-January 26, 1942. He is one of the founders of modern topology and contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis. He is known for the Hausdorff Dimension, the Hausdorff space, the Hausdorff paradox, and other achievements. RIP

ELIE CARTAN- April 9, 1869-May 6, 1951. He made significant contributions to mathematical physics, differential geometry, and group theory. He is known for Lie groups and differential forms.

BOREL- january 7, 1871-february 3, 1956. He is known for founding work in the areas of measure theory and probability.

Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Steinitz’s 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the proof of Kőnig’s theorem for regular bipartite graphs, phrased in the language of configurations.

In 1910 Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields, Crelle’s Journal (1910), 167–309). In this paper he axiomatically studies the properties of fields and defines important concepts like prime field, perfect field and the transcendence degree of a field extension. Steinitz proved that every field has an algebraic closure. He also made fundamental contributions to the theory of polyhedra: Steinitz’s theorem for polyhedra is that the 1-skeletons of convex polyhedra are exactly the 3-connected planar graphs. His work in this area was published posthumously as a 1934 book, Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie,[1] by Hans Rademacher.

ERNST ZERMELO- July 27, 1871-May 21, 1953. A logician, his work had major implications in the foundation of mathematics, and he is best known for developing the Zermelo-Fraenkel axiomatic set theory, and his proof of there well-ordering theorem.

BERTRAND RUSSELL- May 18, 1872-February 2, 1970. He is known for his work in mathematical logic, paradoxes of set theory, Russell’s paradox, the barber’s paradox, prepositional logic, and much more.

Issai Schur (January 10, 1875 – January 10, 1941) was a Jewish mathematician who worked in Germany for most of his life. As a student of Frobenius, he worked on group representations (the subject with which he is most closely associated), but also in combinatorics and number theory and even theoretical physics. He is perhaps best known today for his result on the existence of the Schur decomposition and for his work on group representations (Schur’s lemma).

TEIJI TAKAGI- April 21, 1875-February 28, 1960. He is best know for proving the Takagi existence theorem in class field theory. He worked on the Blanomange curve, a graph that is nowhere differentiable but a uniformly continuous function.

HENRI LEBESGUE- june 28, 1875-july 26, 1941. He is most famous for his theory of integration, which is summing the area between an axis and the curve of a function defined for that axis.

HARDY- february 7, 1877-december 1, 1947. He is known for his achievements in number theory and analysis, and for the hardy-weinberg principle, hardy-ramanujan asymptotic formula, and the hardy-littlewood circle method. He is the mathematician who discovered ramanujan.

EDMUND LANDAU- February 14, 1877-February 19, 1938. He worked in number theory and complex analysis. He is known for his work in the distribution of prime numbers and the Landau prime ideal theorem.

Maurice Fréchet (French: [moʁis ʁəne fʁeʃɛ]; 2 September 1878 – 4 June 1973) was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness. Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions.

Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini’s theorem and the Fubini–Study metric.

LIPOT FEJER- February 9, 1880-october 15, 1959. He is known for research in harmonic analysis, in particular, Fourier series.

OSWALD VEBLEN- June 24, 1880-August 10, 1960. A geometer and topologist, he applied these to atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.

LUITZEN BROUWER- february 27, 1881- December 2, 1966. He worked in topology, set theory, measure theory, and complex analysis.he was the founder of mathematical intuitionism. He is known for the Brouwer fixed-point theory and the hairy ball theorem

WACLAW SIERPINSKI- march 14, 1882-october 21, 1969. He is known for outstanding contributions to set theory, research on the axiom of choice and the continuum hypothesis, number theory, the theory of functions, and topology.

EMMY NOETHER- March 23, 1882-April 14, 1935. She is known for her landmark contributions to abstract algebra and theoretical physics. One of the leading mathematicians, she developed the theories of rings, fields, and algebras, and explained the connection between, symmetry and conservation laws.

GEORGE BIRKHOFF- March 21, 1884-November 12, 1944. He is best known for what is now known as the ergodic theorem. He also proved Poincare’s ‘Last Geometric Theorem’, a special case of the three-body problems, and proved that the Schwarzschild geometry is the unique symmetric solution of the Einstein field equation.

SOLOMON LEFSCHETZ- September 3, 1884-October 5, 1972. He did fundamental work on algebraic topology and its applications to algebraic geometry, and on the theory of non-linear ordinary differential equations. He is known for the Lefschetz hyperplane theorem, the Lefschetz number. The Lefschetz fixed point theorem, among other acheivements.

JOHN LITTLEWOOD- june 9, 1885-september 6, 1977. He is best known for achievements in analysis, number theory, and differential equations. He attempted to solve the Riemann hypothesis by showing that if it were true then the prime number theorem follows and obtains the zero term. He is known for the 1st and 2nd hardy-littlewood theorems, the 1st is a strong form of the twin prime conjecture.

HERMANN WEYL- November 9, 1885-december 8, 1955. He is known for well algebra, well transform, the well lemma (a very weak form of the Laplace transform), the well tensor, and other discoveries.

LUDWIG BIEBERBACH- december 4, 1886-september 1, 1982. German mathemetician known for bieberbach conjecture.

GEORGE POLYA- December 13, 1887-september 7, 1985. He made fundamental contributions to combinatorics, number theory, numerical analysis, and probability theory. He is known for the poly conjecture, the ploy enumeration theorem, the Hilbert-polya theorem, among other accomplishments.

RAMANUJAN- december 22, 1887-april 26, 1920. A mathematical genius. With almost no formal training in pure mathematics, made substantial contributions to analysis, number theory, infinite series, and continued fractions, including solving problems considered unsolvable.

RICHARD COURANT- January 8, 1888-January 27, 1972. He is known for the Courant number and the Courant minimax principle. He published in 1943 the finite element method, which is his numerical treatment of the plain torsion problem for multiply-connected domains, and is now one of the ways to solve partial differential equations numerically.

LOUIS MORDELL- january 28, 1888-march 12, 1972. he is known for pioneering research in number theory.

STEFEN BANACH-

March 30, 1892-August 31, 1945. He is generally considered one one of the world’s most important and influential mathematicians of the 20th century. He is known for the Banach-Tarski paradox, the Banach-Steinhaus theorem, and for functional analysis, which he founded.

GASTON JULIA- february 3, 1893-march 19, 1978. he devised the formula for the Julia set.

NORBERT WIENER- November 26, 1894-March 18, 1964. He developed Tauberian theorems ( which deals with infinite series) in summability theory, most of which could be encapsulated in a principle from harmonic analysis. He is also known for abstract Wiener space, a mathematical object in measure theory, and real-valued continuous paths on the unit interval known as classical Wiener space.

EMIL ARTIN- march 3, 1898-december 20, 1962. One of the leading mathematicians in the 20th century known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. he also contributed to the pure theories of rings, groups, and fields.

OSCAR ZARISKI- April 24, 1899-July 4, 1986. One of the most influential algebraic geometers of the 20th century. He is known for Zariski theory on holomorphic functions.

MARY CARTRIGHT- december 17, 1900-april 3, 1998. she was the first to analyze a dynamical system of chaos.

ANTONI ZYGMUND- decemeber 25, 1900-may 30, 1992. Considered one of the greatest analysts of the 20th century. his main interest was harmonic analysis.

ALFRED TARSKI-

January 14, 1901-October 26, 1983. A prolific author who did work in model theory, metamathematics, algebraic logic, abstract algebra, topology, geometry, measure theory, mathematical logic, and set theory. He is known for his work on the foundations of modern logic, Tarski’s undefinability theory, and the Banach-Tarski paradox, which states that a ball can be decomposed into a finite number of point sets and reassembled into 2 balls of the original.

Bartel Leendert van der Waerden (Dutch: [vɑn dər ˈʋaːrdə(n)]; February 2, 1903 – January 12, 1996) was a Dutch mathematician and historian of mathematics. Van der Waerden is mainly remembered for his work on abstract algebra. He also wrote on algebraic geometry, topology, number theory, geometry, combinatorics, analysis, probability and statistics, and quantum mechanics (he and Heisenberg had been colleagues at Leipzig). In his later years, he turned to the history of mathematics and science.

FRANK RAMSEY- february 22, 1903-january 19, 1930. He is known for Ramsey theory, a branch of mathematics that studies the conditions under which order must appear.

VON NEUMANN- December 28, 1903-february 8, 1957. He had a phenomenal memory. He made major contributions to functional analysis, topology, numerical analysis, quantum mechanics, quantum statistical analysis, game theory, computing, linear programming, self-replicating machines, and statistics. He is known for a great numbers of achievements.

ANDREY KOLMOGOROV- April 25, 1903-October 20, 1987. He is known for work in probability theory, topology, intuitional logic, turbulence studies, classical mechanics, mathematical analysis, Kolmogorov complexity, KAM theorem, and then KPP equation.

ALONZO CHURCH- June 14, 1903-august 11, 1995. Made major contributions to mathematical logic and to the foundation of theoretical computer science. He is best known for the lambda calculus and the church-rosser theorem.

W.V.D. HODGE- June 17, 1903-july 7, 1975. He was specifically a geometer. He discovered far reaching topological relations between algebraic geometry and differential geometry, and an area now called hodge theory, and pertaining more generally to Mahler manifolds.

RENATO CACCIOPPOLI-

january 20, 1904-may 8, 1959. he is known for his contributions to mathematical analysis, including the theory of functions of several complex variable, functional analysis, and measure theory. RIP.

HENRI CARTAN- July 8, 1904-August 13, 2008. He made substantial contributions to algebraic topology. He is known for Cartan’s theorems A and B.

KURT GODEL- April 28, 1906-January 14, 1978. Considered the greatest logician in history along with Aristotle, Tarski, and Frege. He published 2 incompleteness theorems in 1931 in mathematical logic, which he proved, that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. the theorems show that finding a complete and consistent set of axioms for all of mathematics is impossible.

OLGA TAUSSKY-TODD- august 30, 1906-october 7, 1995. She wrote more than 300 research papers in algebraic number theory, integral matrices, and matrices in algebra and analysis.

HAROLD COXETER- February. 9, 1907-march 31, 2003. he is regarded as one of the greatest geometer of the 20th century.

STANISLAW ULAM- april 13, 1909-may 13, 1984. He invented the Monte Carlo method of computation, suggested nuclear pulse space propulsion, proved some theorems, and proposed several conjectures.

PAL TURAN- august 18, 1910-september 26, 1976. Worked primarily in number theory. he is known for the power sum method and extremal graph theory.

SHIING-SHEN CHERN- october 26, 1911-december 3, 2004.- known for the cern-simony theory, chern-weil theory, and the Chern class.

ALEN TURING- June 23, 1912-June 7, 1954. He provided a formalization of the concept of the algorithm and computation with a turning machine, the model of a general purpose computer. He is also considered the father on theoretical computer science and artificial intelligence. He is known for the Turning proof and the Turing test.

PAUL ERDOS- march 26, 1913-september 20, 1996. One of the most prolific mathematicians ever, he is known for his social approach to mathematics and he collaborated with over 500 mathematicians in proving and conjecturing.

PAUL ERDOS GROUP FROM- N IS A NUMBER FILM—

FAN CHUNG, RONALD GRAHAM, BELLA BELLOBAS, VERA SOS, JOEL SPENCER,

HERB WILF

SAMUEL EILENBERG-

September 30, 1913-January 30, 1998. Co-founded category theory with Saunders Mac Lane. He is also known for the Eilenberg-Steenrod axioms.

MARJORIE LEE BROWNE- september 9, 1914-october 19, 1979. One of the first african-american women to earn a Ph.D in mathematics. her work on classical groups demonstrated simple proofs of important topological properties of the relations between classical groups. Her work in general focused on linear and matrix algebra.

MARTIN GARDNER- October 21, 1914-may 22, 2010. An American mathematics popularizer and regarded as the dean of mathematical puzzles.

GEORGE DANTZIG-

november 8, 1914-may 13, 2005. Known for his development of the simplex method, and algorithm for solving linear problems. he also solved 2 open problems in statistical theory.

LAURENT SCHWARTZ- march 5, 1915-july 4, 2002. He pioneered the theory of distributions, which gives a well-defined meaning to objects such as the Dirac delta function.

PAUL HALMOS-

march 3, 1916-october 2, 2006. he made fundamental advances in mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis, especially hilbert spaces.

CLAUDE SHANNON-

april 30, 1918-february 24, 2001. Cryptographer known as the father of information theory. He demonstrated that electrical applications of Boolean algebra could construct any logical, numerical relationship. He is also known for the Shannon number, Shannon expansion, power inequality, binary code, and much more.

ABRAHAM ROBINSON- October 6, 1918-april 11, 1974. He is widely known for the development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorporated into modern mathematics.

RAYMOND SMULLYAN- may 25, 1919-february 6, 2017. A mathematical logician who popularized mathematical logic problems.

JULIA ROBINSON- December 8, 1919,July 30, 1985. She is best known for her work on decision theory and Hilbert’s 10th problem.

LOTFI A. ZADEH- february 4, 1921-. Proposed fuzzy concepts:ets, logic, algorithms, semantics, languages, control, systems, probabilities, events, and information.

ALFRED REYNI-

march 20, 1921-february 1, 1970. Made contributions in combinatorics, graph theory, number theory, and mostly probability theory.

JOSEPH B. KELLER- july 31, 1923-september 7, 2016. he specialized in applied math and is known for his work on the geometrical theory of diffraction, and the einstein-brilouin-keller method.

RENE THOM- september 2, 1923-october 25, 2002. Made a reputation as a topologist and was involved in singularity theory. Founder of catastrophe theory.

EVELYN BOYD GRANVILLE- may 1, 1924. Second african-American woman to earn a Ph.D. in mathematics.

ISADORE SINGER- may 3, 1924-. Proved the atiyah-singer index theorem, which paved the way for new interactions between pure math and theoretical physics.

CHRISTOPHER ZEEMAN- February 4, 1925-february 13, 2016. known for catastrophe theory, geometric topology, and singularity theory. he described a new theory named dihomology, an algebraic structure associated to a topological space, containing both homology and cohomology, introducing what is now known as the Zeeman spectral sequence, to see how singularities in a space perturb poincare duality.

LOUIS NIRENBERG- February 28, 1925-. One of the outstanding analysts of the 20th century. he made fundamental contributions to linear and partial differential equations and their application to complex analysis and geometry. His contributions include the gagliardo-nirenberg interpolation inequality, which is important in the solution of the elliptical partial differential equations, that arise in many areas of math, and the formalization of the bounded mean oscillation known as john-nirenberg space, which is used to study the behavior of both elastic materials and games of chance as martingales. his work on partial differential equations was described as about the best that can be done towards solving the naiver-stokes existence and smoothness problem in fluid mechanics and turbulence.

JOHN TATE- March 13, 1925-. he is known for many fundamental contributions in algebraic number theory, arithmetic geometry, and algebraic geometry. he is known also for Fourier analysis in number fields which is one of the ingredients in the modern theory of automorphic forms and their l-function, notably by its use in the Adele ring and ts self-duality and harmonic analysis on it. He gave a cohomological treatment of global class field theory using techniques of group cohomology applied to the dale class group and Galois cohomology. This treatment made more transparent some of the algebraic structures in the previous approaches in class field theory which used central field algebras to compute the Brauer group of a global field. he also made a number of important contributions to p-adic theory. he invented rigid analytic spaces which led to the field of rigid analytic geometry. he found a p-adic analogue of hodge theory, known as hodge-tare theory, a central technique of modern algebraic number theory. he also created the Tate curve parametrization for certain p-adic elliptic curves and the p-divisible (tate-barsotti) groups. the classification of abelian varieties over finite fields and led to the honda-tate theorem. he is known for the Tate conjecture and the Tate module.

a special case of the Tate conjecture was involved in the proof of the morsel conjecture.

PETER LAX- may 1, 1926-. has worked in pure and applied mathematics and made important contributions to integratabtle systems, fluid dynamics and shock waves, solijtronic physics, hyperbolic conservation laws, and mathematics and computer science.

JEAN-PIERRE SERRE- september 15, 1926-. He made contributions to algebraic topology, algebraic geometry, and algebraic number theory. he was awarded the fields medal, won the wolf prize (2000( and the Abel prize (2003). Together with cartoon, serve established the technique of using eilenberg-maclane spaces for computing homotopy groups of seres, at that time a major problem in topology.

YUTAKA TANIYAMA-

novemeber 12, 1927-novemebr 17, 1958. he is known for the taniyama-shimura conjecture. RIP.

LENNART CARLESON- march 18, 1928-. Known as a leader in the field of harmonic analysis. He proved lusin’s conjecture. he also solved the probability problem of stopping times. In the theory of hardy spaces, he contributed the corona theorem, and established the almost everywhere convergence of fourier series for square-integratabtle functions, now known as Carleson’s theorem. he is also known for the theory of Carleton measures. In the theory of dynamical systems, he has worked in complex dynamics.

ALEXANDER GROTHENDIECK- March 28, 1928-November 13, 2014. Leading figure in the creation of algebraic geometry. He is considered by many to be the greatest mathematician in the 20th century.

JOHN NASH- june 13, 1928-may 23, 2015. He won the 1994 Nobel prize in economics for his producing mathematics of the highest quality. RIP

MICHAEL ATIYAH- april 22, 1929-.-known for the Atiyah-Singer index theorem

GORO SHIMURA-

february 23, 1930-. he is known for the modularity theorem, previously known as the taniyama-shimura conjecture.. he extended the theory of complex multiplication and modular forms to higher dimensions.he brought to the concept of higher dimension equivalent of modular curve. They bear the same relation to general hodge structures as modular curves do to elliptic curves.

STEPHEN SMALE- July 15, 1930-. He researches toplogy, dynamical systems, and mathematical economics.

JACQUES TITS- august 12, 1930-. -works on group theory and incidence geometry, and introduced the tits buildings, the tits alternative, and the tits group.

JOHN MILNOR- february 20, 1931-. he is known for his work in differential topology, k-theory, dynamical systems, exotic spheres, fary-milnor theorem, milnor’s theorem, milnor-thurston kneading theory, and surgery theory.

HEISUKE HIONAKA- april 9, 1931-. he proved that singularities of algebraic varieties admit resolutions in characteristic zero. he also showed that a deformation of kahler manifolds need not be kahler.

KLAUS ROTH- october 29, 1925-november 10, 2015. Known for his work on diophantine approximation, the large sieve, discrepancy theory, and irregularities of distribution.

HERBERT WILF- June 13, 1931-january 7, 2012. Specialized in combinatorics and graph theory.

ROGER PENROSE- August 8, 1931-. he is known for his work in mathematical physics and contributions to general relativity and cosmology.

VIVIENNE MALONES-MAYES-

1932-1995. the 5th african-american woman to earn a Ph.D in math. she studied the properties of functions.

TAIRA HONDA-

June 2, 1932-may 15, 1975. he worked in number theory and proved the honda-tate theorem classifying abelian varieties over finite fields. RIP.

KENNETH APPEL- October 8, 1932-april 19, 2013. he solved the 4-color problem in 1976.

JOHN GRIGGS THOMPSON- october 13, 1932-. noted for his work in the field of finite groups, introducing new techniques in solving the nilpotency of Frobenius kernels problem. he made major contributions to the inverse Galois problem, and found a criterion for a finite group to be a Galois group, which implies that the monster simple group is a Galois group. the Thompson group the is the 26 sporadic finite simple group.

PAUL COHEN-

april 2, 1934-march 23, 2007. He is best known for the proofs that the continuum hypothesis and the axiom of choice are independent from zermelo-fraenkel set theory.

NICOLAS BOURBAKI- a group of mainly French mathematicians aimed at reformulating mathematics on an extremely abstract and formal but self-contained basis in set theory by writing a series of books beginning in 1935.

YAKOV SINAI- september 21, 1935-. Contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. he has worked on mathematical physics and probability theory and his efforts have provided the groundwork for advances in the physical sciences.

DONALD KNUTH- january 10, 1936-. a mathematical and computer scientist, he is known for the knuth-morris-pratt algorithm, knuth-bendix completion algorithm, and the robinson-schensted-knuth correspondence.

ROBERT LANGLANDS- october 6, 1936-.-known for the ganglands program

C.T.C. WALL- december 14, 1936-.-known for the Brauer-wall group, Wall’s conjecture, Surgery on compact manifold.

YURI MANIN- 1937-. has worked in algebraic geometry and diophantine geometry.

DAVID MUMFORD- june 11, 1937-. he is known for distinguished work in algebraic geometry, the mumford-shah functional, and research into vision and pattern theory.

BARRY MAZUR- december 19, 1937-. he is known for diophantine geometry, generalized schoenflies conjecture, Mazur swindle, and Mazur torsion theorem.

JAMES HARRIS SIMONS- 1938-. He developed the chern-simons form, and contributed to the development of string theory by providing a theoretical framework to combine geometry and topology with quantum field theory.

C.P. RAMANUJAM-

january 9, 1938-october 27, 1974. he worked in the fields of number theory and algebraic geometry. He worked on waring’s problem in algebraic number fields and made contributions to number theory. RIP.

SERGEI NOVIKOV- march 20, 1938-. noted for work in algebraic topology and soliton theory. he has worked in ciborium theory. he showed how the Adams spectral sequence, used in calculating homotopy groups, could be adapted to cohomology theory typified by cobordism and k-theory. This required the development of the idea of cohomology operations in the general setting, since the basis of spectral sequence is the initial data of ext functors taken with respect to a ring of such operations, generalizing the steered algebra. the resulting adams-novikov spectral sequence is now a basic tool in stable homotopy theory.

ALAN BAKER- august 19, 1939-. Known for his work on effective methods in number theory, in particular those arising from transcendental number theory. He is also known for work in diophantine equations and for baker’s theorem.

SRINIVAS VARADHAN- january 2, 1940-. Known for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.

ENDRE SZEMERELI- august 21, 1940-.-known for Szemeredi’s theorem, the Szemeredi regularity lemma, the Erdos-Szemeredi theorem, the Hajnal-Szemeredi theorem, and the Szemeredi-Trotter theorem

ENRICO BOMBIERI- november 26, 1940-. He is known for large sieve method in analytic number theory, bombieri-lang conjecture, bomber norm, bombieri-vinogtadov theorem, heights in diophantine geometry, siege’s lemma, and bomieri-friedlander-iwaniec theorem.

KAREN UHLENBECK- august 24, 1942. Known for work in the calculus of variations.

MIKHAIL GROMOV- December 23, 1943-. he has made revolutionary contributions to geometry.

PIERRE DELIGNE- October 3, 1944-. he is known for work on the Weil conjectures, leading to a complete proof in 1973.

MITCHELL FEIGENBAUM- december 19, 1944. Pioneering work in chaos theory and discoverer of the Feigenbaum constants.

PERCI DIACONIS- january 31, 1945-. – known for tackling math problems involving randomization, such as coin flipping and shuffling playing cards.

GRIGORY MARGULIS- february 24, 1946-. Known for work on lattices in lie groups, and the introduction to methods from ergodic theory into diophantine approximations.he is also known for the super rigidity theorem, arithmeticity theorem, expander graphs, and the Oppenheimer conjecture.

WILLIAM THURSTON- October 30, 1946-August 21, 2012. he has made contributions to the study of 3-manifolds.

ALAIN CONNES- april 1, 1947-. Known for the baum-connes conjecture, noncompetitive geometry, and operator algebras.

JOHN BALL- 1948-. He has researched elasticity, the calculus of variations, and infinite-dimensional dynamical systems.

LASZLO LOVASZ- march 9, 1948-. he is best known for his work in combinatorics.

SHING-TUNG BYAU- april 4, 1949-. he has worked mainly in differential geometry, especially geometric analysis. his proof of the positive energy theorem in general relativity demonstrated that einstein’s theory is consistent and stable. his proof of the calami conjecture allowed physicists to show, using calai-yau compactification, that string theory is a viable candidate for a unified theory of nature. Calai-yau manifolds are part of the standard toolkit for string theorists today.

CHARLES FEFFERMAN- april 18, 1949-.-he received his phd from princeton at age 20 in computer, mathematical, and natural sciences. his primary field of research is mathematical analysis. He contributed innovations that revised the study of multidimensional complex analysis by finding fruitful generalizations of classical low-dimensional results. His work on partial differential equations, fourier analysis, in particular convergence, multipliers, divergence, singular integrals and hardy spaces earned him the fields medal in 1978. His work also included a study of the asymptotics of the Bergman kernel off the boundaries of pseudo convex domains in c^m. He has studied mathematical physics, harmonic analysis, fluid dynamics, neural networks, geometry, and spectral analysis.

FAN CHUNG-

October 9, 1949. Works mainly in spectral graph theory, extremal graph theory, and random graphs, in particular in generalizing the erdos-renyi model for graphs with general distribution (including power-law graphs in the study of large information networks.)

FRANK KELLY- December 28, 1950-. he specializes in optimization, queueing theory, and network theory, and has researched random processes, networks, and optimization, especially in very large-scale systems such as telecommunications or transportation networks.

MICHAEL FREEDMAN- april 21, 1951-. He worked on the poincare conjecture in dimension 4, and showed that exotic R^4 manifolds exist.

EDWARD WITTEN- August 26, 1951-. he is known for work in mathematical physics dealing with string theory, quantum gravity, and supersymmetric quantum field theories.

PETER HALL- november 20, 1951-janusry 9, 2016-known for nonparametric statistics and the bootstrap method

BERNARD SILVERMAN- february 22, 1952-. a statistician known for density estimation, nonparametric regression, and functional data analysis.

BRIAN D. RIPLEY- April 29, 1952-. he has made contributions to the fields of spacial statistics and pattern recognition, and his work on artificial neural networks.

VAUGHN JONES- december 31, 1952-. he is known for his work on von Neumann algebras, discovery of the jones polynomial, knot polynomials, and increased interest in low-dimensional topology..

ANDREW WILES- april 11, 1953-. He proved fermat’s last theorem, which was proposed in the 1600s, in 1994. he is also known for proving the Taniyama-Shimura conjecture for semistable elliptic curves, proving the main conjecture of Iwasawa theory

PETER SARNAK- December 18, 1953-. he is known for his work in analytic number theory, and the hafner-samak-mccurley constant.

VLADIMIR DRINFELD- february 14, 1954-. his work is in algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric ganglands correspondence. he is known for quantum groups, geometric ganglands correspondence, the drinfelf-sokolov-wilson equation, and the manin-drinfeld theorem.

JEAN BOURGAIN- febraury 28, 1954-. he is known for work in analytic number theory, harmonic analysis, ergodic theory, Banach spaces, and partial differential equations.

GERD FALTING- july 28, 1954-. Known for his work in algebraic geometry, and for the morsel conjecture and the faltings’ product theorem.

INGRID DAUBECHIES- august 17, 1954-.-known for wavelets.

EFIM ISAAKOVICH ZELMANOV- september 7, 1955-. he is known for his work on combinatorial problems in non associative algebra and group theory, including his solution of the restricted burnside problem. he has worked on jordan algebras in the case of infinite dimensions, and has shown that the glennie’s identity in a certain sense generated all identities that hold. he then showed that the engel identity for lie algebras implies nilpotent, in the case of infinite dimensions.

DORIN ANDRICA- march 12, 1956. Known for the andrica’s conjecture, which deals with the gaps between prime numbers.

NOGA ALON- february 17, 1956-. he contributed to combinatorics and computer science.

PIERRE-LOUIS LIONS- august 11, 1956-. he has worked on partial differential equations. he was the first to give a complete solution to the boltzmann equation with proof. he is known for the mean field game theory.

JEAN-CHRISTOPHE YOCCOZ- may 29, 1957-september 3, 2016. he is known for dynamical systems and the yoccoz puzzle.

SIMON DONALDSON- August 20, 1957-. he is known for his work on topology of smooth (differentiable) 4-dimensional manifolds, Donaldson theory, and donaldson-thomas theory.

CURT MCMULLEN- may 21, 1958-. he is known for work in complex dynamics, hyperbolic geometry, and telchuller theory.

RICHARD BORCHERDS- November 29, 1959-. Currently working on quantum field theory. he is known for work on lattices, number theory, group theory, and infinite dimensional algebras. he is also known for Borcherds algebra. he pioneered the classification of unimodular lattices, and introduced new algebraic objects, most notably vertex algebras and Borchers-Kac-Moody algebras. these ideas came together in his vertex-algebraic construction and analysis of the fake monster lie algebra. He resolved the conway-norton monstrous moonshine conjecture, which describes an intricate relation between the monster group and modular functions on the complex upper half-plane. To prove this conjecture, he drew on theories of vertex algebras and borcherds-kac-moody algebras along with string theory, and applied them to the moonshine module, a vertex operator algebra with monster symmetry.additional work in moonshine concerned mod p variants of this conjecture, and were known as modular moonshine. He also produced the theory of borcherds product, which are holomorphic automorphic forms on O(n,2) that have well-behaved infinite product expansions at cusps.he used this theory to resolve some long standing conjectures about affineness of certain moduli spaces of algebraic surfaces. he has also put perturbative renormalization, in particular the 1-hooft-veltman proof of perturbative renormalizability of gauge theory, into rigorous mathematical language.

JOHN BAEZ- June 12, 1961-. a mathematical physicist.

TIM GOWERS- november 20, 1963-. he is known for work in functional analysis and combinatorics.

MAXIM KONTSEVICH- august 25, 1964- his work concentrates on geometric aspects of mathematical physics, most notably knot theory, quantization, and mirror symmetry. one of his results is a formal deformation quantization that holds for any poisson manifold. he also introduced knot invariants defined by complicated integrals analogous to feynman integrals. In topological field theory, he introduced the moduli space of stable maps, a rigorous formulation of the feynman integral for topological string theory.

VLADIMIR VOEVODSKY- june 4, 1966-. Worked on developing homotopy theory for algebraic varieties and formulated motivic cohomology. he proved the minor conjecture and motivic bloch-kato conjecture. He is also known for the invariant foundations of mathematics and homotopy type theory.

PERELMAN- june 13, 1966. Made landmark contributions to Riemann geometry and geometric topology. He proved the poincare conjecture in 2003, one of the millennial prizes, but refused to accept the 1 million dollar award for solving the problem.

NOAM DAVID ELKIES- august 25, 1966-. he extended school’s algorithm to create the school-elkies-atkin algorithm, proved that an elliptic curve over the rational numbers is super singular at infinitely many primes, and he found a counterexample of ruler’s sum of powers conjecture for fourth powers.

LAURENT LAFFORGUE- november 6, 1966-. made outstanding contributions to langlands’ program in the fields of number theory and analysis. He proved the ganglands conjectures for the automorphism group of a function field. he proved the question of the construction of compactifications of certain moduli stacks of shtukas.

WENDELIN WERNER- September 23, 1968-. Works on random walks, brownian motion, schramm-loewner evolution, and related theories in probability theory and mathematical physics. he won the fields medal for his contributions to the development of stochastic loewner evolution, the geometry of 2-dimensional brownian motion, and conformal field theory.

SHINICHI MOCHIZUKI- march 29, 1969-. He is the leader of and the main contributor to one of the major parts of modern number theory, anabelian geometry. he solved the grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. He initiated and developed absolute anabelian geometry, mono-anabelian geometry, and combinatorial anabelian geometry. He introduced p-adic teichmuller theory and hodge-arakelov theory. his recent theories include the theory of trobenioids, anabelioids and the stale theta-function theory. He is author of the inter-universal theichmuller theory also referred to as the arithmetic deformation theory or mochizuki theory. It supplies a new conceptual view on numbers, by using groups of symmetries such as the full absolute Galois groups and arithmetic fundamental groups. Its applications provide solutions to problems such as the spire conjecture, the hyperbolic Volta conjecture, and the abs conjecture and its generalization over arbitrary number fields.

RAVI VAKIL- February 22, 1970-.-algebraic geometer, known for Gromov-Witten theory, worked on Schubert calculus, he proved that all Schubert problems are enumerative over the real numbers

ANDREI OKOUNKOV- july 26, 1969-. Works on representation theory of infinite symmetric groups and its applications to algebraic geometry, mathematical physics, probability theory and special functions, the statistics of plane partitions, and quantum cohomology of the hilbert scheme of points in the complex plane. He formulated well-known conjectures relating the gromov-witten invariants and donaldson-thomas invariants of threefold. he bridged probability, representation theory, and algebraic geometry.

ELON LINDERSTRAUSS- august 1, 1970-. Works in the area of dynamics, particularly the area of ergodic theory and its applications to number theory, has made progress on the Littlewood conjecture and major progress on Peter saran’s arithmetic quantum unique ergodicity conjecture, studied the distributions of torus periodic orbits of some arithmetic spaces, generalizing theorems by Hermann Minkowski and Yuri link, and has studied systematically the invariant of mean dimension introduced in 1999 by mikhail gromov.

ISLAV SMIMOV- september 3, 1970-. His research includes complex analysis, dynamical systems, and probability theory. he did work on critical percolation theory, where he proved the cary’s formula for critical percolation on the triangular lattice, and deduced conformal invariance. the smimov’s theorem has led to a fairly complete theory for percolation on the triangular lattice and its relationship to the schramm-loewner evolution, and established conformality for there random-cluster model and Ising model in 2 dimensions.

NGO BAO CHAU- june 28, 1972-. Best known for proving the fundamental lemma for unitary groups/automorphic forms.the general strategy was to understand the local orbital integrals appearing in the fundamental lemma in terms of affine springer fibers arising in the hitching vibration. This allowed the use of geometric representation theory, the theory of perverse sheaves, to study what was initially a combinatorial problem in a number-theoretic nature. he proved the fundamental lemma for lie algebras. he completed the proof of the fundamental lemma in all cases.

CEDRIC VILLANI- october 5, 1973-. Works primarily on partial differential equations, Riemann geometry, and mathematical physics. he is known for work on boltzmann equation, kinetic theory, landau damping, transportation theory, and otto-villain theorem.

MANJUL BHARGAVA- august 8, 1974-.- known for the gauss composition laws, 290 theorems, factorial function, ranks of elliptic curves, geometric number theory, representation theory, p-adic analysis

TERRENCE TAO- july 17, 1975-.- know for the Green-Tao theorem, Tao’s inequality, Kakeya conjecture, Horn conjecture

MARTIN HAIRER- november 14, 1975-. works in the field of stochastic analysis, particularly stochastic partial differential equations. He made fundamental advances in many important directions such as the study of variants of hormander’s theorem, systematization of the construction of Lyapunov functions for stochastic systems, development of a general theory of egodicity for non-markovian systems, multi scale analysis techniques, theory of homogenization, theory of path sampling and, most recently, theory of rough paths and the newly introduced theory of regularity structures.

BEN GREEN- february 27, 1977-.-known for the Green-Tao theorem, proof of the Cameron-Erods conjecture, combinatorics, and number theory

MARYAM MIZAKHANI- may 3, 1977-july 14, 2017. Research topics included teichmuller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. she won the fields medal for her work in the dynamics and geometry of Riemann surfaces and their moduli spaces.

1978-LEONARD ALDEMAN december 31, 1945-.,RONALD RIVEST- may 6, 1947-., ADI SHAMIR- july 6, 1952-. creators of the Ursa encryption algorithm.

ARTUR AVILA- june 29, 1979-. Works primarily on dynamical systems and special theory. he proved the conjecture of the 10 martinis, which explains mathematical physicist Barry Simon’s theory about the behavior of schrodinger operators, mathematical tools related to quantum physics.

1994- PAUL WOLFSKEHL- June 30, 1856-September 13, 1906, TANIYAMA- november 12, 1927-november 17, 1958, GORO SHIMURA- february 23, 1930-., BARRY MAZUR- December 19, 1937-., NICK KATZ- December 7, 1943-., JOHN COATES- january 26, 1945-., KEN KIBET- june 28, 1948-., ANDREW WILES- april 11, 1953-., RICHARD TAYLOR- may 19, 1962-. The people mainly responsible for the proving of fermat’s last theorem.

COMPLETE LIST OF MATHEMATICIANS

MATHEMATICS FIELDS

ARITHMETIC

ARITHMETIC-the branch of mathematic dealing with the properties and manipulation of numbers using addition, subtraction, multiplication and division, fractions, decimals, percentages, powers, exponents, roots/radicals, ratios, proportions, variations, metric system, various number systems other than base 10, fundamental theorem of arithmetic

ALGEBRA

ALGEBRA-he part of mathematics in which letter and other general symbols are used

to represent numbers and quantities in formulas and equations. The study of generalizations of arithmetic operations. rules in algebra (I.e. commutative), relations, function, linear equations, slope equation, point slope and slope intersect formulas, parallel lines and perpendicular lines from the slope, quadratic equation, quadratic formula, discriminants and number of solutions, completing the square, composition, factoring and expanding equations, inverse of a function, laws of exponents, laws of logarithms, change of log base, systems of linear and quadratic equations and higher powers, matrices of systems of equations, systems of inequalities, methods of elimination, substitution, and equality, fundamental theorem of algebra, inequalities, finding roots to equations degree 3 or higher, approximating the zeros of a function, irrational roots of polynomial equations, groups, rings, fields, vector spaces, matrices, matrix polynomials, Boolean algebra, partial fractions, sequences and progressions and series, determinants, parametric equations, polar and spherical and cylindrical coordinate systems, synthetic division, permutations, combinations, binomial theorem, imaginary and complex numbers, symmetry and transformations, polynomial division, matrix algebra

ABSTRACT ALGEBRA-occasionally called modern algebra, is study of algebraic structures. These structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. RING THEORY- the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Also studied are associative algebra, lattices, loops, semigroups, quasi groups, magma, monoids, vector spaces, hilbert spaces, group theory, algebraic over fields, lie algebra, and boolean algebra, sets, induction, binary operations, groups, fundamental theory of groups, power of elements cyclical groups, subgroups direct groups, functions, symmetric groups, equivalent relations, cosets, counting elements of finite group, normal subgroup, homomorphism, homomorphism and normal subgroups, stow theory, rings, subrings, ideals, quotient rings, ring homeomorph, polynomial, form polynomial to field, unique factorization fields, complex numbers, groups, relations and operations, sets, notation numbers, integers, properties of integers, rational numbers, real numbers, rings, integral domains, division rings, fields, polynomials, vector spaces, matrices, matrix polynomials, linear algebra, boolean algebra

GROUP THEORY-in math and abstract algebra, studies algebraic structures known as groups. Groups are central to abstract algebra. Other well-known algebraic structures , such as rings, fields, and vector spaces, can all be seen as groups endowed with the additional operations and axioms, groupoid, isomorphism theorem, finite groups, abelian groups, free groups, lie algebra, associative algebra, fedora groups, crystallographic groups, continuous groups

LINEAR ALGEBRA- the specific properties of linear equations, vector spaces, and matrices

BOOLEAN ALGEBRA- computation with truth values true and false

RING THEORY-In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.

COMPUTER ALGEBRA-algebraic methods as algorithms and computer programs

HOMOLOGICAL ALGEBRA-study of algebraic structures that are fundamental to study of topological spaces

ALGEBRAIC NUMBER THEORY-properties of numbers are studied from algebraic point of view

ALGEBRAIC GEOMETRY-geometry in its primitive form specifying curves and surfaces as solutions of polynomial equations

ALGEBRAIC COMBINATORICS-algebraic methods are used to study combinatorial questions

RELATIONAL ALGEBRA-set of finitely relations that is closed under certain operators

COMMUTATIVE ALGEBRA-branch of abstract algebra that studies commutative rings, their ideals, and modules such as rings. Both algebraic geometry and algebraic number theory build on commutative algebra.

GALOIS THEORY-abstract algebra.. connection between field theory and group theory. With Galois theory, certain problems in field theory can be reduced to group theory, which is simpler and better understood.

LIE ALGEBRA THEORY

LIE GROUP THEORY-is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

HOMOLOGY THEORY-associates a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

HOMOTOPY THEORY-

FIELD THEORY-branch of abstract algebra studying fields.

MULTILINEAR ALGEBRA-extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multi vectors with grassmann algebra.

COMPLEX ALGEBRA-algebra dealing with complex numbers.

UNIVERSAL ALGEBRA-studies algebraic structures themselves, not examples/models of algebraic structures

HIGHER DIMENSIONAL ALGEBRA- can be used in quantum algebra and topology, which rely on concepts from higher-dimensional algebra such as n-categories and n-vector spaces.

REPRESENTATION THEORY-subfield of abstract algebra. it studies algebraic structures representing their elements as linear transformations of vector spaces.

GEOMETRY

GEOMETRY-branch of mathematics concerned with the properties and relations of points, lines, solids, and higher dimensional analogs. The study of shape. Points, lines, planes, space, line segment, ray, angles, triangles, circles, distance, parallel lines, logical arguments, proofs, indirect. proof, axioms, postulates, corollary, definition, assumptions, theorems, polygons, quadrilaterals, Pythagorean theorem, non-euclidean geometry, conics, areas, perimeters, volumes, surface areas, distance and midpoint formulas, orthogonal projection

SOLID GEOMETRY-geometry of 3-dimensional Euclidean space.includes measuring volumes of various solids or polyhedrons (3 dimensional figures), including pyramids, cylinders, cones, truncated cones, spheres, and prisms.

ANALYTIC GEOMETRY—a branch of algebra that is used to model geometry objects-points, straight lines, lines, circles, being the most basic of these. Coordinates (cartesian, cylindrical, polar), tangents in lines and planes, normal line and vector, parametric equations

ALGEBRAIC GEOMETRY

NON-EUCLIDEAN GEOMETRY-flat geometry of everyday intuition is called euclidean geometry, or parabolic geometry,, and the non-Euclidean geometries are called hyperbolic geometry and elliptical geometry, or riemannian geometry is a non-euclidean geometry.

Elliptic- given a line and point outside the line, there is no parallel line

passing through the point, as all lines in elliptic geometry intersect. The sum of

any interior angles of any triangle is always greater than 180 degrees.

Hyperbolic/bolyai-labachevskian- given a line and a point not on the line, in the

plane containing them both, there are at least 2 distinct lines through the

point that do not intersect the line. Geometry of saddle surface or pseudo

spherical surfaces. Negative gaussian curvature.

AXIOMATIC GEOMETRY-the study of axioms of geometry.

PROJECTIVE GEOMETRY-study of the projective properties of geometric figures.

CONVEX GEOMETRY- studies convex sets, mainly in Euclidean space. Convex set occur naturally in many ares, such as computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, etc.

PARABOLIC GEOMETRY-formally name of euclidean geometry, a comprehensive and deductive mathematical system.

SPHERICAL GEOMETRY—geometry of the 2 dimensional surface of a sphere. It is non-euclidean. Applications are navigation and astronomy.

DISCRETE GEOMETRY-combinatorial geometry may be loosely defined as study of geometric objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity.

DIOPHANTINE GEOMETRY-study of integral and rational points to systems of polynomial equations using ideas and techniques from algebraic number theory.

COMBINATORIAL GEOMETRY- like discrete geometry, studies combinatorial methods of discrete geometric objects.

ENUMERATIVE GEOMETRY-concerned with counting numbers of solutions to algebraic questions.

DIFFERENTIAL GEOMETRY-uses techniques of differential calculus, integral calculus, linear algebra, and multilinear algebra to study problems in geometry.

DIFFERENTIAL ALGEBRAIC GEOMETRY-area of differential algebra that adapts concepts and methods from algebraic geometry and applies them to systems of differential equations/algebraic differential equations.

DISCRETE DIFFERENTIAL GEOMETRY-study of polygons, meshes, and simplistic complexes instead of curves and lines.

COMPLEX DIFFERENTIAL GEOMETRY-study of complex manifolds (topological space that locally resembles euclidean space near each point).

COMPLEX ANALYTICAL GEOMETRY-application of complex numbers to plane geometry.

FRACTAL GEOMETRY-geometry of fractals, geometry shapes that have the same degree of non-regularity on all scales. Facials are the kind of shapes we see in nature.

ARAKELOV GEOMETRY-an approach to diophantine geometry. it is used to study diophantine equations in higher dimensions.

AFFINE GEOMETRY-geometry in which properties are preserved by parallel projection from one plane to another. 3rd and 4th euclidean postulates are meaningless. Type of geometry studied by euler.

TRIGONOMETRY

TRIGONOMETRY-branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. Sine, cosine, target and their inverses, hyperbolic trigonometric function and their inverses, graphs of all the trigonometry functions, phase shifts, amplitude change, inversion, stretching, identities, interpolation, polar coordinates, heron’s and mollweide’s formulas, laws of sines, cosines, and tangents

SPHERICAL TRIGONOMETRY-branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons, especially spherical triangles, defined by a number of intersecting great circles on a sphere.

HYPERBOLIC TRIGONOMETRY-study of hyperbolic triangles in hyperbolic space.

GYRO TRIGONOMETRY-used for studying hyperbolic geometry in analogy to the way vector spaces are used in euclidean geometry. also used so that a gyrovector addition can be found which operates according to the parallelogram law.

PRECALCULUS

PRECALCULUS—

Formal algebra rules

Rational and irrational numbers

Functions

Introduction to graphes

basic graphs

Vocabulary polynomial functions

root/zeros of polynomials

slope of straight line

Linear functions-straight line equations

Quadratic-polynomials 2nd degree

Completing the square

Synthetic division

Root polynomial degree >2

Multiple roots, point of inflection

Reflection of graph

Symmetry of graph

Translation of graph

Rational function

Inverse function

Logs

Logs and exponential functions

Sigma notation for sums

Factorials

Permutations

Combinations

Binomial theorem

Multiplying of sums

Mathematical induction

SETS—-

SET THEORY-deals with formal properties of sets, w/o regard to nature their individual. Constituents, and express. Other branches math in terms sets, corn’s lemma, well ordering theorem, axiom of choice

AXIOMATIC SET THEORY-study of the systems of axioms in context relevant to set theory and mathematical logic.

CLASSICAL MATH- standard approach to math based on classical logic and ZFC set theory.

COMBINATORIAL SET THEORY- infinitary combinatorics

LOGIC—-

MATHEMATICAL LOGIC-application Formal logic to math.. is mathematical proof. Sound approaches to solving quantifiable and abstract. Study of expressive power of formal systematic. And deductive power of formal proof systems.

SYMBOLIC LOGIC-by far the simplest type of logic. Boolean algebras. Mathematical logic. Deals with values of variables which have truth values of true or false, usually denoted by 1 and 0 respectively.

PROPOSITIONAL CALCULUS- branch of symbolic logic that deals with propositions and the relations between them, without examination of their content.

ETC. MATH—

REAL AND COMPLEX VARIABLES-functions of a real and complex variables are those whose domains are real and complex numbers, more specifically, the subsets of the real and complex numbers for which the functions are defined.

MATRICES AND VECTORS-

MATRICES-a matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined, hermitian forms, lambda matrices

VECTOR ANALYSIS-a branch of math that deals with quantities that have both magnitude and direction. Some physical and geometric quantities, called scalars, can be defined by specifying magnitude in suitable units of measure.
LINEAR ALGEBRA-branch of mathematics concerning vector spaces and linear mappings between such spaces, but is also concerned with properties common to all vector spaces. Such equations are naturally represented using the formalism of matrices and vectors. (Linear equations, matrices, determinants, vector spaces).

Includes-vectors, matrices, linear systems, dimensions, rank, linear transformation, vector spaces, determinants, eigenvalues, eigenvectors, orthogonality, change of basis, applications and computing eigenvalues, complex scalars, solving large linear systems,

Quadratic forms

MULTILINEAR ALGEBRA-extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multi vectors with grassmann algebra.

SUPER LINEAR ALGEBRA-a generalization of linear algebra to include z2-gradation on all objects. the main objects of study are super vector spaces, (or more generally, supermodels over a commutative super algebra) and linear transformations between them.

CALCULUS, VECTOR CALCULUS, DIFFERENTIAL EQUATIONS, ANALYSIS

CALCULUS—

CALCULUS-a branch of mathematics concerned with the determination, properties, and application of derivatives and differentials. Concerned with rates of change and slopes of curves. It is also concerned with the determination, properties, and applications of integrals which gives the algebraic sum of areas under and between curves. Calculus sties limits, differentiation, integration, and infinite series. Calculus is the basis for classical analysis., , includes- polar and cylindrical and spherical coordinates, infinite sequences and series, power series, integration by parts, limits, continuity, inverse functions, finite and infinite sequences and series, polar coordinates, complex numbers, integration of rational functions by partial fractions, parametric curves, applications derivatives, application integration

VECTOR CALCULUS/VECTOR ANALYSIS-a branch of mathematics concerned with differentiation and integration of vector fields in 3-dimension euclidean space, includes-vectors, lines, and planes, vector valued functions, partial derivatives, multiple integrals, calculation of vector fields, functions of several variables, limits, continuity, partial derivatives, total derivatives, chain rule, extrema functions of 2 variables, vector valued functions in space, kepler’s laws, directional derivatives and gradients, tangent planes and normal vectors to surface, Lagrange multipliers, application Lagrange multipliers, triple integrals in cylindrical, triple integration in spherical, curl, diverge, line integral, line integral in force field, fundamental theorem of line literal, optimization, least square regression, vector and dot product space, cross product and 2 vectors in space, lines planes in space, curves in space, iterated integration and area, double integral and volume, double integral and polar coordinates, center of mass, surface area of solid, intermediate form and improper integral, polar coordinates, infinite series, vectors, parametric curves, surfaces and graphing, functions of several variables and their partial derivatives, multiple integration, vector fields and green and stokes

MULTIVARIABLE/MULTIVARIATE CALCULUS- extension of calculus in one variable to the calculus with functions of several variable. The differentiation and integration of functions involving multiple variable, rather than just one.

CALCULUS OF VARIATIONS/FUNCTIONAL ANALYSIS-a form of calculus applied to expressions or functions in which the law relating the quantities is liable to variation, especially to find what relation between the variables makes an integral a maximum or a minimum.

STOCHASTIC CALCULUS OF VARIATIONS-extends the math field of calculus of variations from deterministic functions to stochastic (random probability or pattern analyzed statistically) processes.

OPTIMAL CONTROL THEORY-a generalization of the calculus of variations.

ADVANCED CALCULUS-algebraic and topological structure of the real number system. Rigorous development of 1 variable calculus including continuous, differentiable, and Riemann integrable functions and the fundamental theorem of calculus. Uniform convergence of a sequence of functions.

Set theory—

1. Schroder-bernstein theorem, invjective and bijective functions

2. equivalence relation

3. Double series

Basic linear algebra—

1. Vector spaces- a collection of vectors

2. Subspace-a vector space that is a subset of some other space, spans/bases

3. Linear transformations

4. Determinants, matrices

5. Eigenvalues and eigenvectors-involves scalar multiples

6. Linear spaces-p norms, orthonormal bases, share’s theorem

7. Polar decomposition- deals with factorization of a sort

Sequences—

1. Vector sequences and their limits

2. Sequential compactness

3. Closed sets, open sets

4. Cauchy sequences- a sequence where the elements become arbitrarily close to each other as the sequence progresses

Continuous functions—

1. Definition of continuity

2. EVT-extreme value theorem-if a real-valued function f is continuous in the closed and bounded interval (alb), then f must attain a maximum and a minimum, each at least once.

3. Collected sets-a set that cannot be partitioned into 2 nonempty subsets which are open in the relative topology induced on the set.

4. Sequences of polynomials

5. The operator norm- to measure the size of certain linear operators

6. Ascoli arsella theorem-used to decide whether every sequence of a given family of real-valued continuous functions on a closed and bounded interval has a uniformly convergent subsequence.

Derivatives—

1. Definition, chain rule, matrices involving derivatives

2. Mean value inequality

3. Higher order derivatives-2nd,3rd,4th,etc

4. The Cartesian product

5. Partal derivatives-derivative of a function of 2 or more variables with respect to 1 variable, the others being treated as constant

6. Implicit function theorem-allows functions to be converted to functions of several real variables

7. Taylors formula, 2nd derivative test-learning about concavity

8. Lagrange multipliers-strategy for finding the local maxima and minima of a function subject to equality constraints.

Measurable functions—

1. Compact sets

2. Borel sets-any set in a topological space that can be formed from open sets, or closed sets, through the operations of countable union, countable intersection, and relative complement

3. Lebesgue stieltjes measure

4. Outer measures

The lebesgue integral+its integral for functions of p variables—

1. The monotone convergence theorem

2. Fate’s lemma approximation of it with simple functions-a statement about an inequality. Dominated convergence theorem

3. fubini’s theorem-establishes a connection between a multiple integral and a repeated one

4. Brouwer fixed point theorem

Brouwer degrees—

1. definition

2. borsuk’s theorem

3. Product formula

4. Jordan separation theorem

Integration and differentiation forms—

1. Manifolds-type of space that resembles Euclidean space near each point

2. Eggoroff’s theorem

3. Vitali convergence theorem

4. Binet Cauchy formula-an identity for the determinant of the product of 2 triangular matrices of transpose shapes

5. Stoke’s theorem

6. Green’s theorem

7. Divergence theorem

8. Sphere/spherical coordinates

Laplace and poisson equations—

1. Poisson problem

2. Harmonic functions and its properties

3. lap[lace equation for general sets

4. Subharmonic functions and its properties

5. Jordan curve theorem

Line integrals—

1. Definition, properties

2. Cross product, box product, triple product

3. Cauchy integral

4. Cauchy gourmet theorem

Hausdorff measures and area formulas—

1. Definition, properties

2. Steiner symmetritization

3. Isodiametric inequality

Other topics—

1. Residues

2. Core formula

3. Nonlinear fubini’s theorem

4. Multiple Integegral, line Intelegral ,surface Integral.,integral Formula., infinite series, improper integral, gamma and beta function, Fourier series, Fourier integral, elliptic integrals, fundamentals of complex variables, applied Partial derivative

TENSOR CALCULUS/ANALYSIS-extension of vector calculus to tensor fields (tensors that may vary over a manifold i.e. In spacetime).

EULER CALCULUS-apples algebraic topology and integral geometry to integrate construct functions.

QUANTUM CALCULUS-a form of calculus without the notion of limits as q- calculus and h-calculus.

DIFFERENTIAL EQUATIONS—

DIFFERENTIAL EQUATIONS- is an equation involving a derivative of a function or functions, that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Included are 1st order and 1st degree differential equations, 1st order and 1st degree differential equations variable separation, 1st order and 1st degree differential equations reduction variable separation, 1st order and 1st degree differential equations exact equation and reduction, 1st order and 1st degree differential equations exact equations, 1st order and 1st degree differential equations and those reducible to that form, geometric applications, physical applications, equations 1st order or higher degree, singular solutions-extraneous loci, application of 1st order and higher degree equations, applications 1st order and higher degree, homogenous linear equations with constant coefficient, linear equations with constant coefficients, linear equations with constant coefficients-variable parameters, linear equations with constant coefficients-short method, linear equations with variable coefficients-cauchy and legrendre equations, linear equations with variable coefficients-equations of 2nd order, linear equations with variable coefficients-miscellaneous types, applications of linear equations, system of linear equations, system of simultaneous linear equations, total linear equations, application total and simultaneous equations, numerical approximation and solutions, integration in series, legrendre, bessel, gauss equations, partial differential equations, linear partial differential equations of 1st order, nonlinear partial differential equations of 1st order, homogenous partial differential equations of 1st order, nonhomogeous linear equations with constant coefficient, PDE of order 2 with variable coefficient, linear 1st order DE, separable 1st ODE, exact 1st ODE and euler’s method, 2nd and higher order DE, 2nd order homogenous DE, 2nd order non homogenous DE, 2nd order linear homogenous DE, 2nd order linear nonhomogenous DE, power series and powering points, power through singular points, Laplace transform, system 1st OLDE, 3 fast proofs numerical method, 10 DE solving tools, limited growth population model, classification of equality points, bifurcation-drastic change in solution, methods to explicit solutions, computer solutions DE, system of equations, 2nd order, damped and undamped oscillators, beating modes and resonance oscillators, linear systems DE, linear algebra, complex and zero eginvalues, all possible linear solutions, nonlinear systems viewed globally and nullclines, nonlinear systems new equilibrium-linearization, bifurcation, limited systems and oscillators, nonlinear pendulums, periodic forcing and chaos, chaos and iterated functions, periodic and ordering of iterated functions, chaotic itineraries in a space of all sequences, conquering chaos-mandelbrot and Julia sets

ANALYSIS—

ANALYSIS-deals with limits and related theories, such as differentiation, integration, continuity, measure, infinite series, and analytic functions. These theories are usually in the context of real and complex numbers and functions, and involves some form of the Riemann integral. Part of mathematics of infinitesimal analysis. Includes-real and complex number systems, set theory, numerical sequences and series, continuity, differentiation, riemann-stieltjes integral, sequence and series functions, topics theory series, functions of several variables, lebesgue theory, fuhini’s theorem

FOURIER SERIES- trigonometric series, cylindrical harmonics (bessel function), Laplace equation in curvilinear coordinates, ellipsoidal harmonics

REAL VARIABLE-ANALYSIS/THEORY OF FUNCTIONS OF REAL VARIABLES (CLASSICAL ANALYSIS)-theory of functions of real variables. Real analysis. Branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variables.

COMPLEX VARIABLES-ANALYSIS/THEORY OF FUNCTIONS OF COMPLEX VARIABLES-investigates functions of complex numbers..it is useful in algebraic geometry, number theory, analytic combinatorics, applied math, physics, thermodynamics, hydrodynamics, particularly quantum mechanics.

ALGEBRAIC ANALYSIS-branch of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the properties and generalizations of functions.

LINEAR FUNCTIONAL ANALYSIS-study of vector spaces that have some kind of limit-related structure and linear functions defined on these spaces and respecting these structures in a suitable sense.

NON-LINEAR ANALYSIS examples of this analysis are the generalizations of calculus to banach spaces, and implicit function theorems.

ABSTRACT ANALYSIS-linear and nonlinear ordinary and partial differential equations, optimization theory, and control theory.

FUNCTIONAL ANALYSIS- study of vector spaces endowed with some kind of limit-related structure (I.e. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense, n-dimensional space

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QUATERNIONIC ANALYSIS-applying the number system to the complex numbers and applied to mechanics in 3-dimensional space. Both theoretic and applied mathematics.

NON-STANDARD ANALYSIS-branch of logic which introduces hyperreal numbers to allow for the existence of genuine infinitesimal which are numbers less than 1/2, 1/3, 1/4, 1/5 …, but greater than 1.

HYPERCOMPLEX ANALYSIS-extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number.

TENSOR ANALYSIS-study of tensors, which play a role in subjects such as differential geometry, mathematical physics, algebraic topology, multilinear algebra, homological algebra, and representation theory.

P-ADIC ANALYSIS-branch of number theory that deals with the analysis of functions of p-adic numbers.

NUMERICAL ANALYSIS- deals with the development and use of numerical methods for solving problems, includes-collocation polynomials, finite differences, factorial polynomials, summations, newtons formula, operators, and collocation polynomials, unequal spaced arguments, divided differences, osculating polynomials, taylor polynomial, integral and prediction, numerical differentiation, numerical integration, gaussian integration, singular integration, sum and series, differential equations, differential problems of higher order, least square polynomial approximation, minimax polynomial approximation, approximation by rational functions, trigonometric approximation, nonlinear algebra, linear systems, linear programming, over determined systems, boundary value problems, monte carlo methods

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ADVANCED MATHEMATICS- deals with ordinary and linear differential equations, laplace transforms, vector analysis, multiple line and surface integrals and their theorems, Fourier series and integrals, gamma, beta, and other special functions, Bessel functions, Legendre functions and other orthogonal functions, partial differential equations, complex variables and conformal mapping, complex inversion formula for laplace transforms, matrices, and calculus of variations.

REST OF MATHEMATICS—

NUMBERS (COMBINATORICS)—-

DISCRETE MATH-set of branches of math that all have in common the feature that they are discrete rather than continuous. Not the name of a branch of math, like number theory, algebra, calculus, etc. is a description of a set of branches on math that all have a common feature that they are discrete rather than continuous, includes- arithmetic and grouping, infinite geometric series, induction, binomial theorem, permutations, combinations, probability, determinants order 2 and 3, determinants order n, system linear equations

COMBINATORICS-deals with combinations of objects belonging to a finite set in accordance with certain constraints, such as those of graph theory.

ADDITIVE COMBINATORICS-arithmetic combinatorics that deals with operations of addition and multiplication.

ALGEBRAIC COMBINATORICS-applies method of abstract algebra to problems of combinatorics.

GEOMETRIC COMBINATORICS-it includes subareas such as polyhedral combinatorics, convex geometry, and discrete geometry.

ENUMERATIVE COMBINATORICS- deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations.

ANALYTICAL COMBINATORICS-part of enumerative combinatorics where methods of complex analysis are applied to generating functions.

PROBABILISTIC COMBINATORICS-random structures that play a role in discrete combinatorics

ASYMPTOTIC COMBINATORICS-uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.

EXTREMAL COMBINATORICS-studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc) can be, it it has satisfied certain restrictions.

INFINITORY COMBINATORICS-an expansion of ideas in combinatorics to account for infinite sets.

RAMSEY THEORY- branch of mathematics that studies the conditions under which order must appear.

GRAPH THEORY-the mathematical theory of the properties and application of graphs.

ALGEBRAIC GRAPH THEORY-branch of graph theory in which methods are taken from algebra and employed to problems about graphs.

EXTREMAL GRAPH THEORY-studies maximal and minimal graphs that satisfy a certain property.

NUMBER THEORY—

NUMBER THEORY-branch of mathematics that deals with the properties and relationships of numbers, especially positive integers. The properties of whole numbers.the study of the natural numbers and the integers. Queen of the mathematics.Discovers interesting relationships between different sorts of numbers and to prove that these are true. Studied are counting numbers and recording numbers, properties of numbers division, euclid’s algorithm, prime numbers, aliquot parts, indeterminate problems, theory of linear indeterminate parts, diophantine problems, congruences, analysis of congruences, Wilson’s theorem and its consequences, ruler’s theorem and its consequences, theory of decimal expansion, converse of fermat’s theorem, classical construction problems, natural numbers, triangular numbers and their progression, exponential and geometric growth, recurrence sequence, bite’s formula, tower of hanoi, theory of prime numbers, ruler’s product formula and divisibility, prime number theorem and Riemann, division algorithm and modular arithmetic, cryptography and fermat’s last theorem, rsa encryption scheme, fermat’s method of ascent, fermat’s last theorem, factorization and algebraic number theory, pythagorean triples, algebraic geometry, complex structures of elliptical curves, irrational numbers, transcending algebraic numbers, diophantine approximation, real numbers as continued fractions, and applications of continued fractions

ADDITIVE NUMBER THEORY/COMBINATORIAL NUMBER THEORY-part of number theory that studies subsets of integers and their behavior under addition.

MULTIPLICATIVE NUMBER THEORY-subset of analytic number theory that deals with prime numbers, factorization, and divisors.

GEOMETRIC NUMBER THEORY-the geometry of numbers which has a close relationship with other areas mathematics, especially functional analysis and diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.

ALGEBRAIC NUMBER THEORY-part of algebraic geometry devoted to the study of the points of the algebraic varieties whose coordinates belong to an algebraic number field.

TRANSCENDENTAL NUMBER THEORY-given a complex number a, is there a polynomial p with integer coefficients such as p(a)=0? If no such polynomial exists then the number is called transcendental. This field deals with algebraic independence of numbers.

ANALYTIC NUMBER THEORY-part of number theory using methods of analysis, as opposed to algebraic number theory.

ABSTRACT ANALYTICAL NUMBER THEORY-takes ideas from classical analytic number theory and applies them to various other areas of math.

TOPOLOGY

TOPOLOGY-study of geometric properties and spacial relations unaffected by the continuous change of shape or size of figures, surfaces, manifolds, combinatorial method, vector fields, metric and topological spaces

ALGEBRAIC TOPOLOGY-math that uses tool from abstract algebra to study topological spaces. Basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

SET THEORETIC TOPOLOGY-combines set theory and general toplogy.

HIGHER DIMENSIONAL TOPOLOGY-characteristic classes are a basic invariant, and surgery theory is a key theory. for manifolds of dimensions greater than 3.

THE REST OF MATHEMATICAL FIELDS

ORDER THEORY-studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is less than or precedes another.

CATEGORY THEORY-formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has 2 basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.

MEASURE THEORY-the study of measures. It generalizes the intuitive notions of length, area, and volume. Examples of measures are Jordan measures, lebesgue measures, borel measure, probability measure, complex measure, and Haar measure.

MODEL THEORY-study of classes of mathematical structures (I.e. groups, fields, graphs, universe of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language.

REPRESENTATION THEORY-studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

BIFURCATION THEORY-study of changes in the qualitative to topological structure of a given family. it is part of dynamical systems theory.

PROOF THEORY-mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis of mathematical techniques.

GAUGE THEORY-a type of field theory in which the Lagrangian is invariant under certain lie groups of local transformations.

HOMOLOGY THEORY- deals with a particular type of mathematical object, such as a topological space or a group, and my have one or more associated homology theories. When the underlying object has a geometric interpretation like topological spaces do, the nth homology represents behavior unique to dimension n.

DIMENSION THEORY-is a branch of topology dealing with dimensional invariants of topological spaces.

MATROID THEORY-borrows extensively from the terminology of linear algebra and graph theory. it has applications in geometry, topology, combinatorial optimization, network theory, and coding theory.

COMPLEXITY THEORY-study of complex systems with the inclusion of the theory of complex systems.

KNOT THEORY- the study of mathematical knots where the ends are joined together so it cannot be undone.

ERGODIC THEORY-study of dynamical systems with an invariant measure.

OPERATOR THEORY-part of functional analysis studying operators.

HODGE THEORY-the study of cohomology groups of a smooth manifold.

TYPE THEORY-in mathematical logic and computer science, any class of formal systems, some as alternatives to set theory as the foundation for all of math.

RIBBON THEORY-branch of topology studying ribbons.

LATTICE THEORY-study of lattices, being important in order theory and universal algebra.

K THEORY- in algebraic topology, it is an extraordinary cohomology theory known as topological k-theory.

L THEORY-the k-theory of quadratic forms.

HYPERFUNCTIONAL THEORY- hyperfunctions are generalizations of functions, as a jump from one holomorphic function to another at a boundary, and can be thought of as distributions of infinite order.

KAHLER MANIFOLDS-branch of differential geometry, the union of Riemann geometry, complex differential geometry, and symplectic geometry.

DYNAMICAL SYSTEMS- in which a function describes the time dependence of a point in a geometrical space. Examples include models that describe the swinging of a pendulum clock, the flow of water in a pipe, and the number of fish each springtime in a lake.

POSSIBILITY THEORY-deals with certain types of uncertainty and is an alternative to probability theory.

POTENTIAL THEORY-study of harmonic functions.

CATASTROPHE THEORY- a branch of mathematics concerned with systems displaying abrupt discontinuous change.

CHAOS THEORY- a branch of mathematics which focuses on the behavior of dynamical systems that are highly sensitive to initial conditions. It states that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect is an example of how a small change can result in a large difference in a later state.

SCHEME THEORY-a scheme is a math structure that enlarges the notion of algebraic, strongly based on commutative algebra. Scheme theory allows a systematic use of methods of topology and homological algebra.

MATRIX THEORY-focuses on study of matices. Initially a sub-branch of linear algebra, soon it grew to cover subjects related to graph theory, algebra, combinatorics, and statistics.

PARTITION THEORY- in number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.

K HOMOLOGY-a homology theory on the category of locally compact Hausdorff spaces.

ARAKELOV THEORY-an approach to diophantine geometry which is used to study diophantine equations in higher dimensions.

TRANSFINITE ORDER THEORY

APPLIED MATHEMATICS-

FINITE MATH-collection of topics that are anything but calculus. Survey of mathematical analysis techniques used in the working world and organizing and analyzing information. Refers either to discrete math or business math, including probability, statistics, linear programming with the geometric approach and the simplex method, matrices, determinants, calculus, Markov process, linear equations, counting, probability, statistics

STATISTICS-brach of math dealing with collection, analysis, interpretation, presentation, and organizing data. In applying statistics, it is conventional to begin with a statistical population or a statistical model to be studied, includes-midrange, median, mode, arithmetic mean, summation, measures of variation, elementary probability and binomial distributions, normal distribution, tests of statistical hypothesis, correlation and regression, confidence limits, nonparametric statistics, analysis variance, distributions (t, x^2, f), fisher x values, spearman rank correlation coefficient, willcoxon signed rank values, sum rank critical values

PROBABILITY- is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes. A measure of the chance that the event will occur as a result of an experiment. Probability of event a is the number of ways event a can occur divided by the total number of possible outcomes. A measure of the likelihood that an event will occur. Is quantified as a number between 0 and 1, where, loosely speaking, 0 indicated impossibility and 1 indicates certainty, random variable and probability distribution, mathematical expectation, discrete probability distribution, continuous probability distribution, function random variable, sampling, data, sampling distribution, 1 and 2 sample estimation problems, 1 and 2 sample estimation problems, linear regression and correlation, multiple linear regression, 1 factor experiments, factorial experiments, factorial experiments and fractions, nonparametric statistics

POSSIBILITY THEORY-deals with certain types of uncertainty and is an alternative to probability theory.

OPTIMIZATION-applied math and number. Analysis concerned w/ development. And deterministic algorithms that are capable of guaranteeing convergence in finite time to the optimal solution of a non convex problem.

COMPUTABILITY THEORY-branch of mathematical logic. the study of computable functions and Turing degrees.

THEORY OF COMPUTATION-in mathematics and computer science, it is a branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm.

EXPERIMENTAL MATHEMATICS- an approach in which computation is used to investigate mathematical objects and identify properties and patterns.

MATHEMATICAL PHYSICS- development of mathematical methods for application to problems in physics. Branch of applied math, but deals with physical problems.

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APPLIED MATHEMATICS-deals with mathematical methods that find use in science, engineering, business, computer science, and industry. It is a combination of mathematical science and specialized knowledge. Fields of applied mathematics are mechanics, fluid mechanics, particle mechanics, operational research, and mathematical programming, all using probability theory, category probability theory, statistics, number analysis, symbolic computation, algebraic computation, and more.

ALL THE REST OF MATHEMATICAL FIELDS

• Absolute differential calculus: the original name for tensor calculus developed around 1890.

• Absolute geometry: an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate.

• Abstract algebra: the study of algebraic structures and their properties. Originally it was known as modern algebra.

• Abstract analytic number theory: a branch of mathematics that takes ideas from classical analytic number theory and applies them to various other areas of mathematics.

• Abstract differential geometry: a form of differential geometry without the notion of smoothness from calculus. Instead it is built using sheaf theory and sheaf cohomology.

• Abstract harmonic analysis: a modern branch of harmonic analysis that extends upon the generalized Fourier transforms that can be defined on locally compact groups.

• Abstract homotopy theory: a part of topology that deals with homotopic functions, i.e. functions from one topological space to another which are homotopic (the functions can be deformed into one another).

• Additive combinatorics: the part of arithmetic combinatorics devoted to the operations of addition and subtraction.

• Additive number theory: a part of number theory that studies subsets of integers and their behaviour under addition.

• Affine geometry: a branch of geometry that is centered on the study of geometric properties that remain unchanged by affine transformations. It can be described as a generalization of Euclidean geometry.

• Affine geometry of curves: the study of curves in affine space.

• Affine differential geometry: a type of differential geometry dedicated to differential invariants under volume-preserving affine transformations.

• Ahlfors theory: a part of complex analysis being the geometric counterpart of Nevanlinna theory. It was invented by Lars Ahlfors

• Algebra: a major part of pure mathematics centered on operations and relations. Beginning with elementary algebra, it introduces the concept of variables and how these can be manipulated towards problem solving; known as equation solving. Generalizations of operations and relations defined on sets have led to the idea of an algebraic structure which are studied in abstract algebra. Other branches of algebra include universal algebra, linear algebra and multilinear algebra.

• Algebraic analysis: motivated by systems of linear partial differential equations, it is a branch of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the properties and generalizations of functions. It was started by Mikio Sato.

• Algebraic combinatorics: an area that employs methods of abstract algebra to problems of combinatorics. It also refers to the application of methods from combinatorics to problems in abstract algebra.

• Algebraic computation: see symbolic computation.

• Algebraic geometry: a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties.

• Algebraic graph theory: a branch of graph theory in which methods are taken from algebra and employed to problems about graphs. The methods are commonly taken from group theory and linear algebra.

• Algebraic K-theory: an important part of homological algebra concerned with defining and applying a certain sequence of functors from rings to abelian groups.

• Algebraic number theory: a part of algebraic geometry devoted to the study of the points of the algebraic varieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and is also said to study algebraic structures related to algebraic integers.

• Algebraic statistics: the use of algebra to advance statistics, although the term is sometimes restricted to label the use of algebraic geometry and commutative algebra in statistics.

• Algebraic topology: a branch that uses tools from abstract algebra for topology to study topological spaces.

• Algorithmic number theory: also known as computational number theory, it is the study of algorithms for performing number theoretic computations.

• Anabelian geometry: an area of study based on the theory proposed by Alexander Grothendieck in the 1980s that describes the way a geometric object of an algebraic variety (such as an algebraic fundamental group) can be mapped into another object, without it being an abelian group.

• Analysis: a rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimal calculus. (Then it was known as infinitesimal analysis.) The classical forms of analysis are real analysis and its extension complex analysis, whilst more modern forms are those such as functional analysis.

• Analytic combinatorics: part of enumerative combinatorics where methods of complex analysis are applied to generating functions.

• Analytic geometry: usually this refer to the study of geometry using a coordinate system (also known as Cartesian geometry). Alternatively it can refer to the geometry of analytic varieties. In this respect it is essentially equivalent to real and complex algebraic geometry.

• Analytic number theory: part of number theory using methods of analysis (as opposed to algebraic number theory)

• Applied mathematics: a combination of various parts of mathematics that concern a variety of mathematical methods that can be applied to practical and theoretical problems. Typically the methods used are for science, engineering, finance, economics and logistics.

• Approximation theory: part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials)

• Arakelov geometry: also known as Arakelov theory

• Arakelov theory: an approach to Diophantine geometry used to study Diophantine equations in higher dimensions (using techniques from algebraic geometry). It is named after Suren Arakelov.

• Arithmetic: to most people this refers to the branch known as elementary arithmetic dedicated to the usage of addition, subtraction, multiplication and division. However arithmetic also includes higher arithmetic referring to advanced results from number theory.

• Arithmetic algebraic geometry: see arithmetic geometry

• Arithmetic combinatorics: the study of the estimates from combinatorics that are associated with arithmetic operations such as addition, subtraction, multiplication and division.

• Arithmetic dynamics:Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.

• Arithmetic geometry: the study of schemes of finite type over the spectrum of the ring of integers

• Arithmetic topology: a combination of algebraic number theory and topology studying analogies between prime ideals and knots

• Arithmetical algebraic geometry: an alternative name for arithmetic algebraic geometry

• Asymptotic combinatorics:It uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.

• Asymptotic geometric analysis

• Asymptotic theory: the study of asymptotic expansions

• Auslander–Reiten theory: the study of the representation theory of Artinian rings

• Axiomatic geometry: also known as synthetic geometry: it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods.

• Axiomatic homology theory

• Axiomatic set theory: the study of systems of axioms in a context relevant to set theory and mathematical logic.

• Bifurcation theory: the study of changes in the qualitative or topological structure of a given family. It is a part of dynamical systems theory

• Birational geometry: a part of algebraic geometry that deals with the geometry (of an algebraic variety) that is dependent only on its function field.

• Bolyai-Lobachevskian geometry: see hyperbolic geometry.

• C*-algebra theory: a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties-(i) A is a topologically closed set in the norm topology of operators.(ii)A is closed under the operation of taking adjoints of operators.

• Cartesian geometry: see analytic geometry

• Calculus: a branch usually associated with limits, functions, derivatives, integrals and infinite series. It forms the basis of classical analysis, and historically was called the calculus of infinitesimals or infinitesimal calculus. Now it can refer to a system of calculation guided by symbolic manipulation.

• Calculus of infinitesimals: also known as infinitesimal calculus. It is a branch of calculus built upon the concepts of infinitesimals.

• Calculus of moving surfaces: an extension of the theory of tensor calculus to include deforming manifolds.

• Calculus of variations: the field dedicated to maximizing or minimizing functionals. It used to be called functional calculus.

• Catastrophe theory: a branch of bifurcation theory from dynamical systems theory, and also a special case of the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries.

• Categorical logic: a branch of category theory adjacent to the mathematical logic. It is based on type theory for intuitionistic logics.

• Category theory: the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.

• Chaos theory: the study of the behaviour of dynamical systems that are highly sensitive to their initial conditions.

• Character theory: a branch of group theory that studies the characters of group representations or modular representations.

• Class field theory: a branch of algebraic number theory that studies abelian extensions of number fields.

• Classical differential geometry: also known as Euclidean differential geometry. see Euclidean differential geometry.

• Classical algebraic topology

• Classical analysis: usually refers to the more traditional topics of analysis such as real analysis and complex analysis. It includes any work that does not use techniques from functional analysis and is sometimes called hard analysis. However it may also refer to mathematical analysis done according to the principles of classical mathematics.

• Classical analytic number theory

• Classical differential calculus

• Classical Diophantine geometry

• Classical Euclidean geometry: see Euclidean geometry

• Classical geometry: may refer to solid geometry or classical Euclidean geometry. See geometry

• Classical invariant theory: the form of invariant theory that deals with describing polynomial functions that are invariant under transformations from a given linear group.

• Classical mathematics: the standard approach to mathematics based on classical logic and ZFC set theory.

• Classical projective geometry

• Classical tensor calculus

• Clifford analysis: the study of Dirac operators and Dirac type operators from geometry and analysis using clifford algebras.

• Clifford theory is a branch of representation theory spawned from Cliffords theorem.

• Cobordism theory

• Cohomology theory

• Combinatorial analysis

• Combinatorial commutative algebra: a discipline viewed as the intersection between commutative algebra and combinatorics. It frequently employs methods from one to address problems arising in the other. Polyhedral geometry also plays a significant role.

• Combinatorial design theory: a part of combinatorial mathematics that deals with the existence and construction of systems of fintie sets whose intersections have certain properties.

• Combinatorial game theory

• Combinatorial geometry: see discrete geometry

• Combinatorial group theory: the theory of free groups and the presentation of a group. It is closely related to geometric group theory and is applied in geometric topology.

• Combinatorial mathematics

• Combinatorial number theory

• Combinatorial set theory: also known as Infinitary combinatorics. see infinitary combinatorics

• Combinatorial theory

• Combinatorial topology: an old name for algebraic topology, when topological invariants of spaces were regarded as derived from combinatorial decompositions.

• Combinatorics: a branch of discrete mathematics concerned with countable structures. Branches of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as many more.

• Commutative algebra: a branch of abstract algebra studying commutative rings.

• Complex algebra

• Complex algebraic geometry: the mainstream of algebraic geometry devoted to the study of the complex points of algebraic varieties.

• Complex analysis: a part of analysis that deals with functions of a complex variable.

• Complex analytic dynamics: a subdivision of complex dynamics being the study of the dynamic systems defined by analytic functions.

• Complex analytic geometry: the application of complex numbers to plane geometry.

• Complex differential geometry: a branch of differential geometry that studies complex manifolds.

• Complex dynamics: the study of dynamical systems defined by iterated functions on complex number spaces.

• Complex geometry: the study of complex manifolds and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry.

• Complexity theory: the study of complex systems with the inclusion of the theory of complex systems.

• Computable analysis: the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis.

• Computable model theory: a branch of model theory dealing with the relevant questions computability.

• Computability theory: a branch of mathematical logic originating in the 1930s with the study of computable functions and Turing degrees, but now includes the study of generalized computability and definability. It overlaps with proof theory and effective descriptive set theory.

• Computational algebraic geometry

• Computational complexity theory: a branch of mathematics and theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

• Computational geometry

• Computational group theory: the study of groups by means of computers.

• Computational mathematics: the mathematical research in areas of science where computing plays an essential role.

• Computational number theory: also known as algorithmic number theory, it is the study of algorithms for performing number theoretic computations.

• Computational real algebraic geometry

• Computational synthetic geometry

• Computational topology

• Computer algebra: see symbolic computation

• Conformal geometry: the study of conformal transformations on a space.

• Constructive analysis: mathematical analysis done according to the principles of constructive mathematics. This differs from classical analysis.

• Constructive function theory: a branch of analysis that is closely related to approximation theory, studying the connection between the smoothness of a function and its degree of approximation

• Constructive mathematics: mathematics which tends to use intuitionistic logic. Essentially that is classical logic but without the assumption that the law of the excluded middle is an axiom.

• Constructive quantum field theory: a branch of mathematical physics that is devoted to showing that quantum theory is mathematically compatible with special relativity.

• Constructive set theory

• Contact geometry: a branch of differential geometry and topology, closely related to and considered the odd-dimensional counterpart of symplectic geometry. It is the study of a geometric structure called a contact structure on a differentiable manifold.

• Convex analysis: the study of properties of convex functions and convex sets.

• Convex geometry: part of geometry devoted to the study of convex sets.

• Coordinate geometry: see analytic geometry

• CR geometry: a branch of differential geometry, being the study of CR manifolds.

• Derived noncommutative algebraic geometry

• Descriptive set theory: a part of mathematical logic, more specifically a part of set theory dedicated to the study of Polish spaces.

• Differential algebraic geometry: the adaption of methods and concepts from algebraic geometry to systems of algebraic differential equations.

• Differential calculus: a subfield of calculus concerned with derivatives or the rates that quantities change. It is one of two traditional divisions of calculus, the other being integral calculus.

• Differential Galois theory: the study of the Galois groups of differential fields.

• Differential geometry: a form of geometry that uses techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on differentiable manifolds. It is closely related to differential topology.

• Differential geometry of curves: the study of smooth curves in Euclidean space by using techniques from differential geometry

• Differential geometry of surfaces: the study of smooth surfaces with various additional structures using the techniques of differential geometry.

• Differential topology: a branch of topology that deals with differentiable functions on differentiable manifolds.

• Diffiety theory

• Diophantine geometry: in general the study of algebraic varieties over fields that are finitely generated over their prime fields.

• Discrepancy theory

• Discrete computational geometry

• Discrete differential geometry

• Discrete dynamics

• Discrete exterior calculus

• Discrete geometry

• Discrete mathematics

• Discrete Morse theory: a combinatorial adaption of Morse theory.

• Distance geometry

• Domain theory

• Donaldson theory: the study of smooth 4-manifolds using gauge theory.

• Dynamical systems theory

• Econometrics: the application of mathematical and statistical methods to economic data.

• Effective descriptive set theory: a branch of descriptive set theory dealing with set of real numbers that have lightface definitions. It uses aspects of computability theory.

• Elementary algebra: a fundamental form of algebra extending on elementary arithmetic to include the concept of variables.

• Elementary arithmetic: the simplified portion of arithmetic considered necessary for primary education. It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includes the concept of fractions and negative numbers.

• Elementary mathematics: parts of mathematics frequently taught at the primary and secondary school levels. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry. (calculus is not usually considered a part)

• Elementary group theory: the study of the basics of group theory

• Elimination theory: the classical name for algorithmic approaches to eliminating between polynomials of several variables. It is a part of commutative algebra and algebraic geometry.

• Elliptic geometry: a type of non-Euclidean geometry (it violates Euclid’s parallel postulate) and is based on spherical geometry. It is constructed in elliptic space.

• Enumerative combinatorics: an area of combinatorics that deals with the number of ways that certain patterns can be formed.

• Enumerative geometry: a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of intersection theory.

• Equivariant noncommutative algebraic geometry

• Ergodic Ramsey theory: a branch where problems are motivated by additive combinatorics and solved using ergodic theory.

• Ergodic theory: the study of dynamical systems with an invariant measure, and related problems.

• Euclidean geometry

• Euclidean differential geometry: also known as classical differential geometry. See differential geometry.

• Euler calculus

• Experimental mathematics

• Extraordinary cohomology theory

• Extremal combinatorics: a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.

• Extremal graph theory

• Field theory: branch of abstract algebra studying fields.

• Finite geometry

• Finite model theory

• Finsler geometry: a branch of differential geometry whose main object of study is the Finsler manifold (a generalisation of a Riemannian manifold).

• First order arithmetic

• Fourier analysis

• Fractional calculus: a branch of analysis that studies the possibility of taking real or complex powers of the differentiation operator.

• Fractional dynamics: investigates the behaviour of objects and systems that are described by differentiation and integration of fractional orders using methods of fractional calculus.

• Fredholm theory: part of spectral theory studying integral equations.

• Function theory: part of analysis devoted to properties of functions, especially functions of a complex variable (see complex analysis).

• Functional analysis

• Functional calculus: historically the term was used synonymously with calculus of variations, but now refers to a branch of functional analysis connected with spectral theory

• Fuzzy arithmetic

• Fuzzy geometry

• Fuzzy Galois theory

• Fuzzy mathematics: a branch of mathematics based on fuzzy set theory and fuzzy logic.

• Fuzzy measure theory

• Fuzzy qualitative trigonometry

• Fuzzy set theory: a form of set theory that studies fuzzy sets, that is sets that have degrees of membership.

• Fuzzy topology

• Galois cohomology: an application of homological algebra, it is the study of group cohomology of Galois modules.

• Galois theory: named after Évariste Galois, it is a branch of abstract algebra providing a connection between field theory and group theory.

• Galois geometry: a branch of finite geometry concerned with algebraic and analytic geometry over a Galois field.

• Game theory

• Gauge theory

• General topology: also known as point-set topology, it is a branch of topology studying the properties of topological spaces and structures defined on them. It differs from other branches of topology as the topological spaces do not have to be similar to manifolds.

• Generalized trigonometry: developments of trigonometric methods from the application to real numbers of Euclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, rational trigonometry, universal hyperbolic trigonometry, fuzzy qualitative trigonometry, operator trigonometry and lattice trigonometry.

• Geometric algebra: an alternative approach to classical, computational and relativistic geometry. It shows a natural correspondence between geometric entities and elements of algebra.

• Geometric analysis: a discipline that uses methods from differential geometry to study partial differential equations as well as the applications to geometry.

• Geometric calculus

• Geometric combinatorics

• Geometric function theory: the study of geometric properties of analytic functions.

• Geometric homology theory

• Geometric invariant theory

• Geometric graph theory

• Geometric group theory

• Geometric measure theory

• Geometric topology: a branch of topology studying manifolds and mappings between them; in particular the embedding of one manifold into another.

• Geometry: a branch of mathematics concerned with shape and the properties of space. Classically it arose as what is now known as solid geometry; this was concerning practical knowledge of length, area and volume. It was then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry. The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including projective geometry, differential geometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of topology.

• Geometry of numbers: initiated by Hermann Minkowski, it is a branch of number theory studying convex bodies and integer vectors.

• Global analysis: the study of differential equations on manifolds and the relationship between differential equations and topology.

• Global arithmetic dynamics

• Graph theory: a branch of discrete mathematics devoted to the study of graphs. It has many applications in physical, biological and social systems.

• Group-character theory: the part of character theory dedicated to the study of characters of group representations.

• Group representation theory

• Group theory

• Gyrotrigonometry: a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vector space in Euclidean geometry.)

• Hard analysis: see classical analysis

• Harmonic analysis: part of analysis concerned with the representations of functions in terms of waves. It generalizes the notions of Fourier series and Fourier transforms from the Fourier analysis.

• High-dimensional topology

• Higher arithmetic

• Higher category theory

• Higher-dimensional algebra

• Hodge theory

• Holomorphic functional calculus: a branch of functional calculus starting with holomorphic functions.

• Homological algebra: the study of homology in general algebraic settings.

• Homology theory

• Homotopy theory

• Hyperbolic geometry: also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is a non-Euclidean geometry looking at hyperbolic space.

• hyperbolic trigonometry: the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.

• Hypercomplex analysis

• Hyperfunction theory

• Ideal theory: once the precursor name for what is now known as commutative algebra; it is the theory of ideals in commutative rings.

• Idempotent analysis

• Incidence geometry: the study of relations of incidence between various geometric objects, like curves and lines.

• Inconsistent mathematics: see paraconsistent mathematics.

• Infinitary combinatorics: an expansion of ideas in combinatorics to account for infinite sets.

• Infinitesimal analysis: once a synonym for infinitesimal calculus

• Infinitesimal calculus: see calculus of infinitesimals

• Information geometry

• Integral calculus

• Integral geometry

• Intersection theory: a branch of algebraic geometry and algebraic topology

• Intuitionistic type theory

• Invariant theory: studies how group actions on algebraic varieties affect functions.

• Inversive geometry: the study of invariants preserved by a type of transformation known as inversion

• Inversive plane geometry: inversive geometry that is limited to two dimensions

• Inversive ring geometry

• Itō calculus

• Iwasawa theory

• K-theory: originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II string theory. (In particular twisted K-theory.)

• K-homology

• Kähler geometry: a branch of differential geometry, more specifically a union of Riemannian geometry, complex differential geometry and symplectic geometry. It is the study of Kähler manifolds. (named after Erich Kähler)

• KK-theory

• Klein geometry: More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.

• Knot theory: part of topology dealing with knots

• Kummer theory: provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field

• L-theory

• Large deviations theory: part of probability theory studying events of small probability (tail events).

• Large sample theory: also known as asymptotic theory

• Lattice theory: the study of lattices, being important in order theory and universal algebra

• Lattice trigonometry

• Lie algebra theory

• Lie group theory

• Lie sphere geometry

• Lie theory

• Line geometry

• Linear algebra – a branch of algebra studying linear spaces and linear maps. It has applications in fields such as abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized in operator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to multilinear algebra.

• Linear functional analysis

• Local algebra: a term sometimes applied to the theory of local rings.

• Local arithmetic dynamics: also known as p-adic dynamics or nonarchimedean dynamics.

• Local class field theory

• Low-dimensional topology

• Malliavin calculus

• Mathematical logic

• Mathematical optimization

• Mathematical physics: a part of mathematics that develops mathematical methods motivated by problems in physics.

• Mathematical sciences: refers to academic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarial science.

• Matrix algebra

• Matrix calculus

• Matrix theory

• Matroid theory

• Measure theory

• Metric geometry

• Microlocal analysis

• Model theory

• Modern algebra: see abstract algebra

• Modern algebraic geometry: the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.

• Modern invariant theory: the form of invariant theory that analyses the decomposition of representations into irreducibles.

• Modular representation theory

• Module theory

• Molecular geometry

• Morse theory: a part of differential topology, it analyzes the topological space of a manifold by studying differentiable functions on that manifold.

• Motivic cohomology

• Multilinear algebra: an extension of linear algebra building upon concepts of p-vectors and multivectors with Grassmann algebra.

• Multivariable calculus

• Multiplicative number theory: a subfield of analytic number theory that deals with prime numbers, factorization and divisors.

• Multiple-scale analysis

• Neutral geometry: see absolute geometry

• Nevanlinna theory: part of complex analysis studying the value distribution of meromorphic functions. It is named after Rolf Nevanlinna

• Nielsen theory: an area of mathematical research with its origins in fixed point topology, developed by Jakob Nielsen

• Non-abelian class field theory

• Non-classical analysis

• Non-Euclidean geometry

• Non-standard analysis

• Non-standard calculus

• Nonarchimedean dynamics: also known as p-adic analysis or local arithmetic dynamics

• Noncommutative algebraic geometry: a direction in noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects.

• Noncommutative geometry

• Noncommutative harmonic analysis: see representation theory

• Noncommutative topology

• Nonlinear analysis

• Nonlinear functional analysis

• Number theory: a branch of pure mathematics primarily devoted to the study of the integers. Originally it was known as arithmetic or higher arithmetic.

• Numerical analysis

• Numerical geometry

• Numerical linear algebra

• Operad theory: a type of abstract algebra concerned with prototypical algebras.

• Operator geometry

• Operator K-theory

• Operator theory: part of functional analysis studying operators.

• Operator trigonometry

• Optimal control theory: a generalization of the calculus of variations.

• Orbifold theory

• Order theory: a branch that investigates the intuitive notion of order using binary relations.

• Ordered geometry: a form of geometry omitting the notion of measurement but featuring the concept of intermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclidean geometry, absolute geometry and hyperbolic geometry.

• Oriented elliptic geometry

• Oriented spherical geometry

• p-adic analysis: a branch of number theory that deals with the analysis of functions of p-adic numbers.

• p-adic dynamics: an application of p-adic analysis looking at p-adic differential equations.

• p-adic Hodge theory

• Parabolic geometry

• Paraconsistent mathematics: sometimes called inconsistent mathematics, it is an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic.

• Partition theory

• Perturbation theory

• Picard–Vessiot theory

• Plane geometry

• Point-set topology: see general topology

• Pointless topology

• Poisson geometry

• Polyhedral combinatorics: a branch within combinatorics and discrete geometry that studies the problems of describing convex polytopes.

• Polyhedral geometry

• Possibility theory

• Potential theory

• Precalculus

• Predicative mathematics

• Probability theory

• Probabilistic combinatorics

• Probabilistic graph theory

• Probabilistic number theory

• Projective geometry: a form of geometry that studies geometric properties that are invariant under a projective transformation.

• Projective differential geometry

• Proof theory

• Pseudo-Riemannian geometry: generalizes Riemannian geometry to the study of pseudo-Riemannian manifolds.

• Pure mathematics: the part of mathematics that studies entirely abstract concepts.

• Quantum calculus: a form of calculus without the notion of limits. There are 2 forms known as q-calculus and h-calculus

• Quantum geometry: the generalization of concepts of geometry used to describe the physical phenomena of quantum physics

• Quaternionic analysis

• Ramsey theory: the study of the conditions in which order must appear. It is named after Frank P. Ramsey.

• Rational geometry

• Rational trigonometry: a reformulation of trigonometry in terms of spread and quadrance instead of angle and length.

• Real algebra: the study of the part of algebra relevant to real algebraic geometry.

• Real algebraic geometry: the part of algebraic geometry that studies real points of the algebraic varieties.

• Real analysis: a branch of mathematical analysis; in particular hard analysis, that is the study of real numbers and functions of Real values. It provides a rigorous formulation of the calculus of real numbers in terms of continuity and smoothness, whilst the theory is extended to the complex numbers in complex analysis.

• Real analytic geometry

• Real K-theory

• Recreational mathematics: the area dedicated to mathematical puzzles and mathematical games.

• Recursion theory: see computability theory

• Representation theory: a subfield of abstract algebra; it studies algebraic structures by representing their elements as linear transformations of vector spaces. It also studies modules over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.

• Representation theory of algebraic groups

• Representation theory of algebras

• Representation theory of diffeomorphism groups

• Representation theory of finite groups

• Representation theory of groups

• Representation theory of Hopf algebras

• Representation theory of Lie algebras

• Representation theory of Lie groups

• Representation theory of the Galilean group

• Representation theory of the Lorentz group

• Representation theory of the Poincaré group

• Representation theory of the symmetric group

• Ribbon theory: a branch of topology studying ribbons.

• Riemannian geometry: a branch of differential geometry that is more specifically, the study of Riemannian manifolds. It is named after Bernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus.

• Rough set theory: the a form of set theory based on rough sets.

• Scheme theory: the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaf theory to study algebraic varieties and is considered the central part of modern algebraic geometry.

• Secondary calculus

• Semialgebraic geometry: a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets.

• Set-theoretic topology

• Set theory

• Sheaf theory

• Sheaf cohomology

• Sieve theory

• Single operator theory: deals with the properties and classifications of single operators.

• Singularity theory: a branch, notably of geometry; that studies the failure of manifold structure.

• Smooth infinitesimal analysis: a rigorous reformation of infinitesimal calculus employing methods of category theory. As a theory, it is a subset of synthetic differential geometry.

• Solid geometry

• Spatial geometry

• Spectral geometry: a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators.

• Spectral graph theory: the study of properties of a graph using methods from matrix theory.

• Spectral theory: part of operator theory extending the concepts of eigenvalues and eigenvectors from linear algebra and matrix theory.

• Spectral theory of ordinary differential equations: part of spectral theory concerned with the spectrum and eigenfunction expansion associated with linear ordinary differential equations.

• Spectrum continuation analysis: generalizes the concept of a Fourier series to non-periodic functions.

• Spherical geometry: a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.

• Spherical trigonometry: a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles.

• Statistics: although the term may refer to the more general study of statistics, the term is used in mathematics to refer to the mathematical study of statistics and related fields. This includes probability theory.

• Stochastic calculus

• Stochastic calculus of variations

• Stochastic geometry: the study of random patterns of points

• Stratified Morse theory

• Super category theory

• Super linear algebra

• Surgery theory: a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.)

• Symbolic computation: also known as algebraic computation and computer algebra. It refers to the techniques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulating them by the numerical quantities represented by them.

• Symbolic dynamics

• Symmetric function theory

• Symplectic geometry: a branch of differential geometry and topology whose main object of study is the symplectic manifold.

• Symplectic topology

• Synthetic differential geometry: a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic.

• Synthetic geometry: also known as axiomatic geometry, it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods.

• Systolic geometry: a branch of differential geometry studying systolic invariants of manifolds and polyhedra.

• Systolic hyperbolic geometry: the study of systoles in hyperbolic geometry.

• Tensor analysis: the study of tensors, which play a role in subjects such as differential geometry, mathematical physics, algebraic topology, multilinear algebra, homological algebra and representation theory.

• Tensor calculus: an older term for tensor analysis.

• Tensor theory: an alternative name for tensor analysis.

• Theoretical physics: a branch primarily of the science physics that uses mathematical models and abstraction of physics to rationalize and predict phenomena.

• Time-scale calculus

• Topology

• Topological combinatorics: the application of methods from algebraic topology to solve problems in combinatorics.

• Topological degree theory

• Topological fixed point theory

• Topological graph theory

• Topological K-theory

• Topos theory

• Toric geometry

• Transcendental number theory: a branch of number theory that revolves around the transcendental numbers.

• Transfinite order theory

• Transformation geometry

• Trigonometry: the study of triangles and the relationships between the length of their sides, and the angles between them. It is essential to many parts of applied mathematics.

• Tropical analysis: see idempotent analysis

• Tropical geometry

• Twisted K-theory: a variation on K-theory, spanning abstract algebra, algebraic topology and operator theory.

• Type theory

• Umbral calculus: the study of Sheffer sequences

• Uncertainty theory: a new branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

• Unitary representation theory

• Universal algebra: a field studying the formalization of algebraic structures itself.

• Universal hyperbolic trigonometry: an approach to hyperbolic trigonometry based on rational geometry.

• Valuation theory

• Variational analysis

• Vector algebra: a part of linear algebra concerned with the operations of vector addition and scalar multiplication, although it may also refer to vector operations of vector calculus, including the dot and cross product. In this case it can be contrasted with geometric algebra which generalizes into higher dimensions.

• Vector analysis: also known as vector calculus, see vector calculus.

• Vector calculus: a branch of multivariable calculus concerned with differentiation and integration of vector fields. Primarily it is concerned with 3-dimensional Euclidean space.

• Wavelet

• Windowed Fourier transform

• Window function

PRACTICAL MATHEMATICS FORMULAS

CIRCLE-

Diameter D = 2 x R

Circumference- C = 2 x Pi*R

area- A = Pi x R^2

SPHERE-

Surface area. A = 4 x Pi x R^2

volume V = 4/3 x Pi*R^3

Diameter of a sphere. d=cuberoot(3/4 x Pi x volume) x 2

PYTHAGOREAN THEORM-

a^2+b^2=c^2

a=length of one right angle’s leg

b=length of other right angle’s leg

c=length of hypotenuse

LAW OF SINES-

A/sinA=b/sinB=c/sinC=2 x R=a x b x c/2 x area of triangle

R=(a x b x c)/(squareroot((a+b+c) x (a+b-c) x (b+c-a))

Area of triangle=1/2 x a x b x sinC

LAWS OF COSINE-

c^2=a^2+b^2-2 x a x b x cosC

cosC=(a^2+b^2+c^2)/2 x a x b

AREA OF A TRIANGLE-

area=base x height x 1/2

AREA OF AN EQUALATRAL TRIANGLE-

area=length of a side SQRT(3)/4

HERON’S FORMULA (area of any triangle)-

area=SRQT(s x (s-side 1) x (s-side 2) x (s-side 3))

s=1/2 x (a + b + c)

SLOPE-

m=(y-y1)/(x-x1)

POINT SLOPE EQUATION OF A LINE-

Y-y1=slope(x-x1)

SLOPE INTERCEPT FORM FOR A LINE-

y=slope(x)+(y intercept)

DISTANCE FORMULA-

distance=squareroot((x-x1)^2+(y-y1)^2+(z-z1)^2))

ALGEBRA FORMULAS-

(a+b)^2=a^2+2 x a x b+b^2

(a-b)^2=a^2-2 x a x b+b^2

x^2-a^2=(x+a) x (x-a)

x^3-a^3=(x-a) x (x^2+a x x+a^2)

x^3+a^3=(x+a) x (x^2-a x x+a^2)

a/b+c/d=(a x d+b x c)/b x d)

a/b-c/d=(a x d-b x c)/b x d

a/b x c/d=a x c/b x d

CONIC SECTIONS-

Circle-

(x-g)^2+(y-h)^2=radius^2, (g=x coordinate, h=y coordinate)

Parabola-

y^2+/-4ax, (a=x coordinate)

x^2=+/-4ay, (a=y coordinate)

ellipse-

x^2/a^2+y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

Hyperbola-

x^2/a^2-y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

QUADRATIC EQUATION-

x=(-b+/-squareroot(b^2-4ac))/2a

LAWS OF EXPONENTS-

a^x x a^y=a^(x+y)

a^x/a^y=a^(x-y)

(a^x)^y=a^(x*y)

(a*b)^x=a^x x b^x

a^0=1

a^1=a

LAWS OF LOGARITHMS-

log(base a)(M x N)=log(base a)(M)+log(base a(N)

Log(base a)(M/N)=log(base a)M-log(base a)(N)

logM^r=r x logM

log(base a)(M)=logM/loga

TRIGONOMETRY-

a=adjacent side of right triangle

o=opposite side of right triangle

h=hypotenuse of right triangle

sine-o/h

cosine=a/h

tangent=o/a

cosecant=h/o

secant=h/a

cotangent=a/o

Pythagorean identities-

sin^2(x)-cos^2(x)=1

sec^2(x)-tan^2(x)=1

csc^2-cos^2(x)=1

Product relations-

Sinx-tanx x cosx

cosx=cotx x sinx

tanx=sinx x secx

cotx=cosx x cscx

Secx-cscx x tanx

cscx=secx x cotx

Trigonometry functions-

sinx=x-x^3/3!+x^5/5!-x^7/7!+…

cosx=1-x^2/2!+x^4/4!-x^6/6!+…

Inverse trigonometry functions-

sin-1x=x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+…

cos-1x=pi/2-(x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+…)

tan-1x=x-x^3/3+x^5/5-x^7/7+…

cot-1x=pi/2-x+x^3/3-x^5/5+x^7/7-…

Hyperbolic functions-

sinhx=x+x^3/3!+x^5/5!+x^7/7!+…

coshx=1+x^2/2!+x^4/4!+x^6/6!+…

Inverse hyperbolic functions-

sinh-1x=x-(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5-(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+…

tanh-1x=x+x^3/3+x^5/5+x^7/7+…

Nth TERM OF AN ARITHMETIC SEQUENCE-

Nth term=a+(number of terms-1)*d

(a=1st term, d=common difference)

SUM OF n TERMS OF AN ARITHMETIC SERIES-

Sum-n/2 x (a+nth term)

(a=1st term, d=common difference)

Nth TERMS OF A GEOMETRIC SEQUENCE-

a(n)=ar^(n-1), (r cannot equal 0.)

(a=1st term, r=common ratio)

SUM OF THE n TERMS OF A GEOMETRIC SEQUENCE-

s=a x ((1-r^n)/(1-r)), (r cannot equal 0, 1.)

(n=number of terms, r=common ratio)

SUM OF AN INFINITE SERIES-

s=n/(1-r)

If absolute value of r<1.

n=number star with

r=how much keep multiplying x with forever

s=sum of infinite series

COMBINATIONS-

C(n,r)=n!/r!(n-r)!

PERMUTATIONS-

P(n,r)=n!(n-r)!

BINOMIAL FORMULA-

(a x x-b)^n

CALCULUS (DIFFERENTIATION)-

d/dx (x^n)=n x x^(n-1)

d/dx sinx= cost

d/dx cosx= -sinx

d/dx tanx=sec^2(x)

d/dx cotx=-csc^2(x)

d/dx sex-secs x tanx

d/dx cscx= -cscx x cotx

d/dx e^x=e^x

d/dx lnx=1/x

d/dx (u+v)=du/dx+dv/dx

d/dx(cu)=c x du/dx

dy/dx=dy/dx x du/dx

(chain rule)

d/dx (u x v)=(v x (du/dx)-(u x (dv/dx)

(product rule)

d/dx(u/v)=(v x du/dx-u x dv/dx/)v^2

(quotient rule)

du=du/dx(dx)

CALCULUS (INTEGRATION)-

The definite integral of t from a to b for definite integral f(t)=F(b)-F(a)

Indefinite integral of x^r dx=x^(r+1)/(r+1)+c, (r cannot equal -1)

Indefinite integral of 1/x dx=ln(absolute value (x))+c

Indefinite integral of sinx dx=—cosx+c

Indefinite integral of cosx dx=sinx+c

Indefinite integral of e^x dx=e^x+c

Indefinite integral of (f(x)+g(x))dx=indefinite integral f(x)+indefinite integral g(x)

Indefinite integral of c x f(x) dx=c x (indefinite integral f(x))

indefinite integral of (u)dv=u x v-indefinite integral (v)du

(integration by parts)

CENTER OF MASS-

Center of mass (x)=((mass1) x (center of mass1)+(mass2) x (center of mass x 2))/

(mass-1+mass-2)

(A point representing the mean position of the matter in a body of system.)

VECTOR ANALYSIS

Norm (magnitude of a vector)=sqrt(x^2+y^2+z^2)

Dot product u (dot) v=(u1)x(v1)+(u2)x(v2)+(u3)x(v3)=||u|| ||v|| cos(theta)

theta is the angle between u and v, 0<=theta<=Pi

Cross product u x v=((u2)x(v3)-(u3)x(v2))i-((u1)x(v3)-(u3)x(v1))j+((u1)x(v2)-(u2)x(v1))k

||u x v||=||u|| x ||v|| sin(theta)

Theta is angle between u and v, 0<=theta<=Pi

2 vectors orthogonal if their dot product v and u=0 or transpose vector v and vector u=0.

BEAUTIFUL MATHEMATICAL FORMULAS

1. PYTHAGOREAN THEOREM-

a^2+b^2=c^2

a=length of one right angle’s leg

b=length of other right angle’s leg

c=length of hypotenuse

This theorem, known for well over 2,000 years, is a fundamental relation in euclidean geometry among the 3 sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. The theorem is extensively use in trigonometry one example from physics where the theorem can be use is with kinetic energy, which has the formula-

energy in joules=1/2*mass in kilograms*velocity in meter^2.

The example illustrating the theorem’s use with kinetic energy is-

The energy needs to go 50 miles per hour is equal to the energy needed to go 40 miles per hour plus the energy needed to go 30 miles per hour. another example is that with the energy needed to accelerate a spaceship 50,000 miles per hour, we could accelerate 2 others, one at 30,000 miles per hour and the other at 40,000 miles per hour because 30,000^2+40,000^2=50,000^2. The Pythagorean theorem can be illustrated in many other ways and has many uses and is used extensively in math and the sciences.

2. EULER’S FORMULAS—

e^(ix) = cos(x) + i*sin(x)

This formula is used very, very frequently in math, physics, and engineering. here are

some uses of euler’s formula: 1. The formula means the complex exponential

function can be used to encode both growth/decline rate (real exponential) and a

rotation rate. 2. The use of this formula is in AC electronic devices, where the AC

current is treated like a complex wave, which make the math 10 times easier. 3. It

also makes hardware DL experiments reduce from days to hours. 4. And quantum

mechanics owes its existence to euler’s formula. When the x in this formula equals

pi, the formula becomes-

e^I(I*pi)+1=0,

another very beautiful formula as it contains many of the very useful and ubiquitous

numbers in math- e (exponent e), i (the imaginary number sqrt(-1)), and pi (3.14…),

along with a 0 and a 1. the formula is used in complex analysis to establish the

fundamental relationship between the trigonometric functions and the complex

exponential function. The famous physicist, Richard Feynmann, has called this

formula-‘our jewel’ and ‘the most remarkable formula in math.

3. LAPLACE’S EQUATION-

d^2/dx^2+d^2/dy^2+d^2/dz^2=0

(Laplace operator)^2 times a scalar function=0

Laplace operator times a scalar function=0

Laplace’s equation and poisson’s equation are the simplest examples of elliptic partial differential equations. The general solutions to laplace’s equation is known as potential theory. The solutions of laplace’s equation are thee harmonic functions. Which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

4. POISSON’S EQUATION-

Laplace operator*real complex-valued functions on the manifold=

real complex-valued functions on the manifold

Poisson’s equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution. With the potential field known, one can then calculate the gravitational or electrostatic field. It is a generalization of laplace’s equation, which is also frequently seen in physics.

5. RIEMANN ZETA FUCTION-

zeta(s)=summation (n=1 to infinity)=1/n^s

The Riemann zeta function or Euler-riemann zeta function is a function of a complex variable s that analytically continues the sum of the dirichlet series when the real part is greater than 1. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

6. NAVIER-STOKES EQUATION-

Y(sub t)+A(y)*y(sub x)=0

In physics, this equation describes the motion of viscous fluid substances. These balance equations arise from applying newton’s 2nd law to fluid motion, together with the assumption that the stress in the fluid is the sum of the diffusing vicious term (proportional to the gradient of velocity) and the pressure term—hence describing viscous flow. Navies-stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. The can be used to model weather, ocean currents, water flow in a pipe, and air flow around an aircraft’s wing. In these equations simplified forms, they help with the design of aircraft and cars, the study of blood flow, the deign of power stations, the analysis of pollution, and many other things. Coupled with maxwell’s equations they can be used to model and study magnetohydrodynamics.

7. PYTHAGOREAN TRIPLES-

For generating pythagorean triples.the formula is-a=m*x is one leg of the right triangle. b=m+d is the other leg of the right triangle. m and x are rational numbers. d=mx/(x+2). c=m(1+x)-d. You can generate pythagorean triples with the previous formulas and arrive at pythagorean triples of a^2+b^2=c^2.

the most general solution in a^2+b^2=c^2 with (a,b)=1 and a even, is given by-

a=2*x*y, b=x^2-y^2, c=x^2+y^2

Where x and y are relatively prime of the opposite parity with x>y>0.

8. EULER’S FORMULA CONNECTING THE VERTICES, FACES, AND EDGES OF A

SOLID SHAPE-

vertices+faces=edges+2

9.EULER’S IDENTITY-

e^(i*pi)+1=0

Called one of the most remarkable formulas in mathematics for its single uses of the notions of addition, multiplication, exponentiation, equality, and the uses of the important constants 0, 1, e, and pi. in 1988, it was voted the most beautiful formula ever.

10. DE MOIVRE’S FORMULA-

(cos(x)+i*sin(x))^n=cos(n*x)+i*sin(n*x)

It is important because it connects complex numbers with trigonometry.

11. MINI-MAX METHOD

SIMPLEX METHOD

In mathematical optimization, Dantzig’s simplex algorithm (or simplex method) is a popular algorithm for linear programming.[1]

The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin.[2] Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial cones, and these become proper simplices with an additional constraint.[3][4][5][6] The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function.

Overview

A system of linear inequalities defines a polytope as a feasible region. The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.

Polyhedron of simplex algorithm in 3D

The simplex algorithm operates on linear programs in standard form:

Maximize

c^T⋅x

Subject to

Ax≤b,

xi≥0

with x=(x 1,…,x n) the variables of the problem, c=(c 1,…,c n)

are the coefficients of the objective function, A is a p×n matrix, and b=

(b 1,…,b p)

bj≥0. There is a straightforward process to convert any linear program into one in standard form so this results in no loss of generality.

In geometric terms, the feasible region defined by all values of

x such that Ax≤b,x i≥0

is a (possibly unbounded) convex polytope. There is a simple characterization of the extreme points or vertices of this polytope, namely an element x=(x 1,…,x n)

of the feasible region is an extreme point if and only if the subset of column vectors A i corresponding to the nonzero entries of x(x i≠0)

are linearly independent.[7] In this context such a point is known as a basic feasible solution (BFS).

It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region then it has this value on (at least) one of the extreme points.[8] This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.[9]

It can also be shown that if an extreme point is not a maximum point of the objective function then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point.[10] If the edge is finite then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution. The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values. This continues until the maximum value is reached or an unbounded edge is visited, concluding that the problem has no solution. The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.[10]

The solution of a linear program is accomplished in two steps. In the first step, known as Phase I, a starting extreme point is found. Depending on the nature of the program this may be trivial, but in general it can be solved by applying the simplex algorithm to a modified version of the original program. The possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called infeasible. In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point. The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded below.

12. Euler line

Euler’s line (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

In geometry, the Euler line, named after Leonhard Euler (/ˈɔɪlər/), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.[1]

The concept of a triangle’s Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.

Triangle centers on the Euler line

Individual centers

Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear.[2] This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler’s time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them.

Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector.[1] However, the incenter generally does not lie on the Euler line;[3] it is on the Euler line only for isosceles triangles,[4] for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.

The tangential triangle of a reference triangle is tangent to the latter’s circumcircle at the reference triangle’s vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle.[5]:p. 447 [6]:p.104,#211;p.242,#346 The center of similitude of the orthic and tangential triangles is also on the Euler line.[5]:p. 447[6]:p. 102

Relation to inscribed equilateral triangles

The locus of the centroids of equilateral triangles inscribed in a given triangle is formed by two lines perpendicular to the given triangle’s Euler line.[10]:Coro. 4

In special triangles

Right triangle

In a right triangle, the Euler line contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle’s orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.

Isosceles triangle

The Euler line of an isosceles triangle coincides with the axis of symmetry. In an isosceles triangle the incenter falls on the Euler line.

Automedian triangle

The Euler line of an automedian triangle (one whose medians are in the same proportions, though in the opposite order, as the sides) is perpendicular to one of the medians.[11]

Systems of triangles with concurrent Euler lines

Consider a triangle ABC with Fermat–Torricelli points F1 and F2. The Euler lines of the 10 triangles with vertices chosen from A, B, C, F1 and F2 are concurrent at the centroid of triangle ABC.[12]

The Euler lines of the four triangles formed by an orthocentric system (a set of four points such that each is the orthocenter of the triangle with vertices at the other three points) are concurrent at the nine-point center common to all of the triangles.[6]:p.111

Generalizations

Quadrilateral

In a convex quadrilateral, the quasiorthocenter H, the “area centroid” G, and the quasicircumcenter O are collinear in this order on the Euler line, and HG = 2GO.[13]

Tetrahedron

A tetrahedron is a three-dimensional object bounded by four triangular faces. Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its Monge point; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter. These points define the “Euler line” of a tetrahedron analogous to that of a triangle. The centroid is the midpoint between its Monge point and circumcenter along this line. The center of the twelve-point sphere also lies on the Euler line.

Simplicial polytope

A simplicial polytope is a polytope whose facets are all simplices. For example, every polygon is a simplicial polytope. The Euler line associated to such a polytope is the line determined by its centroid and circumcenter of mass. This definition of an Euler line generalizes the ones above.[14]

Suppose that

P is a polygon. The Euler line E is sensitive to the symmetries of P

in the following ways: 1. If P has a line of reflection symmetry L, then E

is either L or a point on L.

2. If P has a center of rotational symmetry C, then E=C.

3. If all but one of the sides of P have equal length, then E

is orthogonal to the last side.

Related constructions

A triangle’s Kiepert parabola is the unique parabola that is tangent to the sides (two of them extended) of the triangle and has the Euler line as its directix.

13. BAYE’S THEOREM-

BAYE’S THEOREM

Bayes’ theorem

In probability theory and applications, Bayes’ theorem shows the relation between a conditional probability and its reverse form. For example, the probability of a hypothesis given some observed pieces of evidence and the probability of that evidence given the hypothesis. This theorem is named after Thomas Bayes (how to say: /ˈbeɪz/ or “bays”) and often called Bayes’ law or Bayes’ rule.

Formula

The equation used is:

P(A|B)=P(B|A)P(A)P(B).

Where:

• P(A) is the prior probability or marginal probability of A. It is “prior” in the sense that it does not take into account any information about B.

• P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.

• P(B|A) is the conditional probability of B given A. It is also called the likelihood.

• P(B) is the prior or marginal probability of B, and acts as a normalizing constant.

14. HAMILTON QUANTERNION FORMULA-
Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space] or equivalently as the quotient of two vectors.

Quaternions are generally represented in the form:

a + bi + cj + dk

where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision and crystallographic texture analysis.[5] In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by

H

(Unicode U+210D, ℍ). It can also be given by the Clifford algebra classifications Cℓ0,2(R) ≅ Cℓ03,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra.

The unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).

15. EULER-LAGRANGE EQUATION-

Euler–Lagrange equation

Lx(t,q(t),q˙(t))−d/dtLv(t,q(t),q˙(t))=0.

In the calculus of variations, the Euler–Lagrange equation, Euler’s equation,[1] or Lagrange’s equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian-French mathematician Joseph-Louis Lagrange in the 1750s.

Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat’s theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.

In Lagrangian mechanics, because of Hamilton’s principle of stationary action, the evolution of a physical system is described by the solutions to the Euler–Lagrange equation for the action of the system. In classical mechanics, it is equivalent to Newton’s laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

16. 3 interesting mathematical items

1)

1×1=1

11×11=121

111×111=12321

1111×1111=1234321

11111×11111=123454321

111111×111111=12345654321

Etc

2)

1×8+1=9

12×8+2=98

123×8+3=987

1234×8+4=9876

12345×8+5=98765

Etc

3) 1=.9999999…..

0.999…

The repeating decimal continues with infinitely many nines.

In mathematics, 0.999… (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal numbers 0.9, 0.99, 0.999, etc.[1] This number can be shown to equal 1. In other words, “0.999…” and “1” represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999… is commonly defined. (In other systems, 0.999… can have the same meaning, a different definition, or be undefined.)

More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999…), a property true of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplines—math students can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

Elementary proof

The Archimedean property: any point x before the finish line lies between two of the points

Pn

(inclusive).

There is an elementary proof of the equation 0.999… = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise found in Stillwell (1994, p. 42), is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1. The meaning of the notation 0.999… is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so

0.999…=1.

Intuitive explanation

If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.

More precisely, the distance from 0.9 to 1 is

0.1=1/10; the distance from 0.99 to 1 is 0.01=1/10^2. And so on: the distance to 1 from the nth point (the one with n 9 after the decimal point) is 1/10n.

Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc, then there would be a point on the number line that lies between 1 and all these points. This point would be at a distance from 1 that is less than 1/10n for every integer n. In the standard number systems (the rational numbers and the real numbers), there is no number that is less than 1/10n for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc, and so

1=0.999….

Discussion on completeness

Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.

Formal proof

The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number 0.999…9, with n nines after the decimal point, is denoted 0.(9)n. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. As 1/10n = 0.0…01, with n digits after the decimal point, the addition rule for decimal numbers implies

0.(9)n+1/10^n=1, and 0.

(9)n<1,for every positive integer n.

One has to show that 1 is the smallest number that is no less than all 0.(9)n. For this, it suffices to prove that, if a number x is not larger than 1 and no less than all 0.(9)n, then x = 1. So let x such that 0.

(9)n≤x≤1, for every positive integer n. We have 0≤1−x≤1−0.

(9)n=1/10^n.

This implies that the difference between 1 and x is less than the inverse of any positive integer. Thus this difference must be zero, and, thus x = 1; that is 0.999…=1.

This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, which infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999… is generally not used, as there is no smallest number that is no less than all 0.(9)n. (This is implied by the fact that 0.(9)n ≤ x < 1 implies 0.(9)n–1 ≤ 2x – 1 < x < 1).

Algebraic arguments

The matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. Byers (2007, p. 39) discusses the argument that, in elementary school, one learns that

1/3=0.333…, so multiplying this identity by 3 gives 1=0.999…. He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, “It surely does not mean that the number 1 is identical to that which is meant by the notation .999….” Most undergraduate mathematics majors encountered by Byers feel that while 0.999…

is “very close” to 1 on the strength of this argument, with some even saying that it is “infinitely close”, they are not ready to say that it is equal to one. Richman (1999), by contrast, discusses how “this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking,” but also suggests that the argument may lead skeptics to question this assumption.

Byers also presents the following argument. Let x=.999

. Then: 10 x= 9.999 …multiply by 1010 x

= 9 + 0.999 …10 x= 9 + x definition of x9 x= 9subtract x x= 1 divide by 9

Students who did not accept the first argument sometimes accept the second argument, but, in Byers’ opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.

The same argument also appears in Richman (1999), who notes that skeptics may question whether x is cancellable: whether it makes sense to subtract x from both sides.

Analytic proofs

Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form

b0.b1b2b3b4b5….

The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

Infinite series and sequences

Perhaps the most common development of decimal expansions is to define them as sums of infinite series. In general:

b 0.b 1b 2b 3b 4…=b 0+b 1(1/10)+b 2(1/10)2+b 3(1/10)3+b 4(1 10)4+⋯.

For 0.999… one can apply the convergence theorem concerning geometric series:[2]

If |r|<1 then ar+ar^2+ar^3+⋯=ar/1−r.

Since 0.999… is such a sum with a = 9 and common ratio r = 1⁄10, the theorem makes short work of the question:

0.999…=9(1/10)+9(1/10)^2+9(1/10)^3+⋯=9(1/10)1−1/10=1.

This proof appears as early as 1770 in Leonhard Euler’s Elements of Algebra.[3]

Limits: The unit interval, including the base-4 fraction sequence (.3, .33, .333, …) converging to 1.

The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle’s textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999…[4] A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.[5]

A sequence (x0, x1, x2, …) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:[6]

0.999… =def lim n→∞ to 0.99…9⏟n =

def lim n→∞∑k=1 to n 9/10^k =lim n→∞(1−1/10^n)=1−lim n→∞ to 1/10^n=1−0=1.

The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 1⁄10n → 0 as n → ∞, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, “.999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1”; the 1895 Arithmetic for Schools says, “…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small”.[7] Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1.

Nested intervals and least upper bounds

Nested intervals: in base 3, 1 = 1.000… = 0.222…

The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit “2” and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b0, b1, b2, b3, …, and one writes

x=b0.b1b2b3….

In this formalism, the identities 1 = 0.999… and 1 = 1.000… reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the “=” sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.[8]

One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b0.b1b2b3… is defined to be the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.[9]

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b0.b1b2b3… to be the least upper bound of the set of approximants {b0, b0.b1, b0.b1b2, …}.[10] One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,

The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.[11]

Proofs from the construction of the real numbers

Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.[12]

Dedekind cuts

In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x.[13] In particular, the real number 1 is the set of all rational numbers that are less than 1.[14] Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form

1 −(1/10)^n.

Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number

a/b< 1 ,

which implies

a/b< 1 −(1/10)b.

Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.[16] The above approach to assigning a real number to each decimal expansion is due to an expository paper titled “Is 0.999 … = 1?” by Fred Richman in Mathematics Magazine,[17] which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.[18] Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate. He also notes that typically the definitions allow { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) “Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. […] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning.”[19] A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1⁄3 has no representation; see “Alternative number systems” below.

Cauchy sequences

Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)[20]

If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xn − yn) has the limit 0. Truncations of the decimal number b0.b1b2b3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.[21] Thus in this formalism the task is to show that the sequence of rational numbers(1−0,1−9/10,1−99/100,…)=(1,1/10,1/100,…)

has the limit 0. Considering the nth term of the sequence, for n ∈ ℕ, it must therefore be shown that

lim n→∞ to 1/10^n=0.

This limit is plain[22] if one understands the definition of limit. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.[16] The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton’s 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.[23]

Infinite decimal representation

Commonly in secondary schools’ mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 0.999… as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version.[24] With this construction of the reals, all proofs of the statement “1 = 0.999…” can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.

Generalizations

The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.[25]

Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111… equals 1, and in base 3 (the ternary numeral system) 0.222… equals 1. In general, any terminating base b expression has a counterpart with repeated trailing digits equal to b − 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.[26]

Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000…. This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik–Loreti constant q = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the Thue–Morse sequence, which does not repeat.[27]

A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example:[28]

• In the balanced ternary system, 1⁄2 = 0.111… = 1.111….

• In the reverse factorial number system (using bases 2!,3!,4!,… for positions after the decimal point), 1 = 1.000… = 0.1234….

Impossibility of unique representation

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered lexicographically. Indeed, the following two properties account for the difficulty:

• If an interval of the real numbers is partitioned into two non-empty parts L, R, such that every element of L is (strictly) less than every element of R, then either L contains a largest element or R contains a smallest element, but not both.

• The collection of infinite strings of symbols taken from any finite “alphabet”, lexicographically ordered, can be partitioned into two non-empty parts L, R, such that every element of L is less than every element of R, while L contains a largest element and R contains a smallest element. Indeed, it suffices to take two finite prefixes (initial substrings) p1, p2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for L the set of all strings in the collection whose corresponding prefix is at most p1, and for R the remainder, the strings in the collection whose corresponding prefix is at least p2. Then L has a largest element, starting with p1 and choosing the largest available symbol in all following positions, while R has a smallest element obtained by following p2 by the smallest symbol in all positions.

The first point follows from basic properties of the real numbers: L has a supremum and R has an infimum, which are easily seen to be equal; being a real number it either lies in R or in L, but not both since L and R are supposed to be disjoint. The second point generalizes the 0.999…/1.000… pair obtained for p1 = “0”, p2 = “1”. In fact one need not use the same alphabet for all positions (so that for instance mixed radix systems can be included) or consider the full collection of possible strings; the only important points are that at each position a finite set of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow “9” in each position while forbidding an infinite succession of “9”s). Under these assumptions, the above argument shows that an order preserving map from the collection of strings to an interval of the real numbers cannot be a bijection: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof “an instructive exercise in elementary point-set topology”; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.[29]

Applications

One application of 0.999… as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:

• 1⁄7 = 0.142857142857… and 142 + 857 = 999.

• 1⁄73 = 0.0136986301369863… and 0136 + 9863 = 9999.

• 1⁄77 = 0.012987012987… and 012 + 987 = 999.

E. Midy proved a general result about such fractions, now called Midy’s theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If it can be proved that a decimal of the form 0.b1b2b3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.[30] Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.[31]

Positions of 1⁄4, 2⁄3, and 1 in the Cantor set

Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:

• A point in the unit interval lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.[32]

Repeating nines also turn up in yet another of Georg Cantor’s works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.[33] A variant that may be closer to Cantor’s original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.[34]

Skepticism in education

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

• Students are often “mentally committed to the notion that a number can be represented in one and only one way by a decimal.” Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[35]

• Some students interpret “0.999…” (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 “at infinity”.[36]

• Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read “0.999…” as meaning the sequence rather than its limit.[37]

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999…

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that “students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because ‘you haven’t specified how many places there are’ or ‘it is the nearest possible decimal below 1′”.[38]

The elementary argument of multiplying 0.333… = 1⁄3 by 3 can convince reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.[39] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.[40] Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the fractional proof, insist that “logic” supersedes the mathematical calculations.

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who “challenged almost everything I said in class but never questioned his calculator,” and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a “wildly imagined infinite growing process.”[41]

As part of Ed Dubinsky’s APOS theory of mathematical learning, he and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have “not yet constructed a complete process conception of the infinite decimal”. Other students who have a complete process conception of 0.999… may not yet be able to “encapsulate” that process into an “object conception”, like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.[42]

Cultural phenomenon

With the rise of the Internet, debates about 0.999… have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999… is described as a “popular sport”, and it is one of the questions answered in its FAQ.[43] The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999… via 1⁄3 and limits, saying of misconceptions,

The lower primate in us still resists, saying: .999~ doesn’t really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.[44]

A Slate article reports that the concept of 0.999… is “hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums”.[45] In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment’s Battle.net forums that the company issued a “press release” on April Fools’ Day 2004 that it is 1:

We are very excited to close the book on this subject once and for all. We’ve witnessed the heartache and concern over whether .999~ does or does not equal 1, and we’re proud that the following proof finally and conclusively addresses the issue for our customers.[46]

Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical jokes, such as:[47]

Q: How many mathematicians does it take to screw in a lightbulb?

A: 0.999999….

In alternative number systems

Although the real numbers form an extremely useful number system, the decision to interpret the notation “0.999…” as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.[48]

One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of —rather than independent alternatives to— the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as “0.999…” is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

Infinitesimals

Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[49] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[50] Lightstone shows how to associate to each number a sequence of digits, 0.d 1d 2d 3…;…

d ∞−1 d ∞d ∞+1…,

indexed by the hypernatural numbers. While he does not directly discuss 0.999…, he shows the real number 1⁄3 is represented by 0.333…;…333… which is a consequence of the transfer principle. As a consequence the number 0.999…;…999… = 1. With this type of decimal representation, not every expansion represents a number. In particular “0.333…;…000…” and “0.999…;…000…” do not correspond to any number.

The standard definition of the number 0.999… is the limit of the sequence 0.9, 0.99, 0.999, … A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, …)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number uH=0.999…;…999000…, with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality uH < 1. Accordingly, an alternative interpretation for “zero followed by infinitely many 9s” could be 0.

999…⏟H=1−110^H.

All such interpretations of “0.999…” are infinitely close to 1. Ian Stewart characterizes this interpretation as an “entirely reasonable” way to rigorously justify the intuition that “there’s a little bit missing” from 1 in 0.999….[52] Along with Katz & Katz, Robert Ely also questions the assumption that students’ ideas about 0.999… < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[53][54] Jose Benardete in his book Infinity: An essay in metaphysics argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:

The intelligibility of the continuum has been found—many times over—to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999… does not equal 1 but falls infinitesimally short of it. I think that .9999… should indeed be admitted as a number … though not as a real number.[55]

Hackenbush

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1⁄3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the surreal number 1⁄ω, where ω is the first infinite ordinal; the relevant game is LRRRR… or 0.000…2.[56]

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111…2 = 0.11000…2, which are both equal to 3⁄4, but the first representation corresponds to the binary tree path LRLRLLL… while the second corresponds to the different path LRLLRRR….

Revisiting subtraction

Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999… + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.[57]

In the process of defining multiplication, Richman also defines another system he calls “cut D”, which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction d he allows both the cut (−∞, d ) and the “principal cut” (−∞, d ]. The result is that the real numbers are “living uneasily together with” the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut D, but there is “a sort of negative infinitesimal,” 0−, which has no decimal expansion. He concludes that 0.999… = 1 + 0−, while the equation “0.999… + x = 1” has no solution.[58]

p-adic numbers

When asked about 0.999…, novices often believe there should be a “final 9,” believing 1 − 0.999… to be a positive number which they write as “0.000…1”. Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the final 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no “final 9” in 0.999….[59] However, there is a system that contains an infinite string of 9s including a last 9.

The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.

The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to pn, than it is to 1. The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.[60] Another derivation uses a geometric series. The infinite series implied by “…999” does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

…999=9+9(10)+9(10)^2+9(10)^3+⋯=9/1−10=−1.

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher’s limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = …999 then 10x = …990, so 10x = x − 9, hence x = −1 again.[60]

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by “blind faith and unabashed juggling of symbols”[62] one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of “double-decimals” with eventually repeating left ends to represent a familiar system: the real numbers.[63]

Ultrafinitism

The philosophy of ultrafinitism rejects as meaningless concepts dealing with infinite sets, such as idea that the notation

0.999…

might stand for a decimal number with an infinite sequence of nines, as well as the summation of infinitely many numbers

9/10+9/100+⋯

corresponding to the positional values of the decimal digits in that infinite string. In this approach to mathematics, only some particular (fixed) number of finite decimal digits is meaningful. Instead of “equality”, one has “approximate equality”, which is equality up to the number of decimal digits that one is permitted to compute.[64] Although Katz and Katz argue that ultrafinitism may capture the student intuition that 0.999… ought to be less than 1, the ideas of ultrafinitism do not enjoy widespread acceptance in the mathematical community, and the philosophy lacks a generally agreed-upon formal mathematical foundation.[65]

Related questions

• Zeno’s paradoxes, particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.[66]

• Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has a “point at infinity”. Here, it makes sense to define 1⁄0 to be infinity;[67] and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.[68]

• Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where “0” denotes the additive identity and is neither positive nor negative, the usual interpretation of “−0” is that it should denote the additive inverse of 0, which forces −0 = 0.[69] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the sign and magnitude or ones’ complement formats, or floating point numbers as specified by the IEEE floating-point standard).[70][71]

GREAT MATHEMATICAL THEOREMS

1) Hippocrate’s Quadrature of a lune (ca. 440 BC.)

(Constructing a square with the same area as a circle)

An angle inscribes in a semicircle is a right angle.

2) Euclid’s proof of the pythagorean theorem (ca. 300 BC)

3) Euclid’s and the infinitude of primes (ca. 300 BC)

4) Archimedes determination of circular area (c. 225 BC)

The area of a regular polygon is 1/2 h*q

h=distance to a side of the polygon from the center.

q=perimeter

5) Heron’s formula for triangular area (ca. 75)

(area of any triangle)

area=sqrt(s*(s-side 1)*(s-side 2)*(s-side 3))

(s=1/2 of the perimeter of the triangle

6) Cardano and the solution of the cubic equation (1545)

An example of a cubic equation- x^3+6x=20

7) Newton’s binomial theorem and approximation of pi (late 1660s)

8) The bernoullis and the harmonic series (1689)

The harmonic series 1+1/2+1/3+1/4+… is infinite

9) The extraordinary sums of leonhard euler. (1734)

(Calculating the sums of infinite seres)

1. Sinex-x^3/3!+x^5/5!-7^7/7!+9^9/9!-…

2. (pi^2)/6=1+1/4+1/9+1/16+1/25+…1/k^2

(K-1,2,3,4,5,6…)

10) De moivre’s formula-

For any complex number (and, in particular, for any real number) x and integer n it

holds that

(cos(x)+i*sin(x))^n=cos(n*x)+i*sin(n*x)

This formula is important because it connects complex numbers with trigonometry.

11) Euler’s number theory (1736)

1. If p is a prime and a is any whole number, then (a+1)^p-(a^p+1) is evenly

divisible by p.

2. If p is a prime and if a^p -a is evenly divisible by p, then so is (a+1)^p-(a^p+1)

3. if. P is a prime and a is any whole number, then p divides evenly into a^p-a.

4. Fermat’s little theorem- if p is a prime and a is a whole number which does not

have p as a factor, then p divides evenly into a^(p-1)-1.

5. Suppose that a is an even number and p is a prime that is not a factor of a but

does divide evenly into a+1. Then for some whole number k, p=2k+1.

6. Suppose a is an even number and p is a prime that is not a factor of a but such

that p does divide evenly into a^2+1. Then for some whole number, k, p=4k+1.

7. Suppose a is an even number and p is a prime that is not a factor of a such that

p does not divide evenly into a^4+1. Then for some whole number k, p=8k+1.

8. 2^32 is not prime.

12) The non-denumerability of the continuum. (1874)

The continuum is an infinite set which cannot be put into a 1-to-1 correspondence

with the set of natural numbers. (There real numbers have more members (greater

magnitude) that the counting numbers.)

13). fermat’s little theorem-states that if p is a prime number, then for any integer a, the

number a^p-a is the integer multiple of p.

14) Green’s theorem-gives the relationship between a line integral around a simple

closed curve C and a double integral over a plane region D bounded by C. It is a 2-

dimensional special case of a more general kelvin-stokes theorem, and it deals with

electricity and magnetism.

14) Prime number theorem-in number theory, it describes the asymptotic distribution of

the prime numbers among the positive numbers. The first such distribution is pi(n)

approximately equal to (n)*log(n) is the prime counting function and log(n) is the

natural logarithm of n.

15) Wilson’s theorem- in number theory, this states that a natural number n>1 is a prime

number if and only if- (n-1)!= -1 (mod n)

16) Apery’s theorem- is a result in number theory that states the apery constant f(3) is

irrational,

17. f(3)=summation (n=1 to infinity) 1/n^3=1.2010569…

At 3, the zeta function is irrational.

18) Divergence theorem-

Triple indefinite integral (volume) (del (dot product) F) dV= surface double integral F

(dot product) n dS.

Investor calculus, known also as gauss’s theorem and ostrogradsky’s theorem, is a

result that relates the flow (that is, flux) of a vector field through the surface to the

behavior of the vector field inside the surface.

19) Jordan curve theorem- any continuously closed curve in the plane, separates the

plane into 2 disjoint regions, the inside and outside.

20) Marriage theorem- gives the necessary and sufficient conditions for the existence of

a system of distinct representatives for a set system, or for a perfect matching in a

bipartite graph.

21) Ramsey theorem- states that one will find monochromatic cliques in any edge

labeling (with colors) of a sufficiently large complete graph.

22) Stoke’s theorem- in vector calculus, and more generally differential geometry, is a

statement about the integration of differential forms on manifolds, which both

simplifies generalize several theorems from vector calculus

23) Chebyshev’s theorem-there is always a prime number between n and 2n.

24) De moivre-laplace theorem- in probability theory, is a special case of the central limit

theorem (the normal distribution may be used as an approximation to the binomial

distribution under certain conditions). The theorem shows that the probability mass

function of he random number of successes observed in a series of n independent

Bernoulli trials, each having probability p of success (a binomial distribution with n

trials), converges to the probability density function of the normal distribution with

np and standard deviation sqrt((n*p(1-p)), as n grows large, assuming p is not 0

or 1.

25) Dirichelt’s theorem-given an arithmetic progression of terms, for, 2, ….., the series

contains an infinite number of primes if and and are relatively prime. For any such

arithmetic progression, that different arithmetic progressions with the same modulus

have approximately the same proportions of primes.

ODDS AND ENDS IN MATHEMATICS

ARITHMETIC

1. KINDS OF NUMBERS:

A. Natural numbers— 1,2,3…

B. prime numbers— 2,3,5,7,11…

C. Integers— 1,4,8,-1,-3…

D. Real numbers—

1. Rational numbers— fractions-a whole number in numerator and a whole

number in denominator

Examples- 3/4, 344/12

2. Irrational numbers— a number that cannot be expressed as a ratio between 2

numbers. Cannot be written as a simple fraction. Number of decimals go on

forever without repeating.

Example- pi-3.14…..

3. Irrational algebraic numbers— any complex number that is a root of a non-

zero polynomial in one variable with rational coefficients, or equivalently by

clearing denominators-with integer coefficients. All integers and rational

numbers are algebraic, as are all roots of integers.

examples- sqrt(2), cubicroot(3)

4. Transcendental numbers— a real or complex number that is not algebraic-

that is, it is not the root of a nonzero polynomial equation with integer (or

equivalently rational) coefficients

Examples- pi=3.14159265…, and ‘e;’

5. Complex numbers—real numbers plus imaginary numbers

(Imaginary numbers are the sort of square root of a negative number)

Examples- 3+sqrt(-1i), 2-sqrt(-4i), 2-i

6. algebraic numbers— the above numbers except the transcendentals

E. Transfinite numbers—

w (omega) is defined as the lowest transfinite ordinal number and is the order

type of the natural numbers under their usual linear ordering.

Alph-null is the first transfinite cardinal number and is the cardinality of the

infinite set of the natural numbers.

F. Quanternions—complex number extended from 2 dimensions to 4.

4+2i-3j+7k

G. Octonions—numbers that have 8 dimensions

2. FIBONACCI SEQUENCE-a series of numbers in which each number, the Fibonacci number, is the sum of the 2 preceding numbers (1, 1, 2, 3, 5, 8, 13, etc.). the Fibonacci numbers are used to make a fibonacci spiral (golden spiral). The numbers appear in the branches of trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a cone’s bracts.

3. WHAT ARE PRIME NUMBERS? A number that has more than 2 factors is a composite number. (example-The number 6 is composite because the numbers 2 and 3 (the factors) multiplied together equals 6.). prime numbers have no factors when multiplied together will equal the prime number. The first several prime number are: 2,3,5,7,11,13,17,19,23… prime numbers are basic building blocks numbers.

4. ERATOSTHENES SIEVE-in mathematics, this is a simple way and most efficient ways of finding prime numbers up to a given limit. Let us say we want to find prime numbers up to 50. The way to do this is to mark off all the multiples of the primes. Up to the number 20, the primes are: 2, 3, 5, 7, 1, 13, 17, 19, and 23. Let us take one of the prime numbers and mark off all multiples of it up to 50. We will use the number 5. All multiple of 5 up to 50 are: 10, 15, 20, 25, 30, 35, 40, 45, and 50. So we mark these numbers off the list of numbers 1 to 50 as not being prime. To finish the sieve, we mark off each of the other multiples of the primes up to the number 50. When we are finished marking off all the multiples of the primes for the primes 1 to 13, we are finished and only the prime numbers remain. We can find all the primes up to any number.

5. PI- this number is a mathematical constant, the ratio of a circle’s circumference to its diameter, and ic approximated to 3.14159. it is an irrational number, so it cannot be expressed as a fraction, and the decimals lower ends whine never repeating a pattern of numbers. Pi is also a transcendental number, so it is not the root of any non-zero polynomial having rational coefficients. Because it is a transcendental number makes the ancient challenge of squaring a circle impossible with a compass and straightedge.this number is found in many mathematical formulas in trigonometry and geometry, especially those involving circle, ellipses, and spheres. It also appears in number theory and statistics which do not deal with geometry, and the number appears in almost all areas of physics. An example of pi appearing in the physical sciences is one formula in the astronomy of stars—

L=4 x PI x R^2 x (stefan-boltzmann constant) x T^4,

Which give the luminosity of a star (how many times brighter than the sun the star is) when we know the star’s radius in meters (R) and its temperature in kelvin degrees (T).

so, the Pi constant is is ubiquitous in the sciences.

6. EXPONENT ‘e’- a mathematical constant, equal to approximately 2.71828, is unique in that its natural logarithm is equal to 1. This number has great importance in mathematics. The number e is both irrational and transcendental. numbers. Along with 0, 1, pi and I,. this number play an important recurring role across mathematics, especially in statistics. It is the basis for the rate of growth shared by all continually growing processed.it shows up whenever systems grow exponentially and continually: radioactive decay, population, growth or decline, interest calculations, and more.it shows up especially in number theory in mathematics. If we keep adding without stopping he numbers 1 over each factorial as the factorials grow, the resulting number will be e, 2.71828… this number turns addition into multiplication. the number e is about continual growth over time and it is a fundamental constant that is found and used in very many areas.

7. NATURAL LOGARITHM-the natural logarithm has the number e as its base.It is the inverse of the number e, so it reverses whatever the number e does. this number is without question the most important number infall of mathematics. the natural log turns multiplication into addition. It is about the time it would take to grow, how fast growth occurs. For example, assuming 5% growth, how long would it take $1,000 to grow to $2,000.the natural log is also used in Isaac newton’s law of cooling.

8. COMBINATIONS-refers to n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the term k-selection, k-multiset, or k-combination with repetition are often used.

9. PERMUTATIONS-a way, especially one of several possible variations, in which a set or number of things can be ordered or arranged.

10. FACTORIALS-they are just products, indicated by an exclamation point. For example, 5 factorial is written 4! And is equal to 1 x 2 x 3 x 4=24.

11. LOGARITHMS-a quantity representing the power in which a fixed number (the base) must be raised to produce a given number.

12. MATHEMATICAL INDUCTION-a means of proving a theorem by showing that is it is true for any particular case (base case), and to prove the next case in a series (inductive step), then it must be true for the cases after these.

13. MAGIC SQUARES-an arrangement of the numbers from 1 to n^2 of an n x n matrix, with each number occurring exactly once, and such that the sum of the entries of any row, column, or diagonal is the same. This sum is n(n^2+1)/2.

14. GOLDEN RECTANGLE/GOLDEN RATIO-the golden ratio has been claimed to have held special fascination for at least 2,400 years. Ii is the number 1.61803398875 and it is an irrational number. . the inverse of this number gives the decimal part of that number which is good approximation of decimal part of the golden ratio number. a golden rectangle can be constructed by having the two opposite side’s lengths of rectangle have the ratio 1.618 to the other two side’s lengths. Luca pachouli defined the ratio as the divine proportion, and Johannes kepler said that it was a precious jewel. The ancient building the parthenon’s facade is circumscribed by the golden ratio, as does the great mosque of kairouan, which applies the ratio throughout the mosque’s design. Salvador Dali explicitly use the golden ratio in his masterpiece painting, the sacrament of the last supper. Erik Satie used the ratio in several of his musical pieces. The golden ratio operates as a universal law as it is found in the arrangement of leaves and branches along the stems of plants and the veins of leaves, the skeletons of animals and the branching of their veins and nerves, the proportion of chemical compounds, the geometry of crystals, and in artistic endeavors. it also plays a role geometric figure such as the pentagram, the pentagon, the rhombic tricontahedron, and the golden rhombus. The Fibonacci sequence and the golden ratio are intimately interconnected. The golden ratio is the limit of the ratios of successive terms of the Fibonacci numbers where the fibonacci number plus one is used for the numerator and the only the Fibonacci number is used in the denominator and the limit converges to the golden ratio number. The golden ratio occurs in many, many more numerous ways in nature, mathematics, and the arts.

15. SYNTHETIC DIVISION-a shortcut method of polynomial division in the special case of dividing by a linear factor, and it only works in this case.

16. ARITHMETIC SEQUENCE/PROGRESSION/SERIES-sequence of numbers such that the difference between the consecutive terms is constant. For example the number 2,5, 8,11… have a constant of difference of 3.

17. GEOMETRIC PROGRESSION/SEQUENCE/SERIES-sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called a common ratio. For example, the sequence 2,6,18,54,… is a geometric progression with a common ratio of 3.

18. INFINITE SERIES-a series which continues forever to arrive at a number answer. The series can either converge where one gets a numerical answer, or the series can diverge where the answer is either positive infinity or negative infinity.

19. FUNDAMENTAL THEOREM OF ARITHMETIC- in number theory, also called the unique factorization theorem, or unique prime factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors.

20. MODULAR ARITHMETIC- system of arithmetic for integers, where numbers wrap around upon reaching a certain value, the modulus. The modulo operation finds the remainder after division of one number by another, the modulus. Given 2 positive numbers, a (the dividend) and n (the divisor, the modulo n (abbreviated as a mod n) is the remainder of the euclidean division of a by n.

21. REMAINDER THEOREM- in algebra, is an application of euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial is equal to. In particular, is a divisor of if and only if.

22. WHAT ARE GOOGOL AND GOOGOLPEX NUMBERS? A google os a very, very big number- 10^100, or the number 1 followed by 100 zeros. This is much greater than the number of atoms that make up the universe. A googolplex Is 10^google, or 10^10^100. This number is 1 followed by a gooleplex number of zeros.

23. CONTINUED FRACTIONS- an expression obtained throughout the iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

24. TETRATION- hyper-4 is the next hyperoperation after exponentiation, and is defined as the integrated exponentiation.. It is used for the notation of very large numbers

25. KNUTH’S UP-ARROW NOTATION- method of notation of very large numbers. It is based on the idea that multiplication can be viewed as integrated addition and exponentiation as iterated multiplication. Continuing in this manner leads to titration (iterated exponentiation) and to the remainder of the hyper operation sequence, knuth’s arrows. For example, 4 (1 arrow) 4=4^4=256. 4(2 arrows) 4=4^4^4=4.295×10^9. 4 (3 arrows) 4=4^4^4^4=3.4×10^38. This pattern can be continued to calculate larger and larger numbers.

26. HYPEROPERATION- an infinite sequence of arithmetic operations.it starts with with the operation of successor hyper0 (n=0), – 1+b=1+1+1…1=b copes of 1, addition hyper1 (n=1), a+b=a+1+1+1…+1=b copies of a, then multiplication hper2 (n=2), a x b=a+a+a…+a=b copes of a, exponentiation hyper3 (n=3) a x a x a…x a=b copies of a, tetration (hyper4 n=4) a (arrow) a (arrow) a (arrow)… a (arrow) a=b copies of a, pentagon hyer5 (n=5) a (2 arrows) a (2 arrows) a (2 arrows)… a (2 arrows)=b copies of a, and so on. This can produce numbers much larger than those which use scientific notation, such as skewers number, and the googleplexplex, but there are some numbers which even they cannot easily show such as graham’s number and trees(3).

27. GRAHAM’S NUMBER- graham’s number is larger than a googolplex. It is an enormous number that arises as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. Graham’s number is the largest positive integer ever published in a mathematical proof. It is much larger than many other large numbers such as skewe’s number and moser’s number, both of which are much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of graham’s number, assuming that each digit occupies one planck volume, possibly the smallest measurable space. It cannot even be expressed by power towers.

28. SKEWE’S NUMBER- the number above which must fall (assuming the Riemann hypothesis is true), where is the prime counting function and the logarithmic integral. The upper bounds for the smallest natural number x for which pi(x)>li(x), where pi is the prime counting function and li is the logarithmic integral function.these bounds have since been improved by others: there is a crossing near e^727.95133. it is not known whether it is the smallest.

29. LIOUVILLE NUMBER- transcendental number irrationality measure 2 or greater. Liouville numbers have infinite irrationality measure.

30. MERSENNE PRIME- a prime number that is one less than a power of two.

31. ANTILOGARITHM- the number for which a logarithm stands, for example, if the logarithm of x equals y, then x is the antilogarithm of y.

32. INTERPOLATION- an estimation of a value based on extending a known sequence of values or facts beyond the area that is certainly known. Polynomial interpolation is a method of estimating values between known data points.120. GODEL NUMBER-a function which assigns to each symbol and formula of some formal language a unique natural number called a model number. It was used by model for his incompleteness theorem proof.

33. TRANSFINITE INDUCTION- an extension of mathematical induction to well-ordered sets, for example of sets of ordinal numbers or cardinal numbers.

34. MODULAR FUNCTION-a function, like a modular form, is invariant with respect to the modular group, but without the condition that f(z) be holomorphic at infinity. Instead, modular functions are meromorphic with infinity being the only pole.

35. STERLING’S FORMULA- an approximation of factorials.

36. GODEL NUMBER-a function which assigns to each symbol and formula of some formal language a unique natural number called a model number. It was used by model for his incompleteness theorem proof.

37. CARMICHAEL NUMBER- a composite number which satisfies the modular arithmetic congruence relation: for all integers which are relatively prime to.

38. BRUN’S CONSTANT- in number theory, this is the sum of the reciprocals of the twin primes that converges to a finite value known as brun’s constant, approximately 1.90216053104.

39. PERFECT NUMBERS-a positive integer that is equal to the sum of its proper divisors. 6 is a perfect number because it is the sum of its proper divisors 1, 2, and 3.

40. SUNDMAN’S SERIES- analytic methods to prove the existence of a convergent infinite series solution to the 3-body problem.

41. METHOD OF STEEPEST DESCENT (or stationary phase method, saddle point method, or the gradient descent method)- an extension of laplace’s method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent. a first-order iterative optimization algorithm for finding the minimum of a function.

42. SYNTHETIC DIVISION-a shorthand method of polynomial division in the special case of dividing by a linear factor.

43. ANDICA’S CONJECTURE- deals with the gaps between prime numbers. the inequality sqrt(p(n)+a)-sqrt(p(n))<1 holds for all n, where p(n) is the nth prime number.

44. 2nd HARDY-LITTLEWOOD CONJECTURE- in number theory, it means that the number of primes from x+1 to x+y is always less than or equal to the number of primes from 1 to y.

45. 1st HARDY-LITTLEWOOD CONJECTURE-states that the asymptotic number of prime constellations can be computed explicitly.

46. SCHANUEL’S CONJECTURE- in mathematics, specifically transcendental number theory, it concerns the transcendence degree of certain extensions of the rational numbers. The conjecture, if proven, would generalize most known results in transcendental number theory. Euler’s identity states that e^(pi x I)+1=0. If the conjecture is true than this is, in some precise sense involving exponential rings, the only relation between e, pi, and I over the complex numbers.

47. A theorem of number theory, which states that, if one knows that, if one knows the remainder of the division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

48. ERDOS-STRAUS CONJECTURE- in number theory, it states that for all integers n>=2, the rational numbers 4/n can be expressed as the sum of 3 unit fractions.

49. ABC CONJECTURE- in number theory, it states that in terms of 3 positive integers, a, b, and c that are relatively prime and satisfy a+b=c.

50. POLIGNAC’S CONJECTURE-for any positive prime even number n, there are infinitely many prime gaps of size n. (There are infinitely many cases of 2 consecutive prime numbers with difference n.

51. CHEN’S THEOREM- in number theory, for every sufficiently large even number can be written as the sum of either 2 primes, or a prime and a semiprime (the product of 2 primes).

52. DINNER PARY PROBLEM- states how many people must you know at a dinner to ensure that there are a subset of 3 people who all either mutual acquaintances, or mutual strangers? For example, to guarantee at least 3 people will be mutual acquaintances or mutual strangers, you must invite a minimum of 6 people to the party. regardless of how many people you invite to the part, there will always be at least 3 people who are mutual aquaintances or aquaintances. This problem is interesting because it shows that total disorder in a graph is impossible..there will always be an island of order in random, infinite chaos.

53. HAPPY ENDING PROBLEM- a problem of determining for n>=3 the smallest number of points g(n) in general position in the plane (no 3 of which are collinear), such that every possible arrangement of g(n) points will always contain at least one set of n points that are vertices of a convex polygon of n sides.

54. SESSA’S CHESSBOARD-if a chessboard were to have wheat placed upon each square such that one grain were placed on the 1st square, 2 on the 2nd square, 4 on the 3rd, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the Finish? @^64, or 18,446,744,073,708,551,815 grains in total from all the grains that were doubled and put on each of the 64 squares of the chessboard.

uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.

55. PRIME COUNTING FUNCTION- a function counting there number of prime numbers less than or equal to some real number x. It is denoted by the number pi(x) and is unrelated to the number pi.

Its formula is pi(x)=x/lnx, where pi is the number of primes for the numbers up to the number x.

56. NEWTON’S METHOD-in numerical analysis, is a method for finding successively better approximations to the roots, or zeros, of a real-related function. It is an example of a root finding algorithm.

57. CATALAN CONJECTURE- states that 2^3 and 3^2 are 2 powers of natural numbers, whose values 8 and 9 respectively are consecutive. The theorem states that this is the only case of 2 consecutive powers is the only solution to x^a-y^b=1.

58. SIERPINSKI NUMBER- an odd natural number k such that it is composite, for all natural numbers n.

59. PERFECT NUMBERS- a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6 because its proper divisors are 1,2, and 3. Other perfect numbers are 28, 496, and 8,128.

60. TOWER OF HAOI-a math game which consists of 3 rods and a number of disks of different sizes which can side onto any rod. The game stars with the disks on one rod with the largest disks at the bottom and smaller disks moving to the top of the rod. The objective of the game:

1. Only one disk can be moved at a time.

2. Each move consists of taking the upper disk from one of the stacks and placing it onto another stack.

3. No bigger disk may be placed onto a smaller disk.

The object of the game is to transfer all the disks onto another rod obeying steps 1-3.

It takes 7 moves when there are 3 disks. The minimum number of moves requires 2^n-1 for n number of disks

61. COLLATZ CONJECTURE- take any positive integer n. If n is even, divide it by 2 and get n/2. If n is odd, divide it by 3 and add 1 and get n/3+1. Repeat the process indefinitely. No matter what number you start with, you will always eventually reach 1.

62. ULAM SPIRAL- a graphical representation of the prime numbers.

63. SURREAL NUMBERS-the most natural collection of the numbers which include both real numbers and the infinite ordinal numbers. Every real number is surrounded by surreals, which are closer to it than any real numbers.

64. BENFORD’S LAW- in many naturally occurring collections of numbers, the leading significant digit is likely to be small.

65. BERNOULLI’S FORMULA-expresses the sum of the 1st p-th powers of the 1st n positive integers.

66. MOSER’S NUMBER-In mathematics, Steinhaus–Moser notation is a notation for expressing certain extremely large numbers. It is an extension of Steinhaus’s polygon notation.

67. WARING’S PROBLEM-In number theory, Waring’s problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers to the power of k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring’s problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909

68 LAGRANGE’S 4 SQUARE PROBLEM-Lagrange’s four-square theorem, also known as Bachet’s conjecture, states that every natural number can be represented as the sum of four integer squares.

ALGEBRA

1. POINCARE GROUP- an example of the representative theory of a Lie group that is neither a compact group nor a semi simple group. It is fundamental to theoretical physics. In a physical theory having Minkowski space as an underlying spacetime, the space of physical states is typically a representation of the poincare group.

2. HORNER’S METHOD- rule in which is either of 2 things: an algorithm for calculating polynomials, which consists of transforming the monomial form into a computationally efficient form, or a method for approximating the roots of a polynomial.

3. BINOMIAL THEOREM-a formula for finding a quick way of multiplying out a binomial with a large power without multiplying it at length.

4. PASCAL’S TRIANGLE-a triangular array of binomial coefficients. The number at the top point of the array is a 1. Each entry of each subsequent row is constructed by adding the number above and the numbers above to the right and above to the left. The triangle is used to determine the coefficients which arise in binomial expansions.

5. GROUPS, RINGS, FIELDS-

GROUP-an algebraic structure consisting of a set of elements equipped with an operation that combines any 2 elements to form a 3rd element. The operation satisfies 4 conditions called the group axioms, namely closure, associativity, identity, and invertibility.

RING-one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with 2 binary operations that generalize the arithmetic operations of addition and multiplication. A ring whose nonzero elements form a commutative multiplication group is called a field. The simplest rings are the integers, polynomials and in 1 or 2 variables, the square real matrices.

FIELDS-is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus a fundamental algebraic structure, which is widely used in algebra, number theory and many others areas of mathematics.

6. PELL’S EQUATION- any Diophantine equation of the form X^2-nY^2=1, where n is any positive nonsquare integer and solutions are sought for X AND Y.

7. DIOPHANTINE EQUATIONS—an equation in which only integer solutions are allowed. The simplest Diophantine equation is in the form aX+bY=C

8. FACTORING-finding what to multiply together to get an expression. Splitting an expression into multiplication of simpler expressions.

9. FUNDAMENTAL THEOREM OF ALGEBRA- every non-constant single variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.

10. LAW OF QUADRATIC RECIPROCITY- a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers.

11. FUNCTION- a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2.

12. PARAMETRIC EQUATIONS-defines a group of quantities as functions of one or more independent variables called parameters. a form of parameter representation of the unit circle is where t is the parameter.

13. LIE GROUPS- the intersection of 2 fundamental fields of mathematics: algebra and geometry. A Lie group is first of all a group. Secondly it is a smooth manifold which is a specific kind of geometric object. The circle and sphere are examples of smooth manifolds.

14. REMAINDER THEOREM- in algebra, is an application of euclidean division of polynomials. It states that the remainder of the division of a polynomial by a linear polynomial is equal to. In particular, is a divisor of if and only if.

15. METHOD OF INFINITE DESCENT- a proof by contradiction that relies on the least integer principle. One typical application is to show that a given equation has no solutions. It can be used for Diophantine equations.

16. CUBIC EQUATIONS-every cubic equation 1), a x x^3+b x x^2+c x x+d=0, with real coefficients and a cannot equal 0, has 3 solutions (some of which may be equal to each other if they are real, and 2 which may be complex non-rap numbers and at least 1 real solution, this last assertion being a consequence of the intermediate value theorem.

17. QUARTIC EQUATIONS-if g=0, then, you can factor the quartic into y times a cubic. The roots of the original equation are then x=-a/4 and the roots of that cubic with a/4 subtracted from each. If f=0, then the quartic in y is actually a quadratic equation in the variable y^2.

18. QUINTIC EQUATION-unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions.

19. GALOIS THEORY-in abstract mathematics specifically, it provides a connection between field theory and group theory. Certain problems in field theory can be reduced to group theory, which is simpler and better understood.

20. SYSTEM OF EQUATIONS- a set of equations or collection of equations that you deal with all together at once. Linear equations are simpler than non-linear ones. The simplest linear equations have 2 equations and 2 variables.114. ECCENTRICITY-denoted by e, a parameter associated with every conic section. A measure of how much the conic section deviates from being circular.

Eccentricity: circle=0, ellipse >0 and <1, parabola=1, hyperbola>1.

21. SIMPLEX METHOD- standard method of maximizing a linear function of several variables under several constraints on other linear functions.

22. SYLVESTER LAW OF INERTIA-A theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remains invariant under a change of basis.

23. TRIPLE PRODUCT-in vector algebra, is a product of 3-dimensional vectors. It is used for 2 different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.

24. BINOMIAL THEOREM-a formula for finding a quick way of multiplying out a binomial with a large power without multiplying it at length.

25. PASCAL’S TRIANGLE-a triangular array of binomial coefficients. The number at the top point of the array is a 1. Each entry of each subsequent row is constructed by adding the number above and the numbers above to the right and above to the left. The triangle is used to determine the coefficients which arise in binomial expansions.

26. MATRIX ALGEBRA-a matrix with the same number of rows and columns is a square matrix. a matrix with an infinite number of rows and columns is an infinite matrix. an empty matrix has no rows and columns.

27. module-In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module.

Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication.

Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.

28. Lie algebra-In mathematics, a Lie algebra (pronounced /liː/ “Lee”) is a vector spaceg

together with a non-associative, alternating bilinear map

g×g→g;(x,y)↦[x,y], called the Lie bracket, satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie’s third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.

Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.

Lie algebras were so termed by Hermann Weyl after Sophus Lie in the 1930s. In older texts, the name infinitesimal group is used.

29. LATTICE- a method or multiplying large numbers using a grid. it breaks the multiplication process into smaller steps. Digits to be carried are written in the grid, making them harder to miss.

30. HOMOLOGY-a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects.

31. RELATION-a relationship between sets of values, between the ordered pairs x values and y values.

32. LANGLANDS PROGRAM- a web of far reaching and influential conjectures that relate galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and abeles.

33. FINITE FIELD-a galois field is a field that contains a finite number of elements. as with any field, it is a set on which the operations of addition, subtraction, multiplication, and division are defined and satisfy certain basic rules.

34. HOMEOMOPHISM-an instance of topological equivalence to another space or figure.

35. ISOMORPHISM- a homomorphism or morphism (mathematical mapping) that admits an inverse. 2 math objects are isomorphic if an isomorphism exists between them.

36. PELL’S NUMBER-In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2.

37. Affine group-In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is the real or complex field or quaternions.

38. lie group-In mathematics, a Lie group (pronounced /ˈliː/ “Lee”) is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

39. Abelian group-In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

40. MAGMA-

In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure. Specifically, a magma consists of a set, M, equipped with a single binary operation, M × M → M. The binary operation must be closed by definition but no other properties are imposed.

41. quasigroup-

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that “division” is always possible. Quasigroups differ from groups mainly in that they need not be associative. A quasigroup with an identity element is called a loop.

42. monoid-Monoid. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory, because they are semigroups with identity.

43. semigroup-In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

44. ABEL-RUFFINI THEOREM-

In algebra, the Abel–Ruffini theorem (also known as Abel’s impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799,[1] and Niels Henrik Abel, who provided a proof in 1824.

GEOMETRY

1. CONIC SECTIONS- there are 4 types of conic sections: circles, ellipses, parabolas, and hyperbolas. These are the figures produced by the intersection of a plane and a right circular cone, the figure produced depends on the angle of the intersection with respect to the cone. The greatest progress in the study of conics was done by the ancient greeks is due to Apollonius of perga (before 190 BC). Conic sections are important in astronomy. The orbits of 2 massive bodies interact that according to newton’s law of universal gravitation are conic section if their common center of mass is considered to be at rest. Their importance are many. The parabola is used if you bounce light onto a parabolic mirror, then the light will focus onto a point. a few of uses of the parabola are satellite dishes, telescope mirrors, and predicting the path of a projectile. the ellipse is used to predict where the planets will be in their orbits since planets travel around the sun in ellipses. Circles have many uses: wheels and tires and designs of all kinds are circular, an example being a compact disk.hyperbolas show up in for example, a sharpened pencil, and when you hear a sonic boom because the sonic boom forms a curve which intersects the ground in a hyperbola (half a hyperbola). Conic sections show up in life and science in a great many ways.

2. LEMINISCATE- in algebraic geometry, any of several figure eight or shaped like an infinity symbol curves.

3. HAUSENDORFF DIMENSION- a fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher dimensional space. The hausendorff dimension measures the local size of a space taking into account the distance between points, the metric.

4. PAPPUS THEOREM-the surface area of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve’s geometric centroid.

5. ARCHIMEDES METHOD-formulas for areas and volumes set the standard for the rigorous treatment of limits until modern times. It had used cavalier’s principle, which involves slicing solids (whose volumes are to be compared) with a family of parallel planes.

6. ARCHIMEDEAN SPIRAL-a curve defined by a polar equation of the form r= theta a. If a=1, so r= theta, then it is an Archimedean spiral. For a= -1, so r= 1/theta, we get the reciprocal (or hyperbolic) spiral.

7. Platonic solids- Platonic solids are regular, convex polyhedrons. there are 5 of these solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. They have been studied by mathematicians for thousands of years because of their great beauty and symmetry for the reason that these solids have congruent regular polygonal faces with the same number of faces meeting at each vertex. The tetrahedron, cube, and octahedron all occur naturally in crystal structures.many viruses have the shape of a icosahedron. The Platonic solid are used in the dice that people play dice with. In the intermediate of liquid crystals, aluminum has an icosahedral shape. In more than 3 dimensions, polyhedra generalize to polytopes (a geometric object with flat sides and may exist in any general number of dimensions), with higher-dimensional covers regular polytopes being the equivalent of the 3-dimensional Platonic solids. So, platonic solids are 3D shaped objects where each face is the same regular polygon, and the same number of polygons meet at each vertex.

8. ARCHIMEDIAN SOLIDS-13 solids that consist of more than a single kind of regular polygon that form it. they do not include the Platonic solids. these solids were valued for their pure forms with high symmetry.

9. EUCLID’S 5th POSTULATE- if a straight line falling on 2 straight lines makes the interior angles on the same side less than 2 right angles, the 2 straight lines, if produced indefinitely, meet on that side on which are the angles less than the 2 right angle

10. SYMMETRY-agreement in dimensions, due proportion, arrangement. Harmonious and beautiful proportion and balance. Symmetry occurs in geometry when an object has reflectional symmetry, rotational symmetry, translational symmetry, helical symmetry, glide symmetry, and rotorflection symmetry. In mathematical logic, a dyadic relation is symmetric. In general, every kind of structure in math will have its own kind of symmetry. Examples include even and odd functions in calculus, symmetric groups in abstract algebra, symmetric matrices in linear algebra, and Galois group in Galois theory. In statistics, it appears in symmetric probability distributions, and as skewness, asymmetry of distributions.

11. PARALLEL POSTULATE/PARALLEL AXIOM-the axiom in euclidean geometry that only one line can be drawn through a given point so that the line is parallel to a given line that does not contain the point.

12. Non-euclidean geometry-flat geometry of everyday intuition is called euclidean geometry, or parabolic geometry,, and the non-Euclidean geometries are called hyperbolic geometry and elliptical geometry, or riemannkan geometry is a non-euclidean geometry.

13. FRACTALS- a never ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating the same process over and over in an ongoing feedback loop.

14. LOGARITHMIC SPIRAL- growth spiral that often appears in nature.

15. WITCH OF AGNESI-curve studied by Maria Agnes, and earlier by Fermat and guido Grandi.

16. ARCHIMEDES SPIRAL- curve defined by the polar equation of the form r=theta x a, with special names being given to certain values of a. Examples of if a=1, so r=theta, then it is called archimedes’ spiral.

17. TAUTOCHRONE PROBLEM-is an isochrone curve for which the time taken for an object sliding without friction in uniform gravity to its lowest point is independent of the starting point. The curve is a cycloid, and the time is equal to pi times the square root of the radius over the acceleration of gravity.

18. SYLVESTER-GALLAI THEOREM- in geometry, given a finite number of points in the euclidean plane, either:

1. All the points are collinear, or

2. There is a line which contains exactly 2 of the points.

19. PAPPUS THEOREM-the surface area of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve’s geometric centroid.

20. KOCH CURVE/SNOWFLAKE-a mathematical curve and one of the earliest fractal curves to have been described.

21. HILBERT SPACE-an infinite-dimensional analog of euclidean space.

22. MINKOWSKIAN SPACE- in the presence of gravity spacetime is described by a 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. This space is essential in the description of general relativity.

23. BIRKOFF’S THEOREM- states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat.

24. BOLZANO-WEIERSTRASS THEOREM- a fundamental result about convergence in a finite-dimensional Euclidean space R^n. It states that each bounded sequence in R^n has a convergent subsequence.

25. ABELIAN GROUP- in abstract algebra, also called a commutative group, is a group in which the result of applying the group operation to 2 group elements does not depend on the order in which they are Witten.. these groups open the axiom of commutativity.

26. PERIODIC BERNOULLI POLYNOMIAL- a Bernoulli polynomial evaluated at the fractional part of argument x. Are used to provide the remainder term in the Euler-maclaurin formula relating sums to integrals. The first polynomial is am sawtooth.

27. BOLZANO-WEIERSTRASS THEOREM- a fundamental result about convergence in a finite-dimensional Euclidean space R^n. It states that each bounded sequence in R^n has a convergent subsequence.

28. ORTHOGONAL PROJECTION- projections onto subspaces. Visualizing a projection onto a plane. A projection onto a subspace is a linear transformation.

29. RIEMANN SURFACE- a surface-like configuration that covers the complex plane with several, and in infinitely many, sheets. The sheets have very complicated structures and interconnections.

30. RANDOM WALK- a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on same mathematical space such as the integers.

31. HODGE THEORY- a key tool is the laplacian operator associated to a riemannian metric on M. It has major applications in: riemannian manifolds, Mahler manifolds, and algebraic geometry of complex projective varieties. Some of the deepest results of algebraic geometry were only accessible through analytic methods by hodge theory. Results of hodge theory for algebraic varieties also have been proved by arithmetic methods, known as p-adic hodge theory.

32. CAVALIERI’S PRINCIPLE- if, in 2 solids of equal; altitudes, the sections made by planes parallel to and at the same distance from their respective bases are always equal, the the volumes of the 2 solids are equal.

33. ERLANGEN PROGRAMME- a method of characterizing geometries based on group theory and projective geometry.projective geometry was emphasized as the unifying frame for all other geometries. by 1872, non-euclidean geometries had merged. Euclidean geometry was more restrictive than affine geometry with was in turn more restrictive than projective geometry. That group theory, which uses algebraic methods to abstract ideas of symmetry, was the most useful way of organizing geometrical knowledge, at the time it had already been introduced into the theory of equations in the form of Galois theory. Be made much more explicit the idea that each geometrical language had its own, appropriate concepts, so that projective geometry talked about conics sections, not about circles or angles because these notions were not invariant under projective transformations. The way of multiple languages of geometry came back together that could be explained by the way subgroups of a symmetry group related to each other.Riemann geometry was also generalized.

34. FRECHET DISTANCE- a measure of similarity between curves that takes into account the location and ordering of the points along the curve.

35. SPHERICAL COORDINATE SYSTEM-a coordinate system of 3-dimensional space where the position of a point is specified by 3 numbers: the radial distance of that point from the fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection.

36. ORTHOGONAL PROJECTION-projections onto subspace. visualizing a projection onto a plane. the projection onto a subspace is a linear transformation.

37. HYPERBOLIC GEOMETRY-non-euclidean geometry. the geometry of euclidean postulates, except the 5th ,which is replaced by a negation. there exists a line and a point not on such

That at least 2 distinct lines parallel to pass through.

38. ELLIPTICAL GEOMETRY-non-euclidean geometry with positive curvature which replaces the parallel postulate with the statement ‘through any point in a plane, there exists no line parallel to a give line.’

39. ISODIAMETRIC INEQUALITY- states that, of all bodies of a given diameter, the sphere has the greatest volume.

40. CLIFFORD’S THEOREM ON SPECIAL DIVISORS- an algebraic curve showing the constraints on special linear systems on a curve C.

41. HOMOLOGY- a general way os associating a sequence of objects such as abelian groups or modules to other mathematical objects.

42. KEPLER CONJECTURE- a mathematical theorem about sphere packing in 3-dimensional Euclidean space.

43. CARTIOD-a degenerate case of a lima con. It is a heart shaped curve. It is also defined as an epicycloid having a single cusp. As its variable a and b get larger, the carotid gets larger.

44. BARYCENTRIC CALCULUS-method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. It provides excellent insight into triangular centers.

45. LUNE OF HIPPOCRATES- a problem of squaring the circle (constructing a square with a straightedge and compass having the same area as a given circle. This problem has a simple solution.

46. JUNG’S THEOREM- an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set.

TRIGONOMETRY

1. HOW ERATOSTHENES MEASURED THE SIZE OF THE EARTH OVER 2,000 YEARS AGO-he knew that at the summer solstice that the sun shone directly into a well at Syene at noon. At the same time, in Alexandria, Egypt, approximately 488 miles due north of seen, the angle of elevation (deviation of the sun shining directly into the well/the angle of deviation) of the sun’s rays was about 7.2 degrees. 360 degrees contains 50 7.2 degrees parts. Since the going around the earth is a circle, and since a circle contains 360 degrees, and since a circle is 50 times more than 7.2 degrees, we multiply the 488 miles by 50 and this equals 24,400 miles. This result is 98.3 precent accurate.

2. INVERSE TRIGONOMETRIC FUNCTION- the inverse of any angular trigonometric ratio results in its angular equivalent.

3. HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIONS- hyperbolic functions are analogs of their ordinary trigonometric, or circular functions, and the inverse hyperbolic functions are the area hyperbolic functions. Hyperbolic functions occur in the solutions of many linear differential equations, in the calculations of angles and distances in hyperbolic geometry, and of the laplace equation in cartesian coordinates.

4. HERON’S FORMULA- formula for finding the area of any type of triangle.

5. LAW OF SINES-the relationship between the sides and angles of a non-right triangle. Allows one to calculate the remaining sides of a triangle when 2 angle and a side are known, or to calculate the remaining angle when 2 sides are known.

6. LAW OF COSINES-for calculating one side of a triangle when 2 of the other sides are known.

7. LAW OF TANGENTSa statement about the relationship between the tangents of 2 angles of a triangle and the lengths of the opposing sides.

8. POLAR COORDINATES- 2-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

9. RADIANS-standard unit of angular measurement, used in many areas of mathematics. There are 2Pi radians in 360 degrees. A radian is equal to 57.3 degrees.

10. SPHERICAL TRIGONOMETRY-branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by the number of intersecting great circles on the sphere.

11. VIVIANI’S THEOREM- the sum of the distance from any interior point to the sides of an equilateral triangle equals the length of the triangle’s altitude.

12. Osborn’s rule-Osborn’s rule is a rule for converting a trigonometric identity into a corresponding hyperbolic one. The rule states that one replaces every occurrence of sine or cosine with the corresponding hyperbolic sine or cosine, and wherever one has a product of two sines, the product of the hyperbolic sines must be negated.

13.

In trigonometry, Mollweide’s formula, sometimes referred to in older texts as Mollweide’s equations, named after Karl Mollweide, is a set of two relationships between sides and angles in a triangle. It can be used to check solutions of triangles.

LINEAR ALGEBRA, CALCULUS, DIFFERENTIAL EQUATIONS AND ANALYSIS

1. EULER PRODUCT-in number theory, an expansion of the dirichlet series into an infinite product indexed by prime numbers.

2. L-FUNCTION- a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An l-function is a dirichlet series, usually convergent on a half plane, that may give rise to an l-function by way of analytic continuation. L-functions are a substantial but still largely conjectural part of analytic number theory. In it, broad generalizations of the Riemann zeta function and the l-series for the dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.

3. HEAT EQUATION- a parabolic partial differential equation that describes the distribution of heat, or variations in temperature, in a given region over time.

4. WEIERSTRASS FUNCTION- a function being continuous everywhere but differentiable nowhere.

5. SQUEEZE RULE- a function squeezed between 2 functions approaching the same limit L must also approach L

6. LAPLACE TRANSFORM-is an integral transform perhaps second only to Fourier transform in its utility in solving physical problems. It is particularly useful in solving linear ordinary differential equations as those arising in the analysis of electronic circuit. The Laplace transform takes a function of a real variable t (the transform reduces a differential equation into an algebraic equation (often time) to a function of a complex variable s (frequency). So by solving a simpler algebraic equation and then inverting the transform, we can derive a solution to the differential equation. laplace transform make solving differential equations easy by taking the laplace transform of a network and manipulating it algebraically and then taking the inverse transform solves the differential equation.

7. FOURIER TRANSFORM- a function derived from a given function and representing it by a series of sinusoidal functions.

8. DIRICHLET L-FUNCTIONS-in analytic continuation, this function can be extended to

a medomorphic function on the whole complex plane.

9. EULER PRODUCT-an expansion of a dirichlet series into an infinite product indexed by prime numbers.

10. DIRICHLET SERIES-functions of a complex variable s that are defined by certain infinite seres. They are generalizations of the Riemann zeta function, they can be applied to discover and prove identities among arithmetic functions.

11. BERNOULLI NUMBER-a sequence of signed rational numbers that can be defined by the exponential generating function. They arise in the seres expansion of trigonometric functions, and are extremely important in number theory and analysis.

12. LAGRANGE MULTIPLIERS-a powerful method to solve class of problems that arise when one wishes to maximize or minimize a function subject to fixed outside equality constraints without the need to explicitly solve the conditions and use them to eliminate extra variables.

13. LIMITS-a value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus (and to mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

14. ELLIPTICAL FUNCTIONS-in complex analysis, it is a meromorphic function that is periodic in 2 directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptical function is determined by its values on a fundamental parallelogram, which then repeats in a lattice.

15. FUNDAMENTAL THEOREM OF CALCULUS-one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. (The indefinite integral of a function is related to its antiderivative, and can be reversed by differentiation.)

16. DIFFERENTIATION- the essence of calculus. It is the instantaneous rate of change of a function with respect to one variable. This is equivalent to finding the slope of the tangent line of the function at a point.

17. INTEGRATION- an integral assigns numbers to functions in a wa that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of two main operations of calculus, with its inverse, differentiation, being the other.

18. EULER-MASCHERONI CONSTANT/EULER CONSTANT- a constant recurring in analysis and number theory, usually denoted by the greek letter gamma. It appears in the exponential integral, laplace transform of the natural logarithm, Laurent series expansion for the Riemann zeta function, where it is the first of the stieltjes constants, calculations of the digamma function, product formula of the gamma function, in dimensional regularization of Feynman diagrams in quantum field theory, , solution of the second kind of vessel’s equation, and many other places in mathematics.

19. GAMMA FUNCTION- an extension of the factorial function, with its argument shifted down to 1, to real and complex numbers. The function is defined for all complex numbers except the non-positive integers.

20. POWER SERIES- in one variable, is an infinite series of the form-

Summation (n=0 to infinity) for a(n)(x-c)^n-a(0)+a(1)(x-c)^1+a(2)(x-c)^2+…,

Where a(n) represents the coefficient of the nth term and c is a constant. The series usually arises as the taylor series of sone known function. In many situations c (the center of the series) is equal to zero, for instance when considering the maclaurin series. In this case it takes the form-

Summation (n=0 to infinity) for a(n)x^n=a(0)+a(1)x+a(2)x^2+…

These power series arise primarily in analysis, but also in combinatorics and electrical engineering.

There are also power series of several variables.

21. PARTIAL DIFFERENTIAL EQUATION-an equation containing one or more partial derivatives. A differential equation that contains unknown multivariable functions and their partial derivatives. An example is the wave function.

22. ELLIPTICAL INTEGRALS- generalizations of the inverse trigonometric functions and provide solutions to a wider class of problems. For instance, while the arc length of a circle is given as a simple function odd a parameter, computing the arc length of an ellipse requires elliptic integrals.

23. INTERMEDIATE VALUE THEOREM-states that a continuous function, f, with an interval [a,b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

24. INFINITESIMALS-means extremely small. It usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed up to produce an integral.

25. SIMPSON’S RULE-a rule for numerical integration, the numerical approximation of definite integrals.

26. L’HOSPITAL’S RULE-if we have an indeterminate form 0/0, we can differentiate the numerator and the denominator and then take the limit to drive at an answer.

27. LAGRANGE MULTIPLIERS-in optimization, this is a strategy for finding the local maxima and minima of a function subject to equality constraints.

28. ARGAND DIAGRAM- a diagram in which complex numbers are represented geometrically using cartesian axis, the horizontal axis the real part of the number and the vertical the complex part.

29. HYPERGEOMETRICAL FUNCTION- the gaussian or ordinary hypergeometric function, f(alb, c,z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is the solution of a second-order linear differential equation. it is the solution of every second-order linear ordinary differential equation. Every second-order linear ODE with 3 regular singular points can be transformed into this equation.

30. HYPERGEOMETRIC SERIES- basic hypergeometric series are analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series.

31. DETERMINANT-in linear algebra, a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A) or set A.

32. EIGENVALUE- 1) each of a set of values of a parameter for which a differential equation has a nonzero solution (an eigenfunction) under given conditions, and 2) any number that is given matrix minus that number times the identity matrix has a zero determinant.

33. EIGENVECTOR-a vector that when operated on by a gives a scalar multiple of itself.

34. DETERMINANT-in linear algebra, a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A) or set A.

35. EIGENVALUE- 1) each of a set of values of a parameter for which a differential equation has a nonzero solution (an eigenfunction) under given conditions, and 2) any number that is given matrix minus that number times the identity matrix has a zero determinant.

36. EIGENVECTOR-a vector that when operated on by a gives a scalar multiple of itself.

37. BRACHISTOCHRONE PROBLEM- the first problem of this type (calculus of variations) is the curve of the fastest descent.

38. METHOD OF EXHAUSTION- a method off finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape.

39. FOURIER ANALYSIS- the analysis of complex waveforms expressed as a series of sinusoidal functions, the frequencies of which form a harmonic series.

40. GAMMA FUNCTION- an extension of the factorial function, with its argument shifted down to 1, to real and complex numbers. If n is a positive integer, it has no zeros, so its reciprocal is a holomorphic function. It is used in the mathematical and applied sciences.

41. HYPERCOMPLEX NUMBER- an element of a finite-dimensional algebra over the real numbers that is until and distributive, but not necessarily associative.

42. IMPLICIT DIFFERENTIATION- a special case of the chain rule for derivatives.

43. GAUSS-JORDAN ELIMINATION- method to solve a system of linear equations by 1) write the augmented matrix of the system 2) use row operations to transform the augmented matrix in the form of a reduced row echelon form.

44. LAMBERT SERIES- series of the form, s(q)=summation of n=1 to infinity of a(n)*q^n/(1-q^n).

The series may be inverted by means of the mob’s inversion formula, and is an example of a mobs transform.

45. LAPLACIAN- Laplace operator is a differential operator given by the divergence of the gradient of a function on Euclidean space. (d^2/dx^2+d^2/dy^2+d^2/dz^2)

46. GREGORY SERIES/MADHAVA-GREGORY SERIES/LEIBNIZ’S SERIES-an infinite Taylor series expansion of the inverse tangent function.

47. HARMONIC SERIES- an infinite series of the reciprocals of the counting numbers which diverges.

48. TRIGONOMETRIC SERIES-a series in the form of:

A(0)/2=summation of n=1 to infinity of (a(n)cosnx+b(n)sing)

49. MODULAR FORM- a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of a modular group, and also satisfying a growth condition.

50. CROSS PRODUCT- in math and vector algebra, is a binary operation on 2 vectors in 3-dimensional space.

51. DOT PRODUCT- scaler product is an algebraic operation that takes 2 equal length sequences of numbers (usually coordinate vectors) and returns aa single number.

52. VECTOR SPACE- a set that is closed under finite vector addition and scalar multiplication.

53. TENSOR- a mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space.

54. JACOBIAN- a determinant whose constituents are the derivatives of a number of functions (u,v,w,…) with respect to each of the same number of variables (x,y,z,…).

55. WRONSKIAN-a determinant used to study differential equations where it can sometimes show linear independence to a set of solutions.

56. EULER’S METHOD-it is a 1st-order numerical procedure for solving ordinary differential equations with a given initial value and is the simplest range-gutta method.uses a step size of h=.1 to find approximate values of the solutions at t=.1, .2, .3 , .4, and .5. compare them to the exact values of the solutions as these points.. It is the most basic explicit method for numerical integration of ordinary differential equations.

57. JACOBI ELLIPTICAL FUNCTION-a set of basic elliptic functions, and auxiliary functions, that are of historic importance. Many of their features show up in important structures and have direct relevance to some applications (I.e. the equation of a pendulum).they also have useful analogies to the functions of trigonometry as indicated by the matching notation sn for sin.

58. INTEGRATION BY PARTS- a heuristic rather than purely mechanical process of solving integrals. Given a single function to integrate, the typical strategy is to carefully separate this single function into a product of 2 functions u(x)v(x).

59. TRAPEZODIAL RULE-method for approximating a definite integral using linear approximations

of f.

60. DIVERGENCE- net flux per unit volume. in vector calculus, the divergence of a Vector field is the rate at which density exits a given region of space. How a vector field changes its magnitude in the neighborhood of a point.

61. CURL- in vector calculus, has to do with how a vector field how the vector field direction changes. At every point in the field, the curl of that point is represented by a vector. If the vector field represents the flow of a moving fluid, then the curl is the circulation density of the fluid. The vector field whose curl is zero is called irrotational.

62. DEFINITE INTEGRAL-an integral expressed as the difference between the values of the integral at specified upper and lower limits of the independent values.

63. INDEFINITE INTEGRAL- n integral expressed without limits, and so containing an arbitrary constant.

64. BIRKOFF’S THEOREM- states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat.

65. GRADIENT- in vector calculus, nabla/del/gradient, is a vector differential operator defined on a 1-dimensional domain and denotes the standard derivative in calculus, or slope.

66. DIRICHLET SERIES- functions of a complex variable that is defined by certain infinite series. They are important generalizations of the Riemann zeta function, and are connected with the distribution of primes.

67. SIMULTANEOUS DIFFERENTIAL EQUATIONS- one of the mathematical equations for an indefinite function of 1 or more variables that relate the values of the function.

68. LAURENT SERIES-a series of a complex function f(x) is a representation of that function as a power series which include terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

69. INNER PRODUCT- a generalization of the dot product. In a vector space, it is a way of multiplying vectors together, with the result being a scalar.

70. CONVERGENCE AND DIVERGENCE-the former the partial sums of the series of goes onto positive or negative infinity. the later when a number is the result.

71. SHORE’S THEOREM- concerns integer factorization. when given a number, finding its prime factors.

72. LEBESGUE INTEGRALS-defined in terms of upper and lower bounds using the lebesgue measure of a set. it uses a lebesgue sum where is the value of the function in subinterval, and is the lebesgue measure of the set of points for which values are approximately.

73. MONOTONE CONVERGENCE THEOREM-in real analysis, is any of a set of number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded. If a sequence is increasing and bounded above the supremum (of a subset S of a partially ordered set that is the least element in T that is less than or equal to all elements in S, if such an element exists) of that set, them the sequence will converge to the supremum. If the sequence is decreasing and is bounded below by an infimum (of a subset S of a partially ordered set that is the greatest element in T that is less than or equal to all elements in S, if such an element exists), it will converge to the infimum.

74. JORDAN CURVE THEOREM- in topology, a non-self-intersecting continuous loop in a plane (a plane simple closed curve).

75. Parametric equations-In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters.[1] Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrization) of the object.[1][2][3] For example, the equations

x= cos t

y= sin t

form a parametric representation of the unit circle, where t is the parameter.

76. SUBHARMONIC FUNCTION (on a Riemann manifold)- assume that for any open subset, that any harmonic function f on U, such that on the boundary of U, the inequality holds on all U, then f is called subharmonic.

77. LINE INTEGRAL- an integral taken along a line, of any function has a continuously varying value along that line.

78. CAUCHY INTEGRAL- has a central place in complex analysis. it expresses that fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk. It also provides integral formulas for all derivatives of a holomorphic function.

79. IMPROPER INTEGRAL-a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. They cannot be computed using a normal Riemann integral.

80. MULTIPLE INTEGRAL-a definite integral of a function of more than 1 real variable. Examples are, f(x,y) or f(x,y,z).

81. FUBINI’S THEORM- in mathematical analysis, is a result that gives conditions under which it is possible to compute double integrals using iterated integrals.

82. LEGENDRE FUNCTION-are solutions to legend’s differential equations. This ordinary differential equation is frequently encountered in physics and other technical fields.

83. INFINITESIMAL-extremely small. To give it meaning, it is usually compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral.

84. BESSEL FUNCTION- for integer a are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to lap lace’s equation in cylindrical coordinates. Spherical Bessel functions with half-integer a are obtained when the Helmholtz is solved in spherical coordinates.

85. TAYLOR SERIES-representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

86. DARBOUX’S FORMULA-for summing infinite series by using integrals or evaluating integrals using infinite series.

87. EULER-LAGRANGE EQUATION- in the calculus of variations, it is a 2nd-order partial differential equation whose solutions are the functions for which a given functional is stationary.

88. EULER-MACLAURIN FORMULA-provides a powerful connection between integrals and sums.

89. STURM-LIOUVILLE EQUATION-a very physically important class of operators with a weight function.

90. GREEN’S THEOREM-gives the relationship between a line integral around a simple closed curve c and a double integral over the p[lane region d by c. it is a 2-dimensional case of the more general curl (kelvin-stokes theorem).

91. VECTOR LAPLACIAN-a differential operator defined over a vector field. Is similar to the scalar laplacian.

92 HAMILTONIAN EQUATIONS-a function is equal to the total energy of the system. These are 1st-order differential equations. Are often useful alternative to Lagrangian equations, which take the form of 2nd-order differential equations.

93. POISSON-BOLTZMANN EQUATION-a 3-dimensional 2nd-order nonlinear elliptical partial differential equation.

94. CAUCHY-RIEMANN EQUATION-in complex analysis, is a system of 2 partial differential equations with together with certain continuity and differentiability criteria form a necessary and sufficient condition for a complex function to be complex differentiable, or holomorphic.

95. LOENZ EQUATION-to model the modal amplitudes in a nonlinear thermal convection problem. Solution can exhibit chaos.

96. WAVE EQUATION-an important 2nd-order linear hyperbolic differential equation that describes waves, such as sound, light, and water waves that arise in acoustics, electromagnetism, and fluid dynamics.

97. HANKEL TRANSFORM-of order zero ia an integral transform equivalent to the 2-dimensional fourier transform with radially symmetric integral kernel.

98. HILBERT TRANSFORM-important in signal processing, where it derives the analytic representation of a signal u(t). a linear operator that takes a function u(t) of a real variable and produces another function of a real variable h(u)(t).

99. Z-TRANSFORM-important in signal processing, converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It ca be considered as a discrete-time equivalent of the Laplace transform.

100. ELLIPTIC INTEGRAL- a generalization of the inverse trigonometric functions and provide solutions to a wider class of problems. Originally arose in connection with the problem of giving the arc length of an ellipse.

101. LEGRENDRE FUNCTION- are solutions to Legendre differential equations. Frequently encountered in physics.

102. BESSEL FUNCTION- are the canonical solutions to bessel’s differential equations, for an arbitrary complex number a, the order of the Bessel function.

103. GAMMA FUNCTION-an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.

104. EULER’S CONSTANT-0.5772156649. recurs in analysis and number theory.

105. FOURIER-BESSEL SERIES-a particular kind of generalized fourier series based on bessel functions.

106. DIRICHLET SERIES-functions of a complex variable s that are defined by certain infinite series.

107. FOURIER SERIES- an infinite series of trigonometric functions that represent an expansion or an approximation of a periodic function used in fourier analysis.

108. Puiseux series-In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676[1] and rediscovered by Victor Puiseux in 1850.

109. Laurent series-In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.

110. Fourier transform-The Fourier transform decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency.

111. Z transform-

In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency domain representation. It can be considered as a discrete-time equivalent of the Laplace transform.

112. Banach spaces-

A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, which is equipped with a norm and which is complete with respect to that norm, that is to say, for every Cauchy sequence {xn} in X, there exists an element x in X such that.

113. Beta function-In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

B(x,y)=∫01tx−1(1−t)y−1dt

for Re x, Re y > 0.

114. Gamma function-In mathematics, the gamma function (represented by the capital Greek alphabet letter Γ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is a positive integer:

Γ(n)=(n−1)!

115. Legendre function-

In mathematics, Legendre functions are solutions to Legendre’s differential equation: (1) They are named after Adrien-Marie Legendre. This ordinary differential equation is frequently encountered in physics and other technical fields.

116. Orthogonal function-

In mathematics, orthogonal functions belong to a function space which is a vector space (usually over R) that has a bilinear form.

117. P-adic numbers-In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

118. Hilbert space-an infinite-dimensional analog of Euclidean space.

119. Fubini’s theorem-In mathematical analysis Fubini’s theorem, introduced by Guido Fubini (1907), is a result that gives conditions under which it is possible to compute a double integral using iterated integrals.

120. Louisville theorem-

http://www.nyu.edu

In complex analysis, Liouville’s theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that.

121. Cauchy integral theorem-In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere in between the two paths, then the two path integrals of the function will be the same.

122. Cauchy-riemann equations-In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d’Alembert (d’Alembert 1752). Later, Leonhard Euler connected this system to the analytic functions (Euler 1797). Cauchy (1814) then used these equations to construct his theory of functions. Riemann’s dissertation (Riemann 1851) on the theory of functions appeared in 1851.

123. picard theorem-In complex analysis, Picard’s great theorem and Picard’s little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.

124. HESSIAN MATRIX-

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables.

125. ISOMORPHISM-

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos “equal”, and μορφή morphe “form” or “shape”) is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them.

126. JACOBIAN-In vector calculus, the Jacobian matrix (/dʒɪˈkoʊbiən/, /jɪˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a square matrix, both the matrix and its determinant are referred to as the Jacobian in literature.

126. Riemann integration-

math.feld.cvut.cz

Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions.

127. Riemann sum-

In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations.

128. Darboux sum-In real analysis, a branch of mathematics, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.[1] The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral.

129. Lebesgue integration-In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

130. Cauchy integral formula-In mathematics, Cauchy’s integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy’s formula shows that, in complex analysis, “differentiation is equivalent to integration”: complex differentiation, like integration, behaves well under uniform limits – a result denied in real analysis.

131. Harmonic function-

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace’s equation, i.e. everywhere on U. This is usually written as. or.

132. Maclaurin series- In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. f the Taylor series is centered at zero, then that series is also called a Maclaurin series.

133. Fourier series- In mathematics, a Fourier series (English: /ˈfʊəriˌeɪ/)[1] is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem. The study of Fourier series is a branch of Fourier analysis.

134. Residue theorem-

mathworld.wolfram.com

In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy’s residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well.

135. Bolzano-weierstrauss theorem-

http://www.youtube.com

In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states that each bounded sequence in Rn has a convergent subsequence.

136. Intermediate value theorem-

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

137. Heine-borel theorem-In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:

For a subset S of Euclidean space Rn, the following two statements are equivalent:

• S is closed and bounded

• S is compact, that is, every open cover of S has a finite subcover. This is the defining property of compact sets, called the Heine–Borel property.

In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrary metric spaces reads:

A subset of a metric space is compact if and only if it is complete and totally bounded.

138. Mean value theorem-

The Mean Value Theorem is one of the most important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that.

139. Monotone convergence theorem-In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

140. Measure space-In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1.

141. Metric space-In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

The most familiar metric space is 3-dimensional Euclidean space. In fact, a “metric” is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

142. Meromorphic function-

http://www.youtube.com

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function.

143. Cauchy-riemann equations-

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex …

144. Looman-menchoff theorem-In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it satisfies the Cauchy–Riemann equations. It is thus a generalization of a theorem by Édouard Goursat, which instead of assuming the continuity of f, assumes its Fréchet differentiability when regarded as a function from a subset of R2 to R2.

145. Picard’s theorem-

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard’s existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

146. Line integral-

tutorial.math.lamar.edu

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integrals in the complex plane.

147. Riemann surface-In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

148. Liuoville theorem-

galileoandeinstein.physics.virginia.edu

In complex analysis, Liouville’s theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that.

149. Riemann mapping theorem-In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk

D={z∈C:|z|<1}.

This mapping is known as a Riemann mapping.

150. Runge’s theorem-In complex analysis, Runge’s theorem (also known as Runge’s approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following:

Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If A is a set containing at least one complex number from every bounded connected component of C∖K then there exists a sequence ( r n )n∈N

of rational functions which converges uniformly to f on K and such that all the poles of the functions (r n)n∈N

151. Surface integral-

tutorial.math.lamar.edu

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.

152. Riemann-stieltjes integral-Generalized Riemann–Stieltjes integral. A slight generalization, introduced by Pollard (1920) and now standard in analysis, is to consider in the above definition partitions P that refine another partition Pε, meaning that P arises from Pε by the addition of points, rather than from partitions with a finer mesh.

152. Elliptical functions-In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice.

153. Modular forms and functions-In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that f (z) be holomorphic at infinity. Instead, modular functions are meromorphic with infinity being the only pole.

Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups.

154. The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.

It is defined for all x by[1][2][3]

gd x=∫0^x1/coshtdx

SETS , LOGIC, AND TOPOLOGY

1. AXIOM OF CHOICE-aOn axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

2. LAW OF EXCLUDED MIDDLE-in logic, is the 3rd of 3 laws of thought. it states that for any proposition, wither that proposition is true, or its negation is true.

3. TAUBERIAN THEOREMS- established the conditions which determine the sets of series (or sequences) on which for 2 given summation methods A and B the inclusion A is a subset of B holds.

4. 7 bridges of konigsberg- problem that laid the foundations of graph theory and prefigured the idea of topology. The negative resolution was made by Euler in 1736.

5. DEDEKIND CUT-a method of construction of the real numbers. It is a partition of the rational numbers into 2 non-empty sets A and B, such that all the elements of A are less than all the element os B, and contain no greater element.

6. MULTIDIMENSIONAL MATH- n dimensions of mathematical space (or object) is the minimum number of coordinates needed to specify any point within it.

7. MANIFOLD- a topological space that locally resembles Euclidean space near each point. Each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

8. HAIRY BALL PROBLEM-theorem of algebraic topology that states that there is no non vanishing continuous tangent vector field on even-dimensional n-sphere. Every smooth vector field on a sphere has a singular point. This means that it is impossible to comb a hairy sphere.

9. PIGEONHOLE PRINCIPLE- if n items are put into m containers, with n>m>0, them at least one container must contain more than one item.. for example, there must be at least 2 left gloves or 2 right gloves in a group of 3 gloves.

10. ROUTE INSPECTION PROBLEM- in graph theory, a postman tour problem is to find a shortest closed path or circuit that visits every edge of a connected graph. when the graph has an eulerian circuit (a closed walk that covers ever edge once), that circuit is an optimal solution.

11. FIXED POINT THEOREM- a result saying that a function will have at least one fixed point (a point x for which R(x)=x) under some conditions of F that can be stated in general terms.these kinds of results are amongst the most generally useful in mathematics. There are the banach, borel, Brouwer, poincare-birkoff, and lefschetz fixed point theorems, among others.

12. CHURCH’S THEOREM-a proof that in general we cannot calculate whether a given mathematical statement in a formal system is true or false.

13. HOMEOMORPHISM (topological isomorphism/bi continuous function)-in topology,

is a continuous function between topological spaces that has a continuous inverse function.

14. COHOMOLOGY-in homology theory and algebraic topology, it is a general term for a sequence of abelian groups associated to a topological space, often defined from a coaching complex.

15. HOMOTOPY- two continuous functions from one topological space to another. if it can be continuously deformed into the other, such a deformation called a homotopy between two functions.

16.DIMENSIONS- the minimum number needed to specify any point within it.

17. HAIRY BALL THEOREM-in algebraic topology, it states that there is no non vanishing continuous tangent vector field on even-dimensional n-spheres. An example of this theorem is that when one is given a ball of hair all over it, it is impossible to comb the hairs continuously and have all the hairs lay flat.

18. BOUWER’S FIXED-POINT THEOREM- for any continuous function f mapping a compact convex set into itself there is a point x sub 0) such that f(x (sub 0)=x (sub 0).

19. JULIA SET- a set of complex numbers that do not converge to any limit when a given mapping is repeatedly applied to them. In some cases the result is a connected fractal set.

20. SEVEN BRIDGES OF KONIGSBERG PROBLEM-in this problem, there is a town with 4 bridges connected to a places on the sides of the river, 2 bridges connected to one side of the river and 2 bridges to the other side of the river, on a mainland north of a river, on a mainland south of the river there are 2 bridges, one big connected to one side of the river and the other bridge connected to the other side. and 1 bridge connecting the 2 towns. The object of the problem is to visit the 2 places on either side of the river and the 2 towns by crossing each bridge only once. In 1736, Leonard euler proved it was impossible to do this.

21. Hausdorff space-n topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the “Hausdorff condition” (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

22. homeomorphism-In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form.

23. Homotopy equivalent-Two topological spaces and are homotopy equivalent if there exist continuous maps and , such that the composition is homotopic to the identity on , and such that is homotopic to . Each of the maps and is called a homotopy equivalence, and is said to be a homotopy inverse to (and vice versa).

24. E8 lie group-

science.howstuffworks.com

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

25. homology-In mathematics, homology[1] is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

26. Metric space-In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.

The most familiar metric space is 3-dimensional Euclidean space. In fact, a “metric” is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.

27. Topological space-a space that has an associated family of subsets that constitute a topology. The relationships between members of the space are mathematically analogous to those between points in ordinary two- and three-dimensional space.

STATISTICS AND PROBABILITY

1. NULL HYPOTHESIS- In a statistical test, the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error.

2. CONFIDENCE INTERVAL-a range of values so defined that there is a specified probability that the value of a parameter lies within it.

3. P-SCORE- calculated probability of finding the observed, or more extreme, results when the null hypothesis of a study question is true.

4. T-SCORE- in psychometric testing are always positive, with a mean of 50. a difference of 10 (positive or negative) from the mean is a difference of one standard deviation. For example, a score of 70 is 2 standard deviations above the mean, while a score of 0 is 1 standard deviations below the mean.

5. INTERMEDIATE VALUE THEOREM-states that a continuous function, f, with an interval [a,b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

6. JULIA SET-a set of complex numbers that do not converge to any limit when a given mapping is repeatedly applied to them. In some cases the result is a connected fractal set.

7. POWER SET-any set s of any subset of s, including the empty set of s itself.

8. LINEAR REGRESSION-in statistics, an approach for modeling the relationship between a scalar dependent variable y and one or more independent variable denoted by x. In the case of one variable, we have a linear regression.

9. NORMAL DISTRIBUTION CURVE- (also called a gaussian bell curve) A bell shaped curve. The mean (average), median (the middle value of the distribution when the values are listed in numerical order), and there mode (the value that occurs most often), are all equal. The area under the normal curve is equal to 1.0. these curves are denser in their centers and less dense at the tails. Normal distributions are defined by the mean and the standard deviation (a calculated quantity that indicates the extent of deviation for a group as a whole).

10. KINDS OF NON-NORMAL DISTRIBUTIONS (NOTE-this list is not exhaustive)-

Exponential distribution- (example- bacteria growth)

When one variable increases, the dependent variable increase even more.

Beta distribution-is used to study variations in the percentage of something across

samples, such as what fraction of people spend time doing homework. It represents

a probability distribution of probabilities,, that is, it represents all the possible

values of a probability when we don’t know what the probability is.

Binomial-a distribution where a binomial random variable is the number of successes x

in n repeated trials of a binomial experiment. The probability distribution of a

binomial random variable is called a binomial distribution. the distribution is used

when there are exactly 2 mutually exclusive outcomes of a trial. It is one of the

most important distributions in probability theory.

Chi square distribution-a special case of the gamma distribution. A distribution with

degrees of freedom is in the distribution of a sum of the squares of k independent

standard random variables.

Gamma distribution-a 2 parameter family of continuous probability distributions. The

exponential and chi square distributions are special cases of this distribution. The

arrival times in the poisson process have gamma distributions, and the chi-square

distribution in statistics is a special case of the gamma distribution. This

distribution is widely used to model physical quantities that take positive values.

Inverse gamma distribution-a 2 parameter family of continuous distributions on the

positive real line, which is the reciprocal of a variable distributed according to the

gamma distribution. The chief use of the inverse gamma distribution is in

bayesian statistics

Log normal distribution-a continuous probability distribution of a random variable whose

logarithm is normally distributed.

Logistic distribution-a continuous probability distribution which is cumulative. Appears in

logistic regression and feedback neural networks, resembles normal distribution

but heavier tails

Maxwell-boltzmann distribution-the distribution is an excellent approximation of gasses

at ordinary temperatures that behave very nearly like an ideal gas.

Poisson distribution-a discrete probability distribution that expresses the probability of a

given number of events occurring in a fixed interval of time.

Skewed distribution-in a positively skewed distribution, the extreme scores are larger,

this the mean is larger than the median. a negatively skewed distribution is

asymmetrical and points in the negative direction, such as would result with a

very easy test.

Symmetric distribution-distribution in a situation in which the values of variable occur at

regular frequencies, and the mean, median, and mode occur at the same point,

unlike asymmetrical distribution, this one does not skew.

Uniform distribution-sometimes known as a rectangular distribution, it had a constant

probability. The probability density function and cumulative distribution function

are a continuous uniform distribution on the interval are.

Unimodal distribution-has clear peak or most frequent value. The values increase first,

rising to a single peak where they decrease. The mode in unimodal doesn’t refer

to the most frequent number in a data set- it refers to the local maximum in a

chart.

Weibull distribution-widely used lifetime of objects distribution in reliability engineering. It

is a reliable distribution that can take on characteristics of other distributions.

11. STANDARD DEVIATION-a quantity calculated to indicate the extent of deviation for a group as a whole.

12. BAYESIAN STATISTICS-a theory in statistics in which the evidence about the true star of the world is expressed in terms of degrees of belief known as bayesian probabilities. The probability is orderly opinion, and that inference from data is nothing other than the revision of such opinion in light of relevant new information.

13. LAW OF LARGE NUMBERS-principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes.

14. CENTAL LIMIT THEOREM- in statistical theory, given sufficiently large sample size from a population with finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

15. MONTE CARLO METHOD- a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle.

16. CORRELATION COEFFICIENT-a number between -1 and 1 calculated so as to represent the linear dependence of 2 variables or sets of data. It measures the strength and direction of the linear relationship between 2 variables that is defined as the (sample) covariance of the variables divided by the product of their (sample) standard deviations.

17. BAYE’S LAW-describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

18. BOREL-CANTELLI LEMMA- in probability theory, it its a result in measure theory. It states that ,under certain conditions, an event will have the probability of either 0 or 1.

19. FACTOR ANALYSIS- process in which the values of observed data are expressed as functions of a number of possible causes in order to find which are the most important.

20. BAYES’ THEOREM- describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

21. BUFFON’S NEEDLE- one of the oldest problems in the field of geometric probability. It states that is one draped a needle repeatedly onto a lined sheet of paper where the lines are equal to the length of the needle and counting the number of times the needle touched the line, we could obtain the value of pi (3.14…). We consider the number of drops and multiply by 2, then divide the number of times the needle had touched the line.

22. MARKOV CHAINS-a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained by the previous event.

MISCELLANEOUS

1. ZENO’S PARADOX-idea contrary to one’s senses, that change is mistaken, and particularly that motion is nothing but an illusion. This paradox is counterintuitive to continuous space.

2. CHAOS- behavior of dynamical systems that are highly sensitive to initial conditions.

3. GAME THEORY-the study of the mathematical models of conflict and cooperation between intelligent rational decision-makers. It is mainly used in economics, political science, psychology, logic, computer science, and biology. It originally addressed zero-sum games, in which one person’s gains resulted in losses for the other participants. It is the science of logical decision making in humans, animals, and computers.

4. PRISONER’S DILEMMA- a standard example of a game analyzed in game theory that shows why 2 completely rational people might not cooperate, even if it appears in their best interests to do so.

5. LEBESGUE MEASURE- a standard way of assigning a measure to subsets of n-dimensional euclidean space. for any subset A of R^n, we can define the outer measure LAMBDA(A) by : we then define the set A to be legesgue measurable if for every subset S of R^n, the lebesgue measurable sets form a lebesgue sigma-algebra and the lebesgue measure is defined by lambda(A)=LAMBDA(A) for any lebesgue measurable set A.

6. CLIFFORD’S theory- a theory in representation theory. It is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound.

7. INFINITE MONKEY THEOREM- if one had an unlimited number of monkeys and they typed on a typewriter with sufficient time, they could type all the books that have ever been written. Even if one monkey were to type with a sufficient amount of time, it could be dome.

8. HAUSDORF MEASURE-a type of outer measure that assigns a number [0, infinity] to each set R^n or, more generally, the metric space. the zero-dimensional hausdoorf measure is the number of points in the set (if the set is finite) or infinite if the set is infinite.

9. SYMMETRIZATION METHODS- algorithm of transforming a set A subset R to a ball B subset R with equal volume vol(B)=vol(A) and centered at the origin.

MATHEMATICAL PARADOXES

NEWCOMB’S PARADOX- a thought experiment involving a game between 2 players, one of whom purports to be able to predict the future. It is one of the most widely debated paradoxes.

VON NEUMANN PARADOX- the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area preserving affine transformation such that the result is 2 planar figures of the same size as the original.

NASH PARADOX in economic and game theory, it is a stable state of a system involving the interaction of different participants, in which no participant can gain unilateral change it the strategies of the others remain unchanged.

BIRTHDAY PARADOX-it concerns the probability that in a set of n randomly chosen people, some pairs will have the same birthday. The probability reaches 100% in 367 people, but there is a 99.9% probability that in 70 people, 2 people will have the same birthday. There is a 50% probability of 2 people having the same birthday with just a group of 23 people.

ST. PETERSBERG PARADOX- related to probability and decision theory in economics. It is based on a theoretical lottery game that leads to a random variable with infinite expected value (infinite expected payoff) but nevertheless seems to be worth only a very small amount to the participants.

ZENO’S PARADOX- the belief in plurality and change are mistaken, and in particular that motion is nothing but an illusion.

BERRY PARADOX- self-referential paradox arising from an expression like “the smallest positive integer not definable in fewer than 12 words.” It arises because of systematic ambiguity in the word ‘definable’.

BURALI-FORTI PARADOX- in set theory, demonstrates that constructing ‘the set of all ordinal numbers’ leads to a contradiction and therefore shows an antinomy in a system that allows its construction.

CANTOR’S PARADOX-there is no greatest cardinal number (a number denoting quantity)

GOODMAN’S PARADOX-that this is where the fundamental problem lies. Consider the evidence that all emeralds examined thus far have been green. This leads to the conclusion by induction that all future emeralds 3ill be green.

GRELLING-NELSON PARADOX-a semantic self-referential paradox concerning the applicability to itself of the word ‘heterological’. Meaning inapplicable to itself.

UNEXPECTED HANGING PARADOX-about a person’s expectations about the timing of a future event that he or she has been told will occur at an unexpected time. It has been applied to, for example, a surprise school test.

HEMPEL’S PARADOX-brought to light a central paradox in the scientific method as it is commonly understood. The problem is with inductive reasoning, and an example is, suppose you see a rave, and you note that it is black. The paradox arises from the question of what constitutes evidence for a statement

HILBERT’S. GRAND HOTEL PARADOX- illustrates a counterintuitive property of infinite sets. It demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate

additional guests, even infinitely many of them, and that this process may be repeated infinitely often.

SKOLEM’S PARADOX-involves a seeming conflict between 2 theorems from classical logic. The lowenhein-skolem theorem says that if a first-order theory has infinite models, it has models whose domains are only countable. Cantor’s theorem says that some sets are uncountable.

CONDORCET/VOTING PARDOX-in which collective references can be cyclical, even if the preferences of individual voters are not cyclic. it means that majority wishes can be in conflict with each other. When this occurs, it is because the conflicting majorities are each each made up of different groups of individuals.

PROBABILITY PARADOXES- are problems within the classical interpretation of probability theory. An example to show that probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined. Examples include the birthday party problem, the Simpson paradox, and the monte hall problem.

BANACH-TARSKI PARADOX- a theorem in set theoretic geometry, which states that: given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can be put back together in a different way to yield 2 identical copies of the original ball. The reassembly only involves moving the pieces around and rotating them, without changing the shape. The pieces are not solid in the usual sense, but infinite scattering of points. The reconstruction can work with ads few as 5 pieces. This paradox contradicts basic geometric intuition.

BARBER’S PARADOX- the barber is the one who shave all those, and only those, who do not shave themselves. The question- who shaves the barber? Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves.

PROSECUTOR’S FALLACY- a fallacy of statistical reasoning, typically used by the prosecution to argue for the guilt of a defendant during a criminal trial. Although it is named after prosecutors it is not specific to them, and some variants of the fallacy can be used by defense lawyers arguing for the innocence of their client. The basic fallacy results from misunderstanding conditional probability and neglecting the prior odds of a defendant being guilty before that evidence is introduced.

CARD PARADOX- a non-self-refential variant of the liar’s paradox. Suppose there is a card with the statements printed on both sides:

Front-the sentence on the other side of this card is true.

Back-the sentence on the other side of this card is false.

1) Trying to assign a truth to either of them leads to a paradox.

It the 1st statement is true, then so is the 2nd. But if the 2nd statement is true, then the 1st is false. It fallows that is the 1st statement is true, then the 1st statement is false.

2) if the 1st statement is false, the so is the 2nd. But the 2nd is false, then the 1st is true. It follows that if the 1st is false, then the 1st is true.

This is a case of circular reference.

SIMPSON’S PARDOX (yule-paradox effect/reversal paradox/amalgamation paradox)-a phenomenon in probability and statistics, in which a tend appears in different groups of data but disappears or reverses when these groups are combined.

STRANGE LOOPS- may involve self-reference and paradox. a tangled hierarchy is a hierarchy consciousness system in which a strange loop appears.

CURRY’S PARADOX- occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-refering sentence and some apparently innocuous logical deduction rules.

COASTLINE PARADOX- is a counterintuitive observation that the coastline of a landmass does not have a well-defined coastline. This results from the fractal-like properties of coastlines.

PARADOX OF THE HEAP- arises from vague predicates. A typical formulation involves a heap of sand, from which grains are individually removed. under the assumption that removing a single grain does not turn a heap into a non-heap, the paradox is to consider what happens when the process id repeated enough times: is a single remaining grain still a heap? If not, when did it change from a heap into a non-heap?

GAMBLER’S FALLACY/Monte Carlo FALLACY/ (the fallacy of the maturity of chances)-the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, to that, if it happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature).this fallacy is seen commonly with gamblers.

MONTE HALL PROBLEM- suppose you are on a game shoe, and you are given the choice of 3 doors. Behind one is a car, and behind the others goats. You chose a door, say #1, and the host, who knows what is behind the doors, opens another door, say door #3, which has a goat. 5hen he says to you,’do you want to pick door #2? Is it to your advantage to switch your choice?

The standard assumption, say that if you switch, you have a 2/3 chance of winning the car, while the contestant who sticks to their initial choice has 1/3 chance. the best course of action is to switch doors. The problem is a paradox of the vertical type, because the correct result (you should switch doors) is so counter intuitive it can seem absurd, but it is nevertheless demonstrably true that you should switch door for the best possible outcome gives the player a 2/3 probability of winning the car.

100 PRISONER’S PROBLEM-this problem is mathematically equivalent to the monte hall problem with the car and goats being replaced with freedom and execution, and also ease on bertrand’s box paradox.

BERKSON’S PARADOX-is a result in conditional probability and statistics which is counterintuitive for some people. it is a complicating factor arising in statistical tests of proportions.

BERTRAND’S PARADOX IN PROBABILITY-it goes as follows: consider an equilateral triangle inscribed by a circle. What is the probability that the cord is longer than a side of the triangle? Bertrand gave 3 arguments, all apparently valid, yet yielding different results.

BERTRAND’S BOX PARADOX-a paradox in probability theory. there are 3 boxes:1. a box containing 2 gold coins, 2. a box containing 2 silver coins, 3. a box containing 1 gold and 1 silver coin. the paradox is in probability, after choosing a box at random and taking 1 coin at random, if that happens to be a gold coin, of the next drawn from the same box also being a gold coin. their solutions illustrate some basic principles, including the kolmogorov axioms.

BOREL-KOLMOGOROV PARADOX-in probability theory, is a paradox relating to conditional probability with resect to an event of probability zero.

BOY OR GIRL PARADOX-a set of questions in probability theory-

mr. jones has 2 children. the older is a girl. What is the probability that both are girls?

Mr. Jones has 2 children. At least one of them is a boy. What is the probability both are boys?

The ambiguity, depends on the exact wording and possible assumptions.

ELISBERG PARADOX-based in decision theory in which people’s choices violate there postulates of subjective expected utility. it is generally taken to be evidence for ambiguity aversion.

EXCHANGE PARADOX-2 players receive envelopes containing different amounts of money. the assignment of amounts ensures each player has the same probability of receiving each possible amount. However, if they swap envelops, there is an expectation of gain. this is a paradox in symmetry.

LITTLEWOOD’S PARADOX-abstract math and logic problem designed to illustrate the seemingly paradoxical, or at least non-intuitive, nature of infinity.

LITTLEWOOD’S LAW-says that a person can expect to experience an event with odds of one in a million at the rate of about one per month.

NECKTIE PARADOX-a variation of the 2 envelope paradox.

NONTRANSITIVE DICE-with an unusual set of dice, a 2 player game where you always have the advantage. you can even teach your opponent how the game works, yet you still win. You finally describe the game to 3 players where you can potentially beat them both at the same time.

SIEGAL’S PARADOX-a way of investing in foreign investments to make money. the technique works mathematically, but it is called a paradox because it sounds as if it should not.

SLEEPING BEAUTY PROBLEM-a puzzle in decision theory in which an ideally rational epistemic agent is to be woken once to twice according to the toss of a coin, and asked her belief the coin having turned up heads.

ST. PETERSBURG PARADOX-a paradox related to probability and decision theory in economics.

BURALI-FORTI PARADOX-in set theory, demonstrates that constructing the set of al ordinal numbers do not, unlike the natural numbers, form a set.

MORE COMPLETE LIST OF PARADOXES

Logic

• Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved leads to paradoxical consequences. Not to be confused with the Barber paradox.

• What the Tortoise Said to Achilles: “Whatever Logic is good enough to tell me is worth writing down…”, also known as Carroll’s paradox, not to be confused with the “Achilles and the tortoise” paradox by Zeno of Elea.

• Catch-22: A situation in which someone is in need of something that can only be had by not being in need of it. A soldier who wants to be declared insane in order to avoid combat is deemed not insane for that very reason, and will therefore not be declared insane.

• Drinker paradox: In any pub there is a customer of whom it is true to say: if that customer drinks, everybody in the pub drinks.

• Paradox of entailment: Inconsistent premises always make an argument valid.

• Lottery paradox: If there is one winning ticket in a large lottery, it is reasonable to believe of any particular lottery ticket that it is not the winning ticket, but it is not reasonable to believe that no lottery ticket will win.

• Raven paradox: (or Hempel’s Ravens): Observing a green apple increases the likelihood of all ravens being black.

• Ross’ paradox: Disjunction introduction poses a problem for imperative inference by seemingly permitting arbitrary imperatives to be inferred.

• Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a surprise. The surprise examination and Bottle Imp paradox use similar logic.

Self-reference

These paradoxes have in common a contradiction arising from either self-reference or circular reference, in which several statements refer to each other in a way that following some of the references leads back to the starting point.

• Barber paradox: A barber (who is a man) shaves all and only those men who do not shave themselves. Does he shave himself? (Russell’s popularization of his set theoretic paradox.)

• Bhartrhari’s paradox: The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.

• Berry paradox: The phrase “the first number not nameable in under ten words” appears to name it in nine words.

• Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?

• Paradox of the Court: A law student agrees to pay his teacher after (and only after) winning his first case. The teacher then sues the student (who has not yet won a case) for payment.

• Curry’s paradox: “If this sentence is true, then Santa Claus exists.”

• Epimenides paradox: A Cretan says: “All Cretans are liars”. This paradox works in mainly the same way as the Liar paradox.

• Grelling–Nelson paradox: Is the word “heterological”, meaning “not applicable to itself”, a heterological word? (Another close relative of Russell’s paradox.)

• Kleene–Rosser paradox: By formulating an equivalent to Richard’s paradox, untyped lambda calculus is shown to be inconsistent.

• Liar paradox: “This sentence is false.” This is the canonical self-referential paradox. Also “Is the answer to this question ‘no’?”, and “I’m lying.”

• Card paradox: “The next statement is true. The previous statement is false.” A variant of the liar paradox that does not use self-reference.

• Pinocchio paradox: What would happen if Pinocchio said “My nose will be growing”?[1]

• Quine’s paradox: “‘Yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation.” Shows that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals.

• Yablo’s paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are false. While constructed to avoid self-reference, there is no consensus whether it relies on self-reference or not.

• Opposite Day: “It is opposite day today.” Therefore, it is not opposite day, but if you say it is a normal day it would be considered a normal day.

• Petronius’ paradox: “Moderation in all things, including moderation” (unsourced quotation sometimes attributed to Petronius).

• Richard’s paradox: We appear to be able to use simple English to define a decimal expansion in a way that is self-contradictory.

• Russell’s paradox: Does the set of all those sets that do not contain themselves contain itself?

• Socratic paradox: “All I know is that I know nothing.”

Vagueness

• Ship of Theseus: It seems like you can replace any component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship you began with.

• Sorites paradox (also known as the paradox of the heap): If you remove a single grain of sand from a heap, you still have a heap. Keep removing single grains, and the heap will disappear. Can a single grain of sand make the difference between heap and non-heap?

Mathematics

• All horses are the same color: A proof by induction that all horses have the same color.

• Ant on a rubber rope: An ant crawling on a rubber rope can reach the end even when the rope stretches much faster than the ant can crawl.

• Cramer’s paradox: The number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.

• Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured in the middle of the building and being disassembled on the roof and basement.

• Interesting number paradox: The first number that can be considered “dull” rather than “interesting” becomes interesting because of that fact.

• Potato paradox: If you let potatoes consisting of 99% water dry so that they are 98% water, they lose 50% of their weight.

• Russell’s paradox: Does the set of all those sets that do not contain themselves contain itself?

Statistics

• Abelson’s paradox: Effect size may not be indicative of practical meaning.

• Accuracy paradox: Predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy.

• Berkson’s paradox: A complicating factor arising in statistical tests of proportions.

• Freedman’s paradox: Describes a problem in model selection where predictor variables with no explanatory power can appear artificially important.

• Friendship paradox: For almost everyone, their friends have more friends than they do.

• Inspection paradox: Why one will wait longer for a bus than one should.

• Lindley’s paradox: Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to false but highly statistically significant results.

• Low birth weight paradox: Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality rate than other low birth weight babies. This is a special case of Simpson’s paradox.

• Simpson’s paradox, or the Yule–Simpson effect: A trend that appears in different groups of data disappears when these groups are combined, and the reverse trend appears for the aggregate data.

• Will Rogers phenomenon: The mathematical concept of an average, whether defined as the mean or median, leads to apparently paradoxical results—for example, it is possible that moving an entry from an encyclopedia to a dictionary would increase the average entry length on both books.

Probability

The Monty Hall problem: which door do you choose?

• Bertrand’s box paradox: A paradox of conditional probability closely related to the Boy or Girl paradox.

• Bertrand’s paradox: Different common-sense definitions of randomness give quite different results.

• Birthday paradox: What is the chance that two people in a room have the same birthday?

• Borel’s paradox: Conditional probability density functions are not invariant under coordinate transformations.

• Boy or Girl paradox: A two-child family has at least one boy. What is the probability that it has a girl?

• Dartboard Puzzle: If a dart is guaranteed to hit a dartboard and the probability of hitting a specific point is positive, adding the infinitely many positive chances yields infinity, but the chance of hitting the dartboard is one. If the probability of hitting each point is zero, the probability of hitting anywhere on the dartboard is zero.[2]

• False positive paradox: A test that is accurate the vast majority of the time could show you have a disease, but the probability that you actually have it could still be tiny.

• Grice’s paradox: Shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination.

• Monty Hall problem: An unintuitive consequence of conditional probability.

• Necktie paradox: A wager between two people seems to favour them both. Very similar in essence to the Two-envelope paradox.

• Nontransitive dice: You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B is likely to win in a roll against C, and C is likely to win in a roll against A.

• Proebsting’s paradox: The Kelly criterion is an often optimal strategy for maximizing profit in the long run. Proebsting’s paradox apparently shows that the Kelly criterion can lead to ruin.

• Sleeping Beauty problem: A probability problem that can be correctly answered as one half or one third depending on how the question is approached.

• Three cards problem: When pulling a random card, how do you determine the color of the underside?

• Three Prisoners problem: A variation of the Monty Hall problem.

• Two-envelope paradox: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.

Infinity and infinitesimals

• Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number that is smaller than itself.

• Cantor’s paradox: The set of all sets would have its own power set as a subset, therefore its cardinality would be at least as great as that of its power set. But Cantor’s theorem proves that power sets are strictly greater than the sets they are constructed from. Consequently, the set of all sets would contain a subset greater than itself.

• Galileo’s paradox: Though most numbers are not squares, there are no more numbers than squares. (See also Cantor’s diagonal argument)

• Hilbert’s paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.

• Russell’s paradox: Does the set of all those sets that do not contain themselves contain itself?

• Skolem’s paradox: Countably infinite models of set theory contain uncountably infinite sets.

• Zeno’s paradoxes: “You will never reach point B from point A as you must always get half-way there, and half of the half, and half of that half, and so on.” (This is also a physical paradox.)

• Supertasks may result in paradoxes such as

• Benardete’s paradox: Apparently, a man can be “forced to stay where he is by the mere unfulfilled intentions of the gods”.

• Grandi’s series: The sum of 1-1+1-1+1-1… can be either one, zero, or one-half.

• Ross–Littlewood paradox: After alternately adding and removing balls to a vase infinitely often, how many balls remain?

• Thomson’s lamp: After flicking a lamp on and off infinitely often, is it on or off?

Geometry and topology

The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original.

• Banach–Tarski paradox: Cut a ball into a finite number of pieces and re-assemble the pieces to get two balls, each of equal size to the first. The von Neumann paradox is a two-dimensional analogue.

• Paradoxical set: A set that can be partitioned into two sets, each of which is equivalent to the original.

• Coastline paradox: the perimeter of a landmass is in general ill-defined.

• Coin rotation paradox: a coin rotating along the edge of an identical coin will make a full revolution after traversing only half of the stationary coin’s circumference.

• Gabriel’s Horn: or Torricelli’s trumpet: A simple object with finite volume but infinite surface area. Also, the Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving a finite distance along that boundary, which also implies that in a sense you go no further if you walk “the wrong way” around the set to reach a nearby point). This can be represented by a Klein bottle.

• Hausdorff paradox: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two copies of itself.

• Nikodym set: A set contained in and with the same Lebesgue measure as the unit square, yet for every one of its points there is a straight line intersecting the Nikodym set only in that point.

• Sphere eversion: A sphere can, topologically, be turned inside out.

Decision theory

• Abilene paradox: People can make decisions based not on what they actually want to do, but on what they think that other people want to do, with the result that everybody decides to do something that nobody really wants to do, but only what they thought that everybody else wanted to do.

• Apportionment paradox: Some systems of apportioning representation can have unintuitive results due to rounding

• Alabama paradox: Increasing the total number of seats might shrink one block’s seats.

• New states paradox: Adding a new state or voting block might increase the number of votes of another.

• Population paradox: A fast-growing state can lose votes to a slow-growing state.

• Arrow’s paradox: Given more than two choices, no system can have all the attributes of an ideal voting system at once.

• Buridan’s ass: How can a rational choice be made between two outcomes of equal value?

• Chainstore paradox: Even those who know better play the so-called chain store game in an irrational manner.

• Decision-making paradox: Selecting the best decision-making method is a decision problem in itself.

• Fenno’s paradox: The belief that people generally disapprove of the United States Congress as a whole, but support the Congressman from their own Congressional district.

• Fredkin’s paradox: The more similar two choices are, the more time a decision-making agent spends on deciding.

• Green paradox: Policies intending to reduce future CO2 emissions may lead to increased emissions in the present.

• Hedgehog’s dilemma: or Lover’s paradox Despite goodwill, human intimacy cannot occur without substantial mutual harm.

• Inventor’s paradox: It is easier to solve a more general problem that covers the specifics of the sought-after solution.

• Kavka’s toxin puzzle: Can one intend to drink the non-deadly toxin, if the intention is the only thing needed to get the reward?

• Morton’s fork: Choosing between unpalatable alternatives.

• Navigation paradox: Increased navigational precision may result in increased collision risk.

• Newcomb’s paradox: How do you play a game against an omniscient opponent?

• Paradox of tolerance: Should one tolerate intolerance if intolerance would destroy the possibility of tolerance?

• Paradox of voting: Also known as the Downs paradox. For a rational, self-interested voter the costs of voting will normally exceed the expected benefits, so why do people keep voting?

• Parrondo’s paradox: It is possible to play two losing games alternately to eventually win.

• Prevention paradox: For one person to benefit, many people have to change their behavior — even though they receive no benefit, or even suffer, from the change.

• Prisoner’s dilemma: Two people might not cooperate even if it is in both their best interests to do so.

• Voting paradox: Also known as Condorcet’s paradox and paradox of voting. A group of separately rational individuals may have preferences that are irrational in the aggregate.

• Willpower paradox: Those who kept their minds open were more goal-directed and more motivated than those who declared their objective to themselves.

MAJOR MATH PROBLEMS THAT HAVE BEEN SOLVED

A. Godel’s incompleteness theorem- two theorems of mathematical logic that

demonstrate the inherent limitations of every formal axiomatic system

containing basic arithmetic. the theorems show that finding a complete and

consistent set of axioms for all of mathematics is impossible.These results

were proved by Kurt model in 1931.

B. Fermat’s Last Theorem- states that there no three positive integers a, b, and c

satisfy the equation a^n + b^n=c^n for any integer value of n greater than 2.

This problem, was not solved for 350 years until 1994 by Andrew wiles.

C. Prime number theorem- describes the asymptotic distribution of the prime

numbers. The formula for finding the approximate number of primes numbers

up to any number is pi(n)=n x Ln(n). pi(n) is the number within which the total

number of primes that will result, and n=total number numbers of numbers in

which we will be searching for prime numbers. This problem was proved

independently by jacques Hadamard and Charles jean de la poussin in 1896.

D. Continuum hypothesis- this is a hypothesis about the sizes of infinite sets. It

states that no set whose cardinality, the number of elements of the set, is

strictly between that of the integers and the real numbers (continuum). There

is no infinite set between the small infinite set of integers and the large infinite

set of real numbers. Georg cantor advanced this theorem in 1878.

E. Four color problem-give any separate of a plane into contacting regions,

producing a figure called a map, no more than 4 colors are required to color

the regions of the map so that no 2 adjacent regions have the same color.

This problem was proved in 1976 by Kenneth Appel and Wolfgang haken

using a computer.

F. Polyps conjecture-in number theory, it states that most (more than 50%) of the

natural numbers less than any given number have an odd number of prime

factors. It was conjectured in 1919 by George poly and proved false in 1958

bt c. Brian haselegrove.

G. Bierberbach conjecture-in complex analysis, this is a theorem that gives a

necessary condition on a holomorphic function in order for it to map the open

unit disk of the complex plane invectively to a complex plane. It was posed by

Ludwig bierberbach in 1916 and proven by Louis brings in 1985.

H. flying problem-the problem is: a fly, moving at 40 mph, zigzags back and forth

between 2 trains, each moving at 20 mph. If the trains are 100 miles apart

initially, how much distance will the fly travel? John von Neumann solved this

problem immediately by summing the series. the answer is 150 miles.

I. Kepler’s conjecture-posed by Johannes kepler in 1611, it is now a mathematical

theorem proved in 1998 by Thomas hales. The problem is about sphere

packing in 3 dimensional Euclidean space. It says that no arrangement of

equally sized spheres filling space has a greater average density than that of

the cubic close packing (face centered cubic) and hexagonal close packing.

The density of these arrangements is around 74.05%.

J. Taniyama-shimura-weil conjecture-states that elliptical curves over the field of

rationalnumbers are related to modular forms. This was proved partially by

Andrew wiles and fully in 2001 by Christophe breuli, Brian Conrad, Fred

diamond, and richard taylor. The conjecture was posed by Robert langlands

and a preliminary version of it by yutaka taniyama in 1956.

K. honeycomb conjecture-states that a regular hexagonal grid or honeycomb is

the best way to divide a surface into regions of equal area with the least total

perimeter. This conjecture was proven in 1999 by mathematician Thomas c.

hales.

L. poincare conjecture-a theorem about the characterization of the 3-sphere,

which is the hypersphere that bounds the unit ball in 4-dimensional space.

This problem was posed by Henri poincare in 1900 and proven by Grigori

Perelman in November 2002.

M. catalans conjecture-posed in 1844 by Eugene Charles Catalan, it says that

2^3 and 3^2 are 2 power of natural numbers, whose values 8 and 9

respectively are consecutive. The theorem states that this is the only case of

3 consecutive powers. That is to say, the only solutions in the natural numbers

of x^a-y^b=1 for a, b>1, x, y>0 is x=3, a=2, y=2, b=3. This conjecture was

proven in 2002 by Preda mihailescu.

N. strong perfect graph theorem-conjectured in 1961 by Claude Berge, it deals

with graph theory. It states that a forbidden graph characterization of the

perfect graphs as being exactly the graphs that have neither odd holes (odd

length induced cycles) nor odd anti holes (complements of odd holes). It was

proven in 2006 by Maria chudnovsky, Neil Robertson, Paul Seymour, and

robin Thomas.

O. Hilbert problems- 23 problems posed between 1900 and 1902 by Davis

hilbert.

1. 1st problem about the continuum hypothesis solved. (See #11-D

above)

2. 2nd problem-Proved if the axioms of arithmetic are consistent. (See

solved Hilbert

problems)

3. 3rd problem-Given any 2 polyhedra of equal volume, is it always

possible to cut the 1st into finitely many polyhedral pieces that can be

reassembled to yield the 2nd? This problem was resolved and the

answer is no. It was proved using Dehn invariants.

4. 5th problem-are continuous groups automatically differential groups?

Resolved by Andrew season and it depends on how the original

statement is interpreted. If however, it is understood as an equivalent

of the halbert-smith conjecture, it is still unsolved.

5. 7th problem-is a^b transcendental, for algebraic a not equal to 0,1

and irrational algebraic b? Resolved and the answer is yes,

illustrated by gelfond’s theorem or the gelfond-schneider theorem.

6. 10th problem-find an algorithm to determine whether a given

Polynomial Diophantine equation with integer coefficients has an

integer solution. Result was that it is impossible by the matiyasevich

theorem (1970) which says there is no such algorithm. This result

was the work of Martin Davis, Yuri matiyasevich, Hilary Putnam, and

Julia robinson.

7. 14th problem-is the ring of invariants of an algebraic group acting on

a polynomial ring always finitely generated? This problem was

resolved and the answer is no. Proved in 1954 by oscar zariski and

a counterexample found in 1959 by masayoshi nagata.

8. 17th problem-express a nonnegative rational function as quotient of

sums of squares. This was resolved as a yes due to Emil artin in

1927. An upper limit was established for the number of square terms

necessary.

9. 18th problem- resolved (see Hilbert solved problems)

10. 19th problem-Are the solutions of regular problems in the calculus

of variations always necessarily analytic? Resolved with a yes and

proven by ennuo de Giorgi (1957) and John Nash (1958).

11. 20th problem- do all variational problems with certain boundary

solutions? This problem was resolved and culminated in solutions

for the non-linear case. This was a significant topic of research

throughout the 20th century.

12. 22nd problems.-uniformization of analytic relations by means of

automorphic functions. resolved.

13. Squaring a circle using only compass and straight edge.

THE MOST MAJOR UNSOLVED PROBLEMS

1. Reimann’s hypothesis-the real part of any non-trivial zero of the Riemann zeta function is 1/2.

2. Goldbach conjecture-every even integer greater than 2 can be expressed as the sum of 2 primes.

3. N-body problem-predicting the individual motions of a group of celestial objects interacting with each other gravitationally.

MILLENNIUM PRIZE UNSOLVED PROBLEMS

(NOTE-problem #8 was solved)

1. P vs PN problem-asked whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.

2. Reimann’s hypothesis-the real part of any non-trivial zero of the Riemann zeta function is 1/2.

3. Hodge conjecture-says that certain de Tham Cohomology classes are algebraic, that is, they are sums of poincare duals of the homology classes of subvarieties.

4. Yang-mills existence and mass gap-prove that for any compact simple gauge group G, a non-trivial Yang-Mills theory exists on R^4 and has a mass gap (triangle)>0.

5. Navier-stokes existence and smoothness-prove or give a counter-example of: in 3 space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes equation.

6. Birch and swinnerton-dyer conjecture-describes the set of rational solutions to equations defining an elliptic curve.

7. poincare conjecture-a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in 4-dimensional space. This problem was posed by Henri poincare in 1900 and proven by Grigori Perelman in November 2002.

HILBERT’S 23 PROBLEMS

(NOTE-several of hilbert’s solved problems listed here.)

1. Continuum hypothesis- (SOLVED)

this is a hypothesis about the sizes of infinite sets. It states that no set whose cardinality, the number of elements of the set, is strictly between that of the integers and the real numbers (continuum). There is no infinite set between the small infinite set of integers and the large infinite set of real numbers. Georg cantor advanced this theorem in 1878. This problem was proven to be impossible to prove or disprove within zermelo-fraenkel set theory with or without the axiom of choice (provided that zermelo-fraenkel set theory is consistent, I.e. it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.

2. Prove that the axioms of arithmetic are consistent- (SOLVED)

Godel’s incompleteness theorem- two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. the theorems show that finding a complete and consistent set of axioms for all of mathematics is impossible.These results were proved by Kurt model in 1931.

There is no consensus on whether results of model and gentian give a solution to the problem. Godless 2nd incompleteness theorem showed that no proof of its consistency can be carried out within arithmetic itself. Gentian proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal e(0).

3. Given any 2 polyhedra of equal volume, is it always possible to cut the 1st into finitely

many polyhedral pieces that can be reassembled to yield the 2nd? (SOLVED)

This problem was resolved and the answer is no. It was proved using Dehn invariants.

4. Construct all metrics where lines are geodesics.

5. Are continuous groups automatically differential groups? (SOLVED-see solved hilbert problems

above)

6. Mathematical treatment of the axioms of physics.

7. Is a^b transcendental, for algebraic a not equal to 0,1 and irrational algebraic b? (SOLVED)

This problem was resolved and the result was yes. It is illustrated by gelfond’s theorem or the gelfond-schneider theorem.

8. Riemann hypothesis-

is that all the nontrivial zeros of the analytic continuation of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have far reaching implications in number theory, especially the distribution of the prime numbers. The official statement of this problem was given by Enrico bombieri.

9. Find the most general law of the reciprocity theorem in any algebraic number field.

10. Find an algorithm to determine whether a given polynomial Diophantine equation

with integer coefficients has an integer solution. (SOLVED-see solved hilbert problems)

11. Solving quadratic forms with algebraic numerical coefficient.

12. Extend the kronecker-weber theorem on abelian extensions of the rational numbers

to any base number field.

13. Solve 7-th degree equation using algebraic (variant:continuous) functions of 2

parameters.

14. Is the ring of invariants of an algebraic group acting on a polynomial ring always

finitely generated? (SOLVED-see solved hilbert problems)

15. Rigorous foundation of Schubert’s enumerative calculus.

16. Describe relative positions of ovals from a real algebraic curve and as limit cycles of

a polynomial vector field on the plane.

17. Express a nonnegative rational function as quotient of sums of squares. (SOLVED-see solved

hilbert problems)

18. A. Is there a polyhedron that admits only an anisohedral tiling in 3 dimensions? B. What is the

densest sphere packing? (SOLVED-see solved hillier problems)

19. Are the solutions of regular problems in the calculus of variations always necessarily

analytic? (SOLVED-see solved hilbert problems)

20.. Do all variational problems with certain boundary conditions have solutions?

(SOLVED-see solved hilbert problems)

21. Proof of the existence of linear differential equations having a prescribed

monochromic group.

22. Uniformization of analytic relations by means of automorphic functions.

(SOLVED-see solved hilbert problems)

23. Further development of the calculus of variations.

24. There was this problem in proof theory, on the criterion for simplicity and general

methods.

SMALE’S PROBLEMS

1. riemann hypothesis-

is that all the nontrivial zeros of the analytic continuation of the Riemann zeta function have a real part of 1/2. A proof or disproof of this would have far reaching implications in number theory, especially the distribution of the prime numbers. The official statement of this problem was given by Enrico bombieri.

2. poincare conjecture

3. Does p=np?

4. Shub-smale t-conjecture on the integer zeros of a polynomial of 1 variable.

5. Height bounds for diophantine curves

6. Finiteness of the number of relative equilibrium in celestial mechanics

7. Distribution of points on the 2-sphere

8. Extend the mathematical model of equilibrium theory to include price adjustments

9. The linear programming problem: find a strongly-polynomial time algorithm which for

given matrix a (a member of) R(m x n) and b (a member of) R(m) decides whether

there exists x (a member of) R(n) with Ax>=b.

10. Pug’s closing lemma (higher order of smoothness)

11. Is 1-dimensional dynamics generally hyperbolic?

12. Centralizers of diffeomorphisms

13. Halbert’s 16th problem

14. Lorenz attractor

15. Do the navies-stokes equations in R^3 always have a unique smooth solution that

extends for all time?

16. Jacobean conjecture

17. Solving polynomial equations in polynomial time in the average case

18. Limits of intelligence (it talks about the fundamental problems of intelligence and

learning, both from human and machine side)

LANDAU’S PROBLEMS

1. Goldbach’s conjecture- can every even integer greater than 2 be written as the sum of 2 primes?

2. Twin prime conjecture-are there infinitely many primes p such that p+2 is prime?

3. legendre’s conjecture—does there always exist at least 1 prime between

consecutive perfect

squares?

4. Are there infinitely many primes p such that p-1 are perfect squares?

(Are there infinitely many primes of the form n^2+1?)

UNSOLVED MATHEMATICAL PROBLEMS

ALGEBRA—

hadamard conjecture-concerns existence. the conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k.

2. In algebraic topology, it staes that if a group G has cohomological dimension2, then it has a 2-dimensional ellenberg-maclane space K(G,1). For n different from 2, a group G of cohomological dimension n has an n-dimensional ellenberg-maclane space.

3. Regarding galois cohomology of a simple connected semi simple algebraic group, the conjecture is that if G is such a group over a perfect field F of cohomological dimension at most 2, then the galois cohomology set h((F,G) is zero.

ALGEBRAIC GEOMETRY—

bass conjecture- says that certain algebraic k-groups are supposed to be infinitely generated.

2. Delight conjecture on special values of L-functions is a formulation of the hope for algebraically of L(n) where L is an L-function and n is an integer in some set depending on L.

ANALYSIS—

Schanuel’s conjecture in transcendental number theory concerning the transcendence degree of certain field extensions of the rational numbers. the conjecture— given any n complex numbers z1,…zn, that are linearly independent over the rational numbers Q, the extension field Q(z1,…zn,e^z1,…e^zn) has transcendence degree of at least n over.

COMBINATORICS—-

1.frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of an underlying space) belonging to half or more of the sets.

2. finding a function to model n-step self-avoiding walks.

DIFFERENTIAL GEOMETRY—

Hopf conjecture- a compact, even-dimensional rie=mannian manifold with positive curvature has positive Euler characteristics.

DISCRETE GEOMETRY—

1. How many units distance can be determined by a set of n points in the euclidean plane?

EUCLIDEAN GEOMETRY—

1.inscribed square problem-does every jordan curve have an inscribed square?

DYNAMICAL SYSTEMS—

1. Mlc conjecture-is the Mandelbrot set locally connected?

GRAPH THEORY—

1. Barnette’s conjecture that every cubic bipartite 3-connected planar graph has a hamiltonian cycle.

GROUP THEORY—

1. Is every finitely presented periodic group finite?

MODEL THEORY—

1. Determine the structure of keister’s order.

NUMBER THEORY—

1. Are there an infinitely many twin primes (twin prime conjecture)? (Primes either 2 more or 2 less that another prime)

2. Are there infinitely many primes in the form n^2+1?

3. Legendre’s conjecture-does there always exist at least 1 prime between consecutive squares?

4. Are there infinitely many primes in the form n^2+1?

5. Are there infinitely many cousin primes?-(prime numbers that differ by 4)

6. Are there infinitely many sexy primes?-(primes that differ by 6)

7. Are there infinitely many fibonacci primes?-(Fibonacci numbers is a series in which each number is the sum of the 2 preceding numbers.)

8. Are there infinitely many Sophie Germain primes?-(numbers of the form. 2p+1)

9. Legendre’s conjecture-does there always exist at least 1 prime between consecutive squares?

10. Are there an infinite number of Mersenne primes? (Primes of the form Prime=2^n-1)

11. Are there infinitely many perfect numbers?-(a perfect number is a positive integer that is equal to the sum of its proper divisors.)

12. Are there infinitely many perfect numbers?-(a perfect number is a positive integer that is equal to the sum of its proper divisors.)

13. Do any odd perfect numbers exist?-(a perfect integer number is that equals the sum of its proper divisors.)

14. Are there infinitely many real quadratic number fields with unique factorization (class number problem)?

PARTIAL DIFFERENTIAL EQUATIONS—

1. Regularity solutions of ruler’s equations.

RAMSEY THEORY—

1. the values of the Ramsey numbers, particularly R(5,5)

SET THEORY—

1. the problem of finding the ultimate core model, one that contains all large cardinals.

2. Woodlin’s omega-hypothesis.

A FEW OTHER UNSOLVED MATH PROBLEMS—-

1. Borel conjecture

2. School conjecture

3. Zeeman conjecture

4. Erdos-ulam problem

COMPLETE LIST OF UNSOLVED MATHEMATICAL PROBLEMS

Algebra

• Homological conjectures in commutative algebra

• Hilbert’s sixteenth problem

• Hilbert’s fifteenth problem

• Hadamard conjecture

• Jacobson’s conjecture

• Existence of perfect cuboids and associated cuboid conjectures

• Zauner’s conjecture: existence of SIC-POVMs in all dimensions

• Wild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems

• Köthe conjecture

• Birch–Tate conjecture

• Serre’s conjecture II

• Bombieri–Lang conjecture

• Farrell–Jones conjecture

• Bost conjecture

• Uniformity conjecture

• Kaplansky’s conjecture

• Kummer–Vandiver conjecture

• Serre’s multiplicity conjectures

• Pierce–Birkhoff conjecture

• Eilenberg–Ganea conjecture

• Green’s conjecture

• Grothendieck–Katz p-curvature conjecture

• Sendov’s conjecture

Algebraic geometry

• Bass conjecture

• Deligne conjecture

• Fröberg conjecture

• Fujita conjecture

• Hartshorne conjectures

• The Jacobian conjecture

• Manin conjecture

• Nakai conjecture

• Resolution of singularities in characteristic p

• Standard conjectures on algebraic cycles

• Section conjecture

• Tate conjecture

• Virasoro conjecture

• Zariski multiplicity conjecture

Analysis

• Schanuel’s conjecture and four exponentials conjecture

• Lehmer’s conjecture

• Pompeiu problem

• Are γ
(the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan’s constant or Khinchin’s constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[13][14][15]

• Khabibullin’s conjecture on integral inequalities

• Hilbert’s thirteenth problem

• Vitushkin’s conjecture

Combinatorics

• Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets[16]

• The lonely runner conjecture: if k+1
runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be at least a distance 1/(k+1)
from each other runner) at some time?[17]

• Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?[18]

• Finding a function to model n-step self-avoiding walks.[19]

• The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?[20]

• The uniqueness conjecture for Markov numbers[21]

• Give a combinatorial interpretation of the Kronecker coefficients.[22]

Differential geometry

• Filling area conjecture

• Hopf conjecture

Discrete geometry

• Solving the happy ending problem for arbitrary n
[23]

• Finding matching upper and lower bounds for k-sets and halving lines[24]

• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[25]

• The Kobon triangle problem on triangles in line arrangements[26]

• The McMullen problem on projectively transforming sets of points into convex position[27]

• Ulam’s packing conjecture about the identity of the worst-packing convex solid[28]

• Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[29]

• How many unit distances can be determined by a set of n points in the Euclidean plane?[30]

Euclidean geometry

• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?[31]

• Inscribed square problem – does every Jordan curve have an inscribed square?[32]

• Kakeya conjecture

• Moser’s worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?[33]

• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[34]

• Shephard’s problem (a.k.a. Dürer’s conjecture) – does every convex polyhedron have a net?[35]

• The Thomson problem – what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?

• Falconer’s conjecture

• g-conjecture

• Circle packing in an equilateral triangle

• Circle packing in an isosceles right triangle

• Lebesgue’s universal covering problem – what is the convex shape in the plane of least area which provides an isometric cover for any shape of diameter one?

• Bellman’s lost in a forest problem – for a given shape of forest find the shortest escape path which will intersect the edge of the forest at some point for any given starting point and direction inside the forest.

• Find the complete set of uniform 5-polytopes[36]

• Covering problem of Rado

• The strong bellows conjecture – must the Dehn invariant of a self-intersection free flexible polyhedron stay constant as it flexes?

Dynamical systems

• Collatz conjecture (3n + 1 conjecture)

• Lyapunov’s second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?

• Furstenberg conjecture – Is every invariant and ergodic measure for the ×2,×3
action on the circle either Lebesgue or atomic?

• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups

• MLC conjecture – Is the Mandelbrot set locally connected?

• Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?

• Is every reversible cellular automaton in three or more dimensions locally reversible?[37]

• Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.

Graph theory

Paths and cycles in graphs

• Barnette’s conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[38]

• Chvátal’s toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[39]

• The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[40]

• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[41]

• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[42]

• The Lovász conjecture on Hamiltonian paths in symmetric graphs[43]

Graph coloring and labeling

• The Erdős–Faber–Lovász conjecture on coloring unions of cliques[44]

• The Hadwiger conjecture relating coloring to clique minors[45]

• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[46]

• Hedetniemi’s conjecture on the chromatic number of tensor products of graphs[47]

• Jaeger’s Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[48]

• The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[49]

• The Ringel–Kotzig conjecture on graceful labeling of trees[50]

• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[51]

Graph drawing

• The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[52]

• The Blankenship–Oporowski conjecture on the book thickness of subdivisions[53]

• Conway’s thrackle conjecture[54]

• Harborth’s conjecture that every planar graph can be drawn with integer edge lengths[55]

• Negami’s conjecture on projective-plane embeddings of graphs with planar covers[56]

• The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[57]

• Turán’s brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[58]

• Universal point sets of subquadratic size for planar graphs[59]

Miscellaneous graph theory

• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[60]

• The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[61]

• Jørgensen’s conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[62]

• Deriving a closed-form expression for the percolation threshold values, especially pc (square site)

• Does a Moore graph with girth 5 and degree 57 exist?

• What is the largest possible pathwidth of an n-vertex cubic graph?

• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.

• Sumner’s conjecture: does every (2n−2)

-vertex tournament contain as a subgraph every n
-vertex oriented tree?[63]

• Tutte’s conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow

• Vizing’s conjecture on the domination number of cartesian products of graphs[64]

Group theory

• Is every finitely presented periodic group finite?

• The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?

• For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?

• Is every group surjunctive?

• Andrews–Curtis conjecture

• Herzog–Schönheim conjecture

• Does generalized moonshine exist?

• Are there an infinite number of Leinster Groups?

Model theory

• Vaught’s conjecture

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ℵ0
is a simple algebraic group over an algebraically closed field.

• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ℵ1
-saturated models of a countable theory.[65]

• Determine the structure of Keisler’s order[66][67]

• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.

• Is the theory of the field of Laurent series over Zp\mathbb {Z} decidable? of the field of polynomials over C?

• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[68]

• The Stable Forking Conjecture for simple theories[69]

• For which number fields does Hilbert’s tenth problem hold?

• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ℵω1
does it have a model of cardinality continuum?[70]

• Shelah’s eventual Categority conjecture: For every cardinal λ
there exists a cardinal μ(λ)
such that If an AEC K with LS(K)<= λ
is categorical in a cardinal above μ(λ)
then it is categorical in all cardinals above μ(λ)

• Shelah’s categoricity conjecture for Lω1,ω
: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[65]

• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[72]

• If the class of atomic models of a complete first order theory is categorical in the ℵn
, is it categorical in every cardinal?[73][74]

• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)

• Kueker’s conjecture[75]

• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?

• Lachlan’s decision problem

• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?

• Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)

• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[76]

• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[77]

Number theory[

General

• Grand Riemann hypothesis

• Generalized Riemann hypothesis

• Riemann hypothesis

• n conjecture

• abc conjecture (Proof claimed in 2012, currently under review.)

• Hilbert’s ninth problem

• Hilbert’s eleventh problem

• Hilbert’s twelfth problem

• Carmichael’s totient function conjecture

• Erdős–Straus conjecture

• Pillai’s conjecture

• Hall’s conjecture

• Lindelöf hypothesis

• Montgomery’s pair correlation conjecture

• Hilbert–Pólya conjecture

• Grimm’s conjecture

• Leopoldt’s conjecture

• Do any odd perfect numbers exist?

• Are there infinitely many perfect numbers?

• Do quasiperfect numbers exist?

• Do any odd weird numbers exist?

• Do any Lychrel numbers exist?

• Is 10 a solitary number?

• Catalan–Dickson conjecture on aliquot sequences

• Do any Taxicab(5, 2, n) exist for n > 1?

• Brocard’s problem: existence of integers, (n,m), such that n! + 1 = m2 other than n = 4, 5, 7

• Beilinson conjecture

• Littlewood conjecture

• Szpiro’s conjecture

• Vojta’s conjecture

• Goormaghtigh conjecture

• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell’s theorem)

• Lehmer’s totient problem: if φ(n) divides n − 1, must n be prime?

• Are there infinitely many amicable numbers?

• Are there any pairs of amicable numbers which have opposite parity?

• Are there any pairs of relatively prime amicable numbers?

• Are there infinitely many betrothed numbers?

• Are there any pairs of betrothed numbers which have same parity?

• The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?

• Piltz divisor problem, especially Dirichlet’s divisor problem

• Exponent pair conjecture

• Is π a normal number (its digits are “random”)?[78]

• Casas-Alvero conjecture

• Sato–Tate conjecture

• Find value of De Bruijn–Newman constant

• Which integers can be written as the sum of three perfect cubes?[79]

• Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?

• Is there a covering system with odd distinct moduli?[80]

Additive number theory

• Beal’s conjecture

• Fermat–Catalan conjecture

• Goldbach’s conjecture

• The values of g(k) and G(k) in Waring’s problem

• Lander, Parkin, and Selfridge conjecture

• Gilbreath’s conjecture

• Erdős conjecture on arithmetic progressions

• Erdős–Turán conjecture on additive bases

• Pollock octahedral numbers conjecture

• Skolem problem

• Determine growth rate of rk(N) (see Szemerédi’s theorem)

• Minimum overlap problem

Algebraic number theory

• Are there infinitely many real quadratic number fields with unique factorization (Class number problem)?

• Characterize all algebraic number fields that have some power basis.

• Stark conjectures (including Brumer–Stark conjecture)

• Kummer–Vandiver conjecture

Combinatorial number theory

• Singmaster’s conjecture: Is there a finite upper bound on the number of times that a number other than 1 can appear in Pascal’s triangle?

Prime numbers

Prime number conjectures

• Catalan’s Mersenne conjecture

• Agoh–Giuga conjecture

• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?

• New Mersenne conjecture

• Erdős–Mollin–Walsh conjecture

• Are there infinitely many prime quadruplets?

• Are there infinitely many cousin primes?

• Are there infinitely many sexy primes?

• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?

• Are there infinitely many Wagstaff primes?

• Are there infinitely many Sophie Germain primes?

• Are there infinitely many Pierpont primes?

• Are there infinitely many regular primes, and if so is their relative density e^−1/2?

• For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?

• Are there infinitely many Cullen primes?

• Are there infinitely many Woodall primes?

• Are there infinitely many palindromic primes to every base?

• Are there infinitely many Fibonacci primes?

• Are there infinitely many Lucas primes?

• Are there infinitely many Pell primes?

• Are there infinitely many Newman–Shanks–Williams primes?

• Are all Mersenne numbers of prime index square-free?

• Are there infinitely many Wieferich primes?

• Are there any Wieferich primes in base 47?

• Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?

• For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[81]

• Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[82]

• Are there infinitely many Wilson primes?

• Are there infinitely many Wolstenholme primes?

• Are there any Wall–Sun–Sun primes?

• Is every Fermat number 22n + 1 composite for n>4{?

• Are all Fermat numbers square-free?

• For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?

• Artin’s conjecture on primitive roots

• Is 78,557 the lowest Sierpiński number (so-called Selfridge’s conjecture)?

• Is 509,203 the lowest Riesel number?

• Fortune’s conjecture (that no Fortunate number is composite)

• Landau’s problems

• Feit–Thompson conjecture

• Does every prime number appear in the Euclid–Mullin sequence?

• Does the converse of Wolstenholme’s theorem hold for all natural numbers?

• Elliott–Halberstam conjecture

• Problems associated to Linnik’s theorem

• Find the smallest Skewes’ number

Partial differential equations

• Regularity of solutions of Vlasov–Maxwell equations

• Regularity of solutions of Euler equations

Ramsey theory

• The values of the Ramsey numbers, particularly R(5,5)

• The values of the Van der Waerden numbers

• Erdős–Burr conjecture

Set theory

• The problem of finding the ultimate core model, one that contains all large cardinals.

• If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, was obtained by Shelah using his pcf theory.

• Woodin’s Ω-hypothesis.

• Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?

• (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?

• Does there exist a Jónsson algebra on ℵω?

• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?

• Does the Generalized Continuum Hypothesis entail ♢(Ecf
(λ)λ+)
for every singular cardinal λ?

• Does the Generalized Continuum Hypothesis imply the existence of an ℵ2-Suslin tree?

Algebra

• Homological conjectures in commutative algebra

• Hilbert’s sixteenth problem

• Hilbert’s fifteenth problem

• Hadamard conjecture

• Jacobson’s conjecture

• Existence of perfect cuboids and associated cuboid conjectures

• Zauner’s conjecture: existence of SIC-POVMs in all dimensions

• Wild Problem: Classification of pairs of n×n matrices under simultaneous conjugation and problems containing it such as a lot of classification problems

• Köthe conjecture

• Birch–Tate conjecture

• Serre’s conjecture II

• Bombieri–Lang conjecture

• Farrell–Jones conjecture

• Bost conjecture

• Uniformity conjecture

• Kaplansky’s conjecture

• Kummer–Vandiver conjecture

• Serre’s multiplicity conjectures

• Pierce–Birkhoff conjecture

• Eilenberg–Ganea conjecture

• Green’s conjecture

• Grothendieck–Katz p-curvature conjecture

• Sendov’s conjecture

Algebraic geometry

• Bass conjecture

• Deligne conjecture

• Fröberg conjecture

• Fujita conjecture

• Hartshorne conjectures

• The Jacobian conjecture

• Manin conjecture

• Nakai conjecture

• Resolution of singularities in characteristic p

• Standard conjectures on algebraic cycles

• Section conjecture

• Tate conjecture

• Virasoro conjecture

• Zariski multiplicity conjecture

Analysis

• Schanuel’s conjecture and four exponentials conjecture

• Lehmer’s conjecture

• Pompeiu problem

• Are γ
(the Euler–Mascheroni constant), π + e, π − e, πe, π/e, πe, π√2, ππ, eπ2, ln π, 2e, ee, Catalan’s constant or Khinchin’s constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers?[13][14][15]

• Khabibullin’s conjecture on integral inequalities

• Hilbert’s thirteenth problem

• Vitushkin’s conjecture

Combinatorics

• Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets[16]

• The lonely runner conjecture: if k+1
runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be at least a distance 1/
(k+1)
from each other runner) at some time?[17]

• Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?[18]

• Finding a function to model n-step self-avoiding walks.[19]

• The 1/3–2/3 conjecture: does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?[20]

• The uniqueness conjecture for Markov numbers[21]

• Give a combinatorial interpretation of the Kronecker coefficients.[22]

Differential geometry

• Filling area conjecture

• Hopf conjecture

Discrete geometry

• Solving the happy ending problem for arbitrary n
[23]

• Finding matching upper and lower bounds for k-sets and halving lines[24]

• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies[25]

• The Kobon triangle problem on triangles in line arrangements[26]

• The McMullen problem on projectively transforming sets of points into convex position[27]

• Ulam’s packing conjecture about the identity of the worst-packing convex solid[28]

• Kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[29]

• How many unit distances can be determined by a set of n points in the Euclidean plane?[30]

Euclidean geometry

• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?[31]

• Inscribed square problem – does every Jordan curve have an inscribed square?[32]

• Kakeya conjecture

• Moser’s worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?[33]

• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?[34]

• Shephard’s problem (a.k.a. Dürer’s conjecture) – does every convex polyhedron have a net?[35]

• The Thomson problem – what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?

• Falconer’s conjecture

• g-conjecture

• Circle packing in an equilateral triangle

• Circle packing in an isosceles right triangle

• Lebesgue’s universal covering problem – what is the convex shape in the plane of least area which provides an isometric cover for any shape of diameter one?

• Bellman’s lost in a forest problem – for a given shape of forest find the shortest escape path which will intersect the edge of the forest at some point for any given starting point and direction inside the forest.

• Find the complete set of uniform 5-polytopes[36]

• Covering problem of Rado

• The strong bellows conjecture – must the Dehn invariant of a self-intersection free flexible polyhedron stay constant as it flexes?

Dynamical systems

• Collatz conjecture (3n + 1 conjecture)

• Lyapunov’s second method for stability – For what classes of ODEs, describing dynamical systems, does the Lyapunov’s second method formulated in the classical and canonically generalized forms define the necessary and sufficient conditions for the (asymptotical) stability of motion?

• Furstenberg conjecture – Is every invariant and ergodic measure for the ×2,×3
action on the circle either Lebesgue or atomic?

• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups

• MLC conjecture – Is the Mandelbrot set locally connected?

• Weinstein conjecture – Does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?

• Is every reversible cellular automaton in three or more dimensions locally reversible?[37]

• Many problems concerning an outer billiard, for example show that outer billiards relative to almost every convex polygon has unbounded orbits.

Graph theory

Paths and cycles in graphs

• Barnette’s conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle[38]

• Chvátal’s toughness conjecture, that there is a number t such that every t-tough graph is Hamiltonian[39]

• The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice[40]

• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs[41]

• The linear arboricity conjecture on decomposing graphs into disjoint unions of paths according to their maximum degree[42]

• The Lovász conjecture on Hamiltonian paths in symmetric graphs[43]

Graph coloring and labeling

• The Erdős–Faber–Lovász conjecture on coloring unions of cliques[44]

• The Hadwiger conjecture relating coloring to clique minors[45]

• The Hadwiger–Nelson problem on the chromatic number of unit distance graphs[46]

• Hedetniemi’s conjecture on the chromatic number of tensor products of graphs[47]

• Jaeger’s Petersen-coloring conjecture that every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph[48]

• The list coloring conjecture that, for every graph, the list chromatic index equals the chromatic index[49]

• The Ringel–Kotzig conjecture on graceful labeling of trees[50]

• The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree[51]

Graph drawing

• The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[52]

• The Blankenship–Oporowski conjecture on the book thickness of subdivisions[53]

• Conway’s thrackle conjecture[54]

• Harborth’s conjecture that every planar graph can be drawn with integer edge lengths[55]

• Negami’s conjecture on projective-plane embeddings of graphs with planar covers[56]

• The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[57]

• Turán’s brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[58]

• Universal point sets of subquadratic size for planar graphs[59]

Miscellaneous graph theory

• The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[60]

• The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[61]

• Jørgensen’s conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[62]

• Deriving a closed-form expression for the percolation threshold values, especially pc
(square site)

• Does a Moore graph with girth 5 and degree 57 exist?

• What is the largest possible pathwidth of an n-vertex cubic graph?

• The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.

• Sumner’s conjecture: does every (2n−2)
-vertex tournament contain as a subgraph every n
-vertex oriented tree?[63]

• Tutte’s conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow

• Vizing’s conjecture on the domination number of cartesian products of graphs[64]

Group theory

• Is every finitely presented periodic group finite?

• The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?

• For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?

• Is every group surjunctive?

• Andrews–Curtis conjecture

• Herzog–Schönheim conjecture

• Does generalized moonshine exist?

• Are there an infinite number of Leinster Groups?

Model theory

• Vaught’s conjecture

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ℵ0
is a simple algebraic group over an algebraically closed field.

• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ℵ1
-saturated models of a countable theory.[65]

• Determine the structure of Keisler’s order[66][67]

• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.

• Is the theory of the field of Laurent series over Zp
decidable? of the field of polynomials over C{\displaystyle \mathbb {C} } ?

• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[68]

• The Stable Forking Conjecture for simple theories[69]

• For which number fields does Hilbert’s tenth problem hold?

• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ℵω1
does it have a model of cardinality continuum?[70]

• Shelah’s eventual Categority conjecture: For every cardinal λ
there exists a cardinal μ(λ)
such that If an AEC K with LS(K)<= λ
is categorical in a cardinal above μ(λ)
then it is categorical in all cardinals above μ(λ)
.[65][71]

• Shelah’s categoricity conjecture for Lω1,ω
: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[65]

• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[72]

• If the class of atomic models of a complete first order theory is categorical in the ℵn
, is it categorical in every cardinal?[73][74]

• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)

• Kueker’s conjecture[75]

• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?

• Lachlan’s decision problem

• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?

• Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)

• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[76]

• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[77]

Number theory

General

• Grand Riemann hypothesis

• Generalized Riemann hypothesis

• Riemann hypothesis

• n conjecture

• abc conjecture (Proof claimed in 2012, currently under review.)

• Hilbert’s ninth problem

• Hilbert’s eleventh problem

• Hilbert’s twelfth problem

• Carmichael’s totient function conjecture

• Erdős–Straus conjecture

• Pillai’s conjecture

• Hall’s conjecture

• Lindelöf hypothesis

• Montgomery’s pair correlation conjecture

• Hilbert–Pólya conjecture

• Grimm’s conjecture

• Leopoldt’s conjecture

• Do any odd perfect numbers exist?

• Are there infinitely many perfect numbers?

• Do quasiperfect numbers exist?

• Do any odd weird numbers exist?

• Do any Lychrel numbers exist?

• Is 10 a solitary number?

• Catalan–Dickson conjecture on aliquot sequences

• Do any Taxicab(5, 2, n) exist for n > 1?

• Brocard’s problem: existence of integers, (n,m), such that n! + 1 = m2 other than n = 4, 5, 7

• Beilinson conjecture

• Littlewood conjecture

• Szpiro’s conjecture

• Vojta’s conjecture

• Goormaghtigh conjecture

• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell’s theorem)

• Lehmer’s totient problem: if φ(n) divides n − 1, must n be prime?

• Are there infinitely many amicable numbers?

• Are there any pairs of amicable numbers which have opposite parity?

• Are there any pairs of relatively prime amicable numbers?

• Are there infinitely many betrothed numbers?

• Are there any pairs of betrothed numbers which have same parity?

• The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?

• Piltz divisor problem, especially Dirichlet’s divisor problem

• Exponent pair conjecture

• Is π a normal number (its digits are “random”)?[78]

• Casas-Alvero conjecture

• Sato–Tate conjecture

• Find value of De Bruijn–Newman constant

• Which integers can be written as the sum of three perfect cubes?[79]

• Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?

• Is there a covering system with odd distinct moduli?[80]

Additive number theory

• Beal’s conjecture

• Fermat–Catalan conjecture

• Goldbach’s conjecture

• The values of g(k) and G(k) in Waring’s problem

• Lander, Parkin, and Selfridge conjecture

• Gilbreath’s conjecture

• Erdős conjecture on arithmetic progressions

• Erdős–Turán conjecture on additive bases

• Pollock octahedral numbers conjecture

• Skolem problem

• Determine growth rate of rk(N) (see Szemerédi’s theorem)

• Minimum overlap problem

Algebraic number theory

• Are there infinitely many real quadratic number fields with unique factorization (Class number problem)?

• Characterize all algebraic number fields that have some power basis.

• Stark conjectures (including Brumer–Stark conjecture)

• Kummer–Vandiver conjecture

Combinatorial number theory

• Singmaster’s conjecture: Is there a finite upper bound on the number of times that a number other than 1 can appear in Pascal’s triangle?

Prime numbers

Prime number conjectures

• Catalan’s Mersenne conjecture

• Agoh–Giuga conjecture

• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?

• New Mersenne conjecture

• Erdős–Mollin–Walsh conjecture

• Are there infinitely many prime quadruplets?

• Are there infinitely many cousin primes?

• Are there infinitely many sexy primes?

• Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?

• Are there infinitely many Wagstaff primes?

• Are there infinitely many Sophie Germain primes?

• Are there infinitely many Pierpont primes?

• Are there infinitely many regular primes, and if so is their relative density e−1/2 ?

• For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?

• Are there infinitely many Cullen primes?

• Are there infinitely many Woodall primes?

• Are there infinitely many palindromic primes to every base?

• Are there infinitely many Fibonacci primes?

• Are there infinitely many Lucas primes?

• Are there infinitely many Pell primes?

• Are there infinitely many Newman–Shanks–Williams primes?

• Are all Mersenne numbers of prime index square-free?

• Are there infinitely many Wieferich primes?

• Are there any Wieferich primes in base 47?

• Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?

• For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[81]

• Can a prime p satisfy 2p − 1 ≡ 1 (mod p2) and 3p − 1 ≡ 1 (mod p2) simultaneously?[82]

• Are there infinitely many Wilson primes?

• Are there infinitely many Wolstenholme primes?

• Are there any Wall–Sun–Sun primes?

• Is every Fermat number 22n + 1 composite for n>4?

• Are all Fermat numbers square-free?

• For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?

• Artin’s conjecture on primitive roots

• Is 78,557 the lowest Sierpiński number (so-called Selfridge’s conjecture)?

• Is 509,203 the lowest Riesel number?

• Fortune’s conjecture (that no Fortunate number is composite)

• Landau’s problems

• Feit–Thompson conjecture

• Does every prime number appear in the Euclid–Mullin sequence?

• Does the converse of Wolstenholme’s theorem hold for all natural numbers?

• Elliott–Halberstam conjecture

• Problems associated to Linnik’s theorem

• Find the smallest Skewes’ number

Partial differential equations

• Regularity of solutions of Vlasov–Maxwell equations

• Regularity of solutions of Euler equations

Ramsey theory

• The values of the Ramsey numbers, particularly R(5,5)

• The values of the Van der Waerden numbers

• Erdős–Burr conjecture

Set theory

• The problem of finding the ultimate core model, one that contains all large cardinals.

• If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, was obtained by Shelah using his pcf theory.

• Woodin’s Ω-hypothesis.

• Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?

• (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?

• Does there exist a Jónsson algebra on ℵω?

• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?

• Does the Generalized Continuum Hypothesis entail ♢(Ecf(λ^)λ+)
for every singular cardinal λ?

• Does the Generalized Continuum Hypothesis imply the existence of an ℵ2-Suslin tree?

MATHEMATICAL PROBLEMS IMPOSSIBLE TO SOLVE

Using only straight edge and compass:

Creating a cube of twice the volume as the original cube.

Dividing an angle into 3 equal parts.

Constructing a square having an area equal to that given a circle.

MATHEMATICAL AWARDS

Fields medal (the Nobel prize of mathematics)

Abel prize

Carl friedrich gauss prize

Breakthrough prize

Chern medal

The shaw prize

Leroy p. Steele prize

Morgan prize

Nevqnlinna prize

Cole prize

Chauvenet prize

Bother memorial prize

Wolf prize

George poly prize

Whitehead prize

Oswald Veblen prize in geometry

De Morgan medal

Debrah and Franklin halos award

Leelaati award

Adams prize

Albert león Whitman memorial prize

Poly prize

Salem prize

Berwick prize

Dannie Heinemann prize for mathematical physics

Louis bachelor prize

Sloan research fellowship

Senior whitehead prize

SCIENCE FACTS MATH

1. A one bedroom apartment can hold about 4.5 trillion grains of sand.

2. If you were to keep typing in a randomly until you correctly typed the 1st verse of psalms 1–“happy are those who do not follow the advice of the wicked” , the statistical probability of the time it would take you to type it correctly if each try took 20 seconds would take 2.862^18 tries and the time to type the phrase correctly would be a little over 4.5 billion years. If we randomly hit keys on a writing keyboard, to finally write the 23rd psalm of the Bible, it would take 10^2,026 years, and to type John Doe would take over 330 years. To reproduce Shakespeares works by randomly pressing keys on a keyboard (the number of tries necessary till his books got typed correctly would take over 10^5,000,000 years. (Imagine the length of time it would take to type everything that has ever been printed. The time to type this is a drop in the bucket compared to how long we will exist. As immortal souls, we will exist forever.

3. If the United States were a mile across, the US Capitol would be 1 inch tall and people would be the size of grains of sand.

4. Bhaskara 2 (Bhaskara Acharya), an Indian man of science, calculated the time it would take the Earth to go around the Sun in the 12th century to an accuracy of 9 decimal places. That is less than a third of a second off the real time (.284 seconds off).

5. If an atom’s size were doubled, then that size were doubled, and you kept doubling the resultant size, in 113 of these doublings, the final size would be as big as our universe.

6. Here is a math fact that deals with largeness–if we had a drop of water, then added 1/2 of a drop of water to the drop, then added 1/3 of a drop to the previous 2 amounts of water, then added 1/4, 1/5/ 1/6,….. and keep adding up in this pattern water indefinitely, the resulting amount of water would fill up not only a lake for example, or an ocean, or even the whole earth’s volume; not even the whole universe, but the amount of water resulting from adding these small amounts of water together indefinitely would be bigger than anything that could possibly be, and in fact, the resulting amount of water would only get bigger and bigger and never stop getting bigger. The amount of water which is the result is equal to- Ln(drops)=amount of water Natural logarithm(number of drops added together)=amount of water. The number of drops to make a given amount of water is equal to Exponential(amount of water in number of drops)=number of smaller and smaller drops of water added together.

7. Here is an easy to understand explanation of the adding up drops of water which get smaller and smaller, and the result is that the total amount of water that gets bigger and bigger, enough to fill the entire universe (and a proof for this). I want to show you how starting with one raindrop, and adding lesser and lesser parts of the raindrops all together, you can fill the entire universe or any conceivable size of anything with water from the raindrops that are added together. This fact may be hard to believe but it is true.

The harmonic series from math is used to explain how this is done. We start with one raindrop, add to that 1/2 of a raindrop, add to those 1/3 of a raindrop, then add 1/4, 1/5, and keep adding lesser and lesser amounts of raindrops. Here is an easy to understand proof of this—
We start with 1 drop of water. We need to add up the next 3 terms (1/2, 1/3, 1/4) together in the harmonic series to equal another 1 drop, so we have this 1 added to the first drop to equal 2 drops. Next, we add the next 13 terms together and this also is equal to 1 drop, adding this 1 to the previous 2 drops to equal 3 drops. To equal 4 drops. we need to add 20 drops, and again, this equals 1 and we add this drop to the previous 3 drops to equal 4 drops. We keep adding successively more of the smaller drops together to equal another 1, and the result is always adding one drop to the drops before it, and the number resulting gets bigger, so the amount of water keeps increasing. Now, since the harmonic series has an infinite number of terms terms, we can keep keep adding successively more and more drops together to equal another drop. And since this goes on forever, the number of drops will keep increasing forever too, and the resulting amount of water will become infinitely large.

To accumulate 10 drops of water would take a little over 22,000 of the smaller drops added together. For 100 drops of water to accumulate, we would need to add together 2.69^43 smaller drops of water, which is an extremely large number of small drop. If the number of smaller drops were equal to the number of atoms that were packed tightly together to fill the universe, (2.912*10^114 atoms), then the amount of water accumulated would only be 263.57 drops. To fill a glass of water would take 2×10^2,044 smaller drops, a bathtub would take 3.9×10^3,845,294 drops, and finally to fill the universe to require 8.7×10^221,932,935 drops. The harmonic series, which models this, says that the increase will never stop. i don’t understand what infinity is other than it is never ending. To me, it means it keeps going on and on and never stops going on. Maybe some very smart and creative people will be able to help us understand infinity better in the future. Our soul essences will experience what infinity is really like because they will live forever. As immortal souls, we will be alive and conscious for all that time, and that will only be the beginning, because we will always BE without time ending. So this, the harmonic series, is a science fact i wanted to present.

8. If one stirs a cup of coffee, there will always be some part of the coffee in the same place after stirring as it was before stirring it.

9. 10 to the 90th power are the number of grains of sand packed together which would fill the universe, 10 to the 100th power of hydrogen atoms it would take to fill the universe, and 10 to the 113th power the number of protons would be needed to fill up the universe. A googol (10 to the 100th power) number of grains of sand would fill up 10 billion universes.

10. 107 billion people have lived on earth during recorded history.

11. Here is a very interesting math fact called the Riemann’s rearrangement theorem. Take the following numbers (which is called a series that goes on forever)-

1-1/2+1/3-1/4+1/5-1/6+1/7-1/8+1/9-1/10……

The Riemann’s rearrangement theorem says that we can rearrange these numbers in any order we want to, and by doing this, we can make an arrangement so that the total gets big up to infinity, or we can rearrange them so that the result adds up to negative infinity, and the numbers can be rearranged to add up to any number possible. This is the theorem proved by Bernhard Riemann, who lived from 1826-1866, and he was a very well known an accomplished mathematician.

12. An inch’s length can be divided up into more parts than all the counting numbers from 1 to infinity.

13. How big is a googol (10^100)? We would need over 10,000,000,000 times the number of grains of sand it takes to fill the universe with to equal a googol.

14. To get an idea of big numbers: 1. A sphere 1 inch big would contain 8,580 millimeter, a 1 foot sphere would contain 148,266,666 millimeter, 1 yard 400,319,875.5 millimeters, 100 yards 3.2X10^15 millimeters, a mile 2.5X10^17 millimeters, 6.9 miles (the distance to my brother’s house) 7.17X10^20 millimeter, and if the Earth were filled with millimeters, there would contain 1.09X10^30 of them.

15. The number of people on Earth, if each one represented a grain of sand size, the size of a sphere for 7 billion people would be 2 feet 7.2 inches in diameter, and if each grain of sand were laid end to end, the grains of sand would stretch a distance of 1,447 miles.

16. Then number of possible chess games is 10^120.

17. If a person kept folding a piece of paper, it would take 26.5 folds and the paper folded would be 5 miles thick. For the thickness of the paper to be the thickness going across the United States would take 37 folds of the paper, for the distance to the moon would take 42 folds, for the distance to the Sun would require 50.56 fold, and to go from one end of the universe to the other would require 103 folds.

18. 100 billion, billion, billion times more than the number of grains of sand that would fill the universe—that is the number is all the possible chess games if each grain of sand were to represent one of the possible chess games.

19. Say we were to have a rope that went all the way around the world in contact with the Earth. By adding 6 feet 4 inches to the length of the rope, the rope going around the Earth would now be 12 inched above the ground.

20. A curve of infinite length could be drawn inside a 2 dimensional square. The is called a Sierpinski curve.

21. If we were to count 1,2,3,4,5…forever, and also count 2,4,6,8…forever, the number of numbers in each set of numbers would be of equal magnitude.

22. James clerk Maxwell memorized the entire bible by the time he was 14 years old.

23. If an inch were equal to an average human lifetime, the distance of the lifetime of a rather short lived star (100 million years) would be about 21 miles long compared to our 1 inch lifespan length.

24. How do we measure the size of the Earth? Eratosthenes, over 2000 years ago, gave a very good approximation of the size of the Earth. He knew that at the summer solstice that the Sun shone directly into a well at Syene at noon. At the same time, in Alexandria, Egypt, approximately 488 miles due north of Syene, the angle of elevation (deviation of the Sun shining directly into the well/the angle of deviation) of the Sun’s rays was about 7.2 degrees. 360 degrees contains 50×7.2 degrees parts. Since the Earth is a circle, and since a circle contains 360 degrees, and since a circle is 50 times more than 7.2 degrees, we multiply the 488 miles by 50 and this equals 24,400 miles. This result has 98.3 precent accuracy for the circumference of the Earth.

25. The average gap between the 1st n consecutive primes is approximately ln(n).

26. Transcendental numbers, (real or complex numbers not algebraic. (Examples- pi=3.14159265…, and ‘e;’), these numbers are by far the most numerous type of number.

27. The largest prime number known as of 1/25/13 contains 17,425,170 digits.

28. The series of the reciprocals of the primes is a series that diverges to infinity.

29. The largest prime gap between 2 prime numbers is 1.3002×10^16. The gap can be extended to 4×10^18 numbers between 2 prime numbers.

30. The monster group is a mind boggling snowflake with more than 10^53 symmetries that exists in a space of 196,884 dimensions.

31. It would take almost 2.5X10^72 atoms to fill the universe.

32. The number of primes up to a given number is approximately equal to 1/Ln(the given number).