Timeline of mathematics

15th century

• 1400 – Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places.
• c. 1400 – Ghiyath al-Kashi "contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots, which is a special case of the methods given many centuries later by [Paolo] Ruffini and [William George] Horner." He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: "On whole number arithmetic", "On fractional arithmetic", "On astrology", "On areas", and "On finding the unknowns [unknown variables]". He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine.
• 15th century – Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general.^[11]
• 15th century – Nilakantha Somayaji, a Kerala school mathematician, writes the Aryabhatiya Bhasya, which contains work on infinite-series expansions, problems of algebra, and spherical geometry.
• 1424 – Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons.
• 1427 – Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
• 1464 – Regiomontanus writes De Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics.
• 1478 – An anonymous author writes the Treviso Arithmetic.
• 1494 – Luca Pacioli writes Summa de arithmetica, geometria, proportioni et proportionalità; introduces primitive symbolic algebra using "co" (cosa) for the unknown.

16th century

• 1501 – Nilakantha Somayaji writes the Tantrasamgraha.
• 1520 – Scipione dal Ferro develops a method for solving "depressed" cubic equations (cubic equations without an x² term), but does not publish.
• 1522 – Adam Ries explained the use of Arabic digits and their advantages over Roman numerals.
• 1535 – Niccolò Tartaglia independently develops a method for solving depressed cubic equations but also does not publish.
• 1539 – Gerolamo Cardano learns Tartaglia's method for solving depressed cubics and discovers a method for depressing cubics, thereby creating a method for solving all cubics.
• 1540 – Lodovico Ferrari solves the quartic equation.
• 1544 – Michael Stifel publishes Arithmetica integra.
• 1545 – Gerolamo Cardano conceives the idea of complex numbers.
• 1550 – Jyeshtadeva, a Kerala school mathematician, writes the Yuktibhāṣā, the world's first calculus text, which gives detailed derivations of many calculus theorems and formulae.
• 1572 – Rafael Bombelli writes Algebra treatise and uses imaginary numbers to solve cubic equations.
• 1584 – Zhu Zaiyu calculates equal temperament.
• 1596 – Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons.

17th century

• 1614 – John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio.
• 1617 – Henry Briggs discusses decimal logarithms in Logarithmorum Chilias Prima.
• 1618 – John Napier publishes the first references to e in a work on logarithms.
• 1619 – René Descartes discovers analytic geometry (Pierre de Fermat claimed that he also discovered it independently).
• 1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra.
• 1629 – Pierre de Fermat develops a rudimentary differential calculus.
• 1634 – Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle.
• 1636 – Muhammad Baqir Yazdi jointly discovered the pair of amicable numbers 9,363,584 and 9,437,056 along with Descartes (1636).^[12]
• 1637 – Pierre de Fermat claims to have proven Fermat's Last Theorem in his copy of Diophantus' Arithmetica.
• 1637 – First use of the term imaginary number by René Descartes; it was meant to be derogatory.
• 1643 – René Descartes develops Descartes' theorem.
• 1654 – Blaise Pascal and Pierre de Fermat create the theory of probability.
• 1655 – John Wallis writes Arithmetica Infinitorum.
• 1658 – Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle.
• 1665 – Isaac Newton works on the fundamental theorem of calculus and develops his version of infinitesimal calculus.
• 1668 – Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbolic segment.
• 1671 – James Gregory develops a series expansion for the inverse-tangent function (originally discovered by Madhava).
• 1671 – James Gregory discovers Taylor's Theorem.
• 1673 – Gottfried Leibniz also develops his version of infinitesimal calculus.
• 1675 – Isaac Newton invents an algorithm for the computation of functional roots.
• 1680s – Gottfried Leibniz works on symbolic logic.
• 1683 – Seki Takakazu discovers the resultant and determinant.
• 1683 – Seki Takakazu develops elimination theory.
• 1691 – Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations.
• 1693 – Edmund Halley prepares the first mortality tables statistically relating death rate to age.
• 1696 – Guillaume de L'Hôpital states his rule for the computation of certain limits.
• 1696 – Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations.
• 1699 – Abraham Sharp calculates π to 72 digits but only 71 are correct.

18th century

• 1706 – John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
• 1708 – Seki Takakazu discovers Bernoulli numbers. Jacob Bernoulli whom the numbers are named after is believed to have independently discovered the numbers shortly after Takakazu.
• 1712 – Brook Taylor develops Taylor series.
• 1722 – Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers.
• 1722 – Takebe Kenko introduces Richardson extrapolation.
• 1724 – Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities on Lives.
• 1730 – James Stirling publishes The Differential Method.
• 1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false.
• 1733 – Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability.
• 1734 – Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations.
• 1735 – Leonhard Euler solves the Basel problem, relating an infinite series to π.
• 1736 – Leonhard Euler solves the problem of the Seven bridges of Königsberg, in effect creating graph theory.
• 1739 – Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.
• 1742 – Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture.
• 1748 – Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana.
• 1761 – Thomas Bayes proves Bayes' theorem.
• 1761 – Johann Heinrich Lambert proves that π is irrational.
• 1762 – Joseph Louis Lagrange discovers the divergence theorem.
• 1789 – Jurij Vega improves Machin's formula and computes π to 140 decimal places, 136 of which were correct.
• 1794 – Jurij Vega publishes Thesaurus Logarithmorum Completus.
• 1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge.
• 1796 – Adrien-Marie Legendre conjectures the prime number theorem.
• 1797 – Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms.
• 1799 – Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers).
• 1799 – Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula.

19th century

• 1801 – Disquisitiones Arithmeticae, Carl Friedrich Gauss's number theory treatise, is published in Latin.
• 1805 – Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations.
• 1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra.
• 1806 – Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram.
• 1807 – Joseph Fourier announces his discoveries about the trigonometric decomposition of functions.
• 1811 – Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration.
• 1815 – Siméon Denis Poisson carries out integrations along paths in the complex plane.
• 1817 – Bernard Bolzano presents the intermediate value theorem—a continuous function that is negative at one point and positive at another point must be zero for at least one point in between.
• 1822 – Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane.
• 1822 – Irisawa Shintarō Hiroatsu analyzes Soddy's hexlet in a Sangaku.
• 1823 - Sophie Germain's Theorem is published in the second edition of Adrien-Marie Legendre's Essai sur la théorie des nombres^[13]
• 1824 – Niels Henrik Abel partially proves the Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
• 1825 – Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis.
• 1825 – Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's Last Theorem for n = 5.
• 1825 – André-Marie Ampère discovers Stokes' theorem.
• 1828 – George Green proves Green's theorem.
• 1829 – János Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry.
• 1831 – Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green.
• 1832 – Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory.
• 1832 – Lejeune Dirichlet proves Fermat's Last Theorem for n = 14.
• 1835 – Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions.
• 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons.
• 1837 – Peter Gustav Lejeune Dirichlet develops Analytic number theory.
• 1841 – Karl Weierstrass discovers but does not publish the Laurent expansion theorem.
• 1843 – Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem.
• 1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative.
• 1847 – George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what is now called Boolean algebra.
• 1849 – George Gabriel Stokes shows that solitary waves can arise from a combination of periodic waves.
• 1850 – Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points.
• 1850 – George Gabriel Stokes rediscovers and proves Stokes' theorem.
• 1854 – Bernhard Riemann introduces Riemannian geometry.
• 1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space.
• 1858 – August Ferdinand Möbius invents the Möbius strip.
• 1858 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions.
• 1859 – Bernhard Riemann formulates the Riemann hypothesis, which has strong implications about the distribution of prime numbers.
• 1870 – Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate.
• 1872 – Richard Dedekind invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers.
• 1873 – Charles Hermite proves that e is transcendental.
• 1873 – Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points.
• 1874 – Georg Cantor proves that the set of all real numbers is uncountably infinite but the set of all real algebraic numbers is countably infinite. His proof does not use his diagonal argument, which he published in 1891.
• 1882 – Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge.
• 1882 – Felix Klein invents the Klein bottle.
• 1895 – Diederik Korteweg and Gustav de Vries derive the Korteweg–de Vries equation to describe the development of long solitary water waves in a canal of rectangular cross section.
• 1895 – Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis.
• 1895 – Henri Poincaré publishes paper "Analysis Situs" which started modern topology.
• 1896 – Jacques Hadamard and Charles Jean de la Vallée-Poussin independently prove the prime number theorem.
• 1896 – Hermann Minkowski presents Geometry of numbers.
• 1899 – Georg Cantor discovers a contradiction in his set theory.
• 1899 – David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry.
• 1900 – David Hilbert states his list of 23 problems, which show where some further mathematical work is needed.

20th century

^[14]

• 1901 – Élie Cartan develops the exterior derivative.
• 1901 – Henri Lebesgue publishes on Lebesgue integration.
• 1903 – Carle David Tolmé Runge presents a fast Fourier transform algorithm
• 1903 – Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem.
• 1908 – Ernst Zermelo axiomizes set theory, thus avoiding Cantor's contradictions.
• 1908 – Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky – Plemelj formulae.
• 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem.
• 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent n = 5.
• 1915 – Emmy Noether proves her symmetry theorem, which shows that every symmetry in physics has a corresponding conservation law.
• 1916 – Srinivasa Ramanujan introduces Ramanujan conjecture. This conjecture is later generalized by Hans Petersson.
• 1919 – Viggo Brun defines Brun's constant B₂ for twin primes.
• 1921 – Emmy Noether introduces the first general definition of a commutative ring.
• 1928 – John von Neumann begins devising the principles of game theory and proves the minimax theorem.
• 1929 – Emmy Noether introduces the first general representation theory of groups and algebras.
• 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution.
• 1930 – Alonzo Church introduces Lambda calculus.
• 1931 – Kurt Gödel proves his incompleteness theorem, which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
• 1931 – Georges de Rham develops theorems in cohomology and characteristic classes.
• 1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk–Ulam antipodal-point theorem.
• 1933 – Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung), which contains an axiomatization of probability based on measure theory.
• 1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory.
• 1942 – G.C. Danielson and Cornelius Lanczos develop a fast Fourier transform algorithm.
• 1943 – Kenneth Levenberg proposes a method for nonlinear least squares fitting.
• 1945 – Stephen Cole Kleene introduces realizability.
• 1945 – Saunders Mac Lane and Samuel Eilenberg start category theory.
• 1945 – Norman Steenrod and Samuel Eilenberg give the Eilenberg–Steenrod axioms for (co-)homology.
• 1946 – Jean Leray introduces the Spectral sequence.
• 1948 – John von Neumann mathematically studies self-reproducing machines.
• 1948 – Alan Turing introduces LU decomposition.
• 1949 – John Wrench and L.R. Smith compute π to 2,037 decimal places using ENIAC.
• 1949 – Claude Shannon develops notion of Information Theory.
• 1950 – Stanisław Ulam and John von Neumann present cellular automata dynamical systems.
• 1953 – Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms.
• 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron.
• 1955 – Enrico Fermi, John Pasta, Stanisław Ulam, and Mary Tsingou numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior.
• 1956 – Noam Chomsky describes a hierarchy of formal languages.
• 1957 – Kiyosi Itô develops Itô calculus.
• 1957 – Stephen Smale provides the existence proof for crease-free sphere eversion.
• 1958 – Alexander Grothendieck's proof of the Grothendieck–Riemann–Roch theorem is published.
• 1959 – Kenkichi Iwasawa creates Iwasawa theory.
• 1960 – C. A. R. Hoare invents the quicksort algorithm.
• 1960 – Irving S. Reed and Gustave Solomon present the Reed–Solomon error-correcting code.
• 1961 – Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
• 1961 – John G. F. Francis and Vera Kublanovskaya independently develop the QR algorithm to calculate the eigenvalues and eigenvectors of a matrix.
• 1961 – Stephen Smale proves the Poincaré conjecture for all dimensions greater than or equal to 5.
• 1962 – Donald Marquardt proposes the Levenberg–Marquardt nonlinear least squares fitting algorithm.
• 1962 – Gloria Conyers Hewitt becomes the third African American woman to receive a PhD in mathematics.
• 1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory.
• 1963 – Martin Kruskal and Norman Zabusky analytically study the Fermi–Pasta–Ulam–Tsingou heat conduction problem in the continuum limit and find that the KdV equation governs this system.
• 1963 – meteorologist and mathematician Edward Norton Lorenz published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and strange attractors or Lorenz Attractor – also the Butterfly Effect.
• 1965 – Iranian mathematician Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of set and he founded the field of Fuzzy Mathematics.
• 1965 – Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions.
• 1965 – James Cooley and John Tukey present an influential fast Fourier transform algorithm.
• 1966 – E. J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix.
• 1966 – Abraham Robinson presents non-standard analysis.
• 1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory.
• 1968 – Michael Atiyah and Isadore Singer prove the Atiyah–Singer index theorem about the index of elliptic operators.
• 1973 – Lotfi Zadeh founded the field of fuzzy logic.
• 1975 – Benoît Mandelbrot publishes Les objets fractals, forme, hasard et dimension.
• 1976 – Kenneth Appel and Wolfgang Haken use a computer to prove the Four color theorem.
• 1978 – Olga Taussky-Todd is awarded the Austrian Cross of Honour for Science and Art, 1st Class, the highest scientific award of the government of Austria.
• 1981 – Richard Feynman gives an influential talk "Simulating Physics with Computers" (in 1980 Yuri Manin proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)).
• 1983 – Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
• 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed.
• 1985 – Louis de Branges de Bourcia proves the Bieberbach conjecture.
• 1986 – Ken Ribet proves Ribet's theorem.
• 1987 – Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places.
• 1991 – Alain Connes and John W. Lott develop non-commutative geometry.
• 1992 – David Deutsch and Richard Jozsa develop the Deutsch–Jozsa algorithm, one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm.
• 1994 – Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat's Last Theorem.
• 1994 – Peter Shor formulates Shor's algorithm, a quantum algorithm for integer factorization.
• 1995 – Simon Plouffe discovers Bailey–Borwein–Plouffe formula capable of finding the nth binary digit of π.
• 1998 – Thomas Callister Hales (almost certainly) proves the Kepler conjecture.
• 1999 – the full Taniyama–Shimura conjecture is proven.
• 2000 – the Clay Mathematics Institute proposes the seven Millennium Prize Problems of unsolved important classic mathematical questions.

21st century

• 2002 – Manindra Agrawal, Nitin Saxena, and Neeraj Kayal of IIT Kanpur present an unconditional deterministic polynomial time algorithm to determine whether a given number is prime (the AKS primality test).
• 2002 – Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241.1 billion digits using a Hitachi 64-node supercomputer.
• 2002 – Preda Mihăilescu proves Catalan's conjecture.
• 2003 – Grigori Perelman proves the Poincaré conjecture.
• 2004 – Ben Green and Terence Tao prove the Green-Tao theorem.
• 2007 – a team of researchers throughout North America and Europe used networks of computers to map E₈.^[15]
• 2009 – Fundamental lemma (Langlands program) had been proved by Ngô Bảo Châu.^[16]
• 2010 – Larry Guth and Nets Hawk Katz solve the Erdős distinct distances problem.
• 2013 – Yitang Zhang proves the first finite bound on gaps between prime numbers.^[17]
• 2014 – Project Flyspeck^[18] announces that it completed proof of Kepler's conjecture.^{[19][20][21][22]}
• 2014 – Using Alexander Yee's y-cruncher "houkouonchi" successfully calculated π to 13.3 trillion digits.^[23]
• 2015 – Terence Tao solved The Erdös Discrepancy Problem
• 2015 – László Babai found that a quasipolynomial complexity algorithm would solve the Graph Isomorphism Problem
• 2016 – Using Alexander Yee's y-cruncher Peter Trueb successfully calculated π to 22.4 trillion digits^[24]