Timeline of mathematics

This is a of  and.

Before 1000 BC

 * ca. – South Africa, ochre rocks adorned with scratched  patterns (see ).
 * ca. to  – Africa and France, earliest known  attempts to.
 * c. 20,000 BC –, : possibly the earliest reference to s and.
 * c. 3400 BC –, the ians invent the first , and a system of.
 * c. 3100 BC –, earliest known allows indefinite counting by way of introducing new symbols.
 * c. 2800 BC – on the, earliest use of decimal ratios in a uniform system of , the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 grams.
 * 2700 BC – Egypt, precision.
 * 2400 BC – Egypt, precise, used even in the for its mathematical regularity.
 * c. 2000 BC – Mesopotamia, the use a base-60 positional numeral system, and compute the first known approximate value of  at 3.125.
 * c. 2000 BC – Scotland, exhibit a variety of symmetries including all of the symmetries of s.
 * 1800 BC – Egypt,, findings volume of a.
 * c. 1800 BC – (Egypt, 19th dynasty) contains a quadratic equation and its solution.
 * 1650 BC –, copy of a lost scroll from around 1850 BC, the scribe presents one of the first known approximate values of π at 3.16, the first attempt at , earliest known use of a sort of , and knowledge of solving first order linear equations.

1st millennium BC

 * c. 1000 BC – s used by the . However, only unit fractions are used (i.e., those with 1 as the numerator) and tables are used to approximate the values of the other fractions.
 * first half of 1st millennium BC – –, in his , describes the motions of the sun and the moon, and advances a 95-year cycle to synchronize the motions of the sun and the moon.
 * 800 BC –, author of the Baudhayana , a geometric text, contains , and calculates the  correctly to five decimal places.
 * c. 8th century BC – the, one of the four s, contains the earliest concept of , and states "if you remove a part from infinity or add a part to infinity, still what remains is infinity."
 * 1046 BC to 256 BC – China, , arithmetic, geometric algorithms, and proofs.
 * 624 BC – 546 BC – Greece, has various theorems attributed to him.
 * c. 600 BC – Greece, the other Vedic "Sulba Sutras" ("rule of chords" in ) use, contain of a number of geometrical proofs, and approximate at 3.16.
 * second half of 1st millennium BC – The, the unique normal of order three, was discovered in China.
 * 530 BC – Greece, studies propositional  and vibrating lyre strings; his group also discovers the  of the.
 * c. 510 BC – Greece,
 * c. 500 BC – grammarian  writes the , which contains the use of metarules,  and s, originally for the purpose of systematizing the grammar of Sanskrit.
 * c. 500 BC – Greece,
 * 470 BC – 410 BC – Greece, utilizes  in an attempt to.
 * 490 BC – 430 BC – Greece, 
 * 5th century BC – India,, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the correct to five decimal places.
 * 5th c. BC – Greece,
 * 5th century – Greece,
 * 460 BC – 370 BC – Greece,
 * 460 BC – 399 BC – Greece,
 * 5th century (late) – Greece,
 * 428 BC – 347 BC – Greece,
 * 423 BC – 347 BC – Greece,
 * 417 BC – 317 BC – Greece,
 * c. 400 BC – India, a mathematicians write the Surya Prajinapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable and . It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
 * 408 BC – 355 BC – Greece,
 * 400 BC – 350 BC – Greece,
 * 395 BC – 313 BC – Greece,
 * 390 BC – 320 BC – Greece,
 * 380- 290 – Greece,
 * 370 BC – Greece, states the  for  determination.
 * 370 BC – 300 BC – Greece,
 * 370 BC – 300 BC – Greece,
 * 350 BC – Greece, discusses al reasoning in .
 * 4th century BC – texts use the Sanskrit word "Shunya" to refer to the concept of "void".
 * 330 BC –China, the earliest known work on, the Mo Jing, is compiled.
 * 310 BC – 230 BC – Greece,
 * 390 BC – 310 BC – Greece,
 * 380 BC – 320 BC – Greece,
 * 300 BC – India, mathematicians in India write the Bhagabati Sutra, which contains the earliest information on.
 * 300 BC – Greece,  in his  studies geometry as an, proves the infinitude of s and presents the ; he states the law of reflection in Catoptrics, and he proves the.
 * c. 300 BC – India, s (ancestor of the common modern )
 * 370 BC – 300 BC – Greece, works on histories of arithmetic, geometry and astronomy now lost.
 * 300 BC –, the invent the earliest calculator, the.
 * c. 300 BC –  writes the Chhandah-shastra, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a, along with the first use of  and.
 * 280 BC – 210 BC – Greece,
 * 280 BC – 220BC – Greece,
 * 280 BC – 220 BC – Greece,
 * 279 BC – 206 BC – Greece,
 * c. 3rd century BC – India,
 * 250 BC – 190 BC – Greece,
 * 262 -198 BC – Greece,
 * 260 BC – Greece, proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3.
 * c. 250 BC – late s had already begun to use a true zero (a shell glyph) several centuries before in the New World. See.
 * 240 BC – Greece, uses  to quickly isolate prime numbers.
 * 240 BC 190 BC– Greece,
 * 225 BC – Greece, writes On  and names the, , and.
 * 202 BC to 186 BC –China, , a mathematical treatise, is written in.
 * 200 BC – 140 BC – Greece,
 * 150 BC – India, mathematicians in India write the Sthananga Sutra, which contains work on the theory of numbers, arithmetical operations, geometry, operations with, simple equations, , quartic equations, and  and combinations.
 * c. 150 BC – Greece,
 * 150 BC – China, A method of appears in the Chinese text .
 * 150 BC – China, appears in the Chinese text .
 * 150 BC – China, appear in the Chinese text .
 * 150 BC – 75 BC – Phoenician,
 * 190 BC – 120 BC – Greece, develops the bases of.
 * 190 BC - 120 BC – Greece,
 * 160 BC – 100 BC – Greece,
 * 135 BC – 51 BC – Greece,
 * 206 BC to 8 AD – China,
 * 78 BC – 37 BC – China,
 * 50 BC –, a descendant of the (the first   ), begins development in.
 * mid 1st century (as late as 400 AD)
 * final centuries BC – Indian astronomer writes the Vedanga Jyotisha, a Vedic text on  that describes rules for tracking the motions of the sun and the moon, and uses geometry and trigonometry for astronomy.
 * 1st C. BC – Greece,
 * 50 BC – 23 AD – China,

1st millennium AD

 * 1st century – Greece,, (Hero) the earliest fleeting reference to square roots of negative numbers.
 * c 100 – Greece,
 * 60 – 120 – Greece,
 * 70 – 140 – Greece,
 * 78 – 139 – China,
 * c. 2nd century – Greece, of  wrote the .
 * 132 – 192 – China,
 * 240 – 300 – Greece,
 * 250 – Greece, uses symbols for unknown numbers in terms of syncopated, and writes , one of the earliest treatises on algebra.
 * 263 – China, computes  using.
 * 300 – the earliest known use of as a decimal digit is introduced by.
 * 234 – 305 – Greece,
 * 300 – 360 – Greece,
 * 335 – 405– Greece,
 * c. 340 – Greece, states his  and his.
 * 350 – 415 – Byzantine Empire,
 * c. 400 – India, the is written by a mathematicians, which describes a theory of the infinite containing different levels of, shows an understanding of , as well as  to , and computes  of numbers as large as a million correct to at least 11 decimal places.
 * 300 to 500 – the is developed by.
 * 300 to 500 – China, a description of is written by.
 * 412 – 485 – Greece,
 * 420 – 480 – Greece,
 * b 440 – Greece, "I wish everything was mathematics."
 * 450 – China, computes  to seven decimal places. This calculation remains the most accurate calculation for π for close to a thousand years.
 * c. 474 – 558 – Greece,
 * 500 – India, writes the Aryabhata-Siddhanta, which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of  and, and also contains the  and cosine values (in 3.75-degree intervals from 0 to 90 degrees).
 * 480 – 540 – Greece,
 * 490 – 560 – Greece,
 * 6th century – Aryabhata gives accurate calculations for astronomical constants, such as the and, computes π to four decimal places, and obtains whole number solutions to  by a method equivalent to the modern method.
 * 505 – 587 – India,
 * 6th century – India,
 * 535 – 566 – China,
 * 550 – mathematicians give zero a numeral representation in the   system.
 * 7th century – India, gives a rational approximation of the sine function.
 * 7th century – India, invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. He also develops methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon.
 * 628 – Brahmagupta writes the , where zero is clearly explained, and where the modern Indian numeral system is fully developed. It also gives rules for manipulating both, methods for computing square roots, methods of solving  and s, and rules for summing , , and the.
 * 602 – 670 – China,
 * 8th century – India, gives explicit rules for the, gives the derivation of the  of a  using an  procedure, and also deals with the  to base 2 and knows its laws.
 * 8th century – India, gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
 * 773 – Iraq, Kanka brings Brahmagupta's Brahma-sphuta-siddhanta to to explain the Indian system of arithmetic  and the Indian numeral system.
 * 773 – translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor.
 * 9th century – India, discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata's tabular.
 * 810 – The is built in Baghdad for the translation of Greek and  mathematical works into Arabic.
 * 820 – –  mathematician, father of algebra, writes the ', later transliterated as ', which introduces systematic algebraic techniques for solving linear and quadratic equations. Translations of his book on  will introduce the   number system to the Western world in the 12th century. The term  is also named after him.
 * 820 – Iran, conceived the idea of reducing  problems such as  to problems in algebra.
 * c. 850 – Iraq, pioneers  and  in his book on.
 * c. 850 – India, writes the Gaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as the.
 * 895 – Syria, : the only surviving fragment of his original work contains a chapter on the solution and properties of s. He also generalized the, and discovered the by which pairs of s can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
 * c. 900 – Egypt, had begun to understand what we would write in symbols as $$x^n \cdot x^m = x^{m+n}$$
 * 940 – Iran, extracts  using the Indian numeral system.
 * 953 – The arithmetic of the at first required the use of a dust board (a sort of handheld ) because "the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded."  modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world.
 * 953 – Persia, is the "first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the s $$x$$, $$x^2$$, $$x^3$$, ... and $$1/x$$, $$1/x^2$$, $$1/x^3$$, ... and to give rules for  of any two of these. He started a school of algebra which flourished for several hundreds of years". He also discovered the  for  s, which "was a major factor in the development of  based on the decimal system".
 * 975 – Mesopotamia, extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formulae: $$ \sin \alpha = \tan \alpha / \sqrt{1+\tan^2 \alpha} $$ and $$ \cos \alpha = 1 / \sqrt{1 + \tan^2 \alpha}$$.

1000–1500

 * c. 1000 – (Kuhi) solves s higher than the.
 * c. 1000 – first states a special case of.
 * c. 1000 – is discovered by, but it is uncertain who discovers it first between , , and.
 * c. 1000 – introduces the  using the  to Europe.
 * 1000 – writes a book containing the first known  by . He used it to prove the, , and the sum of  . He was "the first who introduced the theory of ic ".
 * c. 1000 – studied a slight variant of 's theorem on s, and he also made improvements on the decimal system.
 * 1020 – gave the formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the  and the volume of the.
 * 1021 – formulated and solved  geometrically.
 * 1030 – writes a treatise on the  and  number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.
 * 1070 – begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations.
 * c. 1100 – Omar Khayyám "gave a complete classification of s with geometric solutions found by means of intersecting s". He became the first to find general solutions of cubic equations and laid the foundations for the development of  and . He also extracted  using the decimal system (Hindu-Arabic numeral system).
 * 12th century – have been modified by Arab mathematicians to form the modern  system (used universally in the modern world).
 * 12th century – the Arabic numeral system reaches Europe through the.
 * 12th century – writes the, which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, , the shadow of the , methods to solve indeterminate equations, and.
 * 12th century – (Bhaskara Acharya) writes the ' ('), which is the first text to recognize that a positive number has two square roots.
 * 12th century – Bhaskara Acharya conceives, and also develops , , a proof for the , proves that division by zero is infinity, computes to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places.
 * 1130 – gave a definition of algebra: "[it is concerned] with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known."
 * 1135 – followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations that "represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry".
 * 1202 – demonstrates the utility of  in his  (Book of the Abacus).
 * 1247 – publishes Shùshū Jiǔzhāng ().
 * 1248 – writes , a 12 volume mathematical treatise containing 170 formulas and 696 problems mostly solved by polynomial equations using the method.
 * 1260 – gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning  and  methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been joint attributed to  as well as Thabit ibn Qurra.
 * c. 1250 – attempts to develop a form of non-Euclidean geometry.
 * 1303 – publishes Precious Mirror of the Four Elements, which contains an ancient method of arranging s in a triangle.
 * 14th century – is considered the father of, who also worked on the power series for &pi; and for sine and cosine functions, and along with other  mathematicians, founded the important concepts of.
 * 14th century –, a Kerala school mathematician, presents a series form of the that is equivalent to its  expansion, states the  of differential calculus, and is also the first mathematician to give the radius of circle with inscribed.

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 – "contributed to the development of s not only for approximating s, but also for s 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  notation in  and . 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 – and  introduced  for algebra and for mathematics in general.
 * 15th century –, 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 – 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 – 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 .
 * 1494 – writes ; introduces primitive symbolic algebra using "co" (cosa) for the unknown.

16th century

 * 1501 – writes the.
 * 1520 – develops a method for solving "depressed" cubic equations (cubic equations without an x2 term), but does not publish.
 * 1522 – explained the use of Arabic digits and their advantages over Roman numerals.
 * 1535 – independently develops a method for solving depressed cubic equations but also does not publish.
 * 1539 – learns Tartaglia's method for solving depressed cubics and discovers a method for depressing cubics, thereby creating a method for solving all cubics.
 * 1540 – solves the.
 * 1544 – publishes Arithmetica integra.
 * 1545 – conceives the idea of s.
 * 1550 –, a mathematician, writes the , the world's first  text, which gives detailed derivations of many calculus theorems and formulae.
 * 1572 – writes Algebra treatise and uses imaginary numbers to solve cubic equations.
 * 1584 – calculates.
 * 1596 – computes π to twenty decimal places using inscribed and circumscribed polygons.

17th century

 * 1614 – discusses Napierian s in Mirifici Logarithmorum Canonis Descriptio.
 * 1617 – discusses decimal logarithms in Logarithmorum Chilias Prima.
 * 1618 – John Napier publishes the first references to in a work on.
 * 1619 – discovers  ( claimed that he also discovered it independently).
 * 1619 – discovers two of the.
 * 1629 – Pierre de Fermat develops a rudimentary.
 * 1634 – shows that the area under a  is three times the area of its generating circle.
 * 1636 – jointly discovered the pair of s 9,363,584 and 9,437,056 along with  (1636).
 * 1637 – Pierre de Fermat claims to have proven in his copy of ' Arithmetica.
 * 1637 – First use of the term by René Descartes; it was meant to be derogatory.
 * 1643 – René Descartes develops.
 * 1654 – and Pierre de Fermat create the theory of.
 * 1655 – writes Arithmetica Infinitorum.
 * 1658 – shows that the length of a cycloid is four times the diameter of its generating circle.
 * 1665 – works on the  and develops his version of.
 * 1668 – and  discover an  for the logarithm while attempting to calculate the area under a.
 * 1671 – develops a series expansion for the inverse- function (originally discovered by ).
 * 1671 – James Gregory discovers.
 * 1673 – also develops his version of infinitesimal calculus.
 * 1675 – Isaac Newton invents an algorithm for the.
 * 1680s – Gottfried Leibniz works on symbolic logic.
 * 1683 – discovers the  and.
 * 1683 – Seki Takakazu develops.
 * 1691 – Gottfried Leibniz discovers the technique of separation of variables for ordinary s.
 * 1693 – prepares the first mortality tables statistically relating death rate to age.
 * 1696 – states  for the computation of certain.
 * 1696 – and  solve, the first result in the.
 * 1699 – calculates π to 72 digits but only 71 are correct.

18th century

 * 1706 – develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
 * 1708 – discovers .  whom the numbers are named after is believed to have independently discovered the numbers shortly after Takakazu.
 * 1712 – develops.
 * 1722 – states  connecting s and s.
 * 1722 – introduces.
 * 1724 – Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities on Lives.
 * 1730 – publishes The Differential Method.
 * 1733 – studies what geometry would be like if  were false.
 * 1733 – Abraham de Moivre introduces the to approximate the  in probability.
 * 1734 – introduces the  for solving first-order ordinary s.
 * 1735 – Leonhard Euler solves the, relating an infinite series to π.
 * 1736 – Leonhard Euler solves the problem of the, in effect creating.
 * 1739 – Leonhard Euler solves the general with.
 * 1742 – conjectures that every even number greater than two can be expressed as the sum of two primes, now known as.
 * 1747 –  the   problem (one-dimensional ).
 * 1748 – discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana.
 * 1761 – proves.
 * 1761 – proves that π is irrational.
 * 1762 – discovers the.
 * 1789 – improves Machin's formula and computes π to 140 decimal places, 136 of which were correct.
 * 1794 – Jurij Vega publishes .
 * 1796 – proves that the  can be constructed using only a.
 * 1796 – conjectures the.
 * 1797 – associates vectors with complex numbers and studies complex number operations in geometrical terms.
 * 1799 – Carl Friedrich Gauss proves the (every polynomial equation has a solution among the complex numbers).
 * 1799 – partially proves the  that  or higher equations cannot be solved by a general formula.

19th century

 * 1801 – , Carl Friedrich Gauss's treatise, is published in Latin.
 * 1805 – Adrien-Marie Legendre introduces the for fitting a curve to a given set of observations.
 * 1806 – discovers the two remaining.
 * 1806 – publishes proof of the  and the.
 * 1807 – announces his discoveries about the.
 * 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 – carries out integrations along paths in the complex plane.
 * 1817 – presents the —a  that is negative at one point and positive at another point must be zero for at least one point in between. Bolzano gives a first formal.
 * 1821 – publishes  which purportedly contains an erroneous “proof” that the  of continuous functions is continuous.
 * 1822 – presents the  for integration around the boundary of a rectangle in the.
 * 1822 – Irisawa Shintarō Hiroatsu analyzes in a.
 * 1823 - is published in the second edition of  Essai sur la théorie des nombres
 * 1824 – partially proves the  that the general  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 s in.
 * 1825 – and Adrien-Marie Legendre prove Fermat's Last Theorem for n = 5.
 * 1825 – discovers.
 * 1826 – gives counterexamples to ’s purported “proof” that the  of continuous functions is continuous.
 * 1828 – George Green proves.
 * 1829 –, , and invent hyperbolic.
 * 1831 – rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green.
 * 1832 – presents a general condition for the solvability of s, thereby essentially founding  and.
 * 1832 – Lejeune Dirichlet proves Fermat's Last Theorem for n = 14.
 * 1835 – Lejeune Dirichlet proves about prime numbers in arithmetical progressions.
 * 1837 – proves that doubling the cube and  are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons.
 * 1837 – develops.
 * 1838 – First mention of in a paper by ; later formalized by . Uniform convergence is required to fix  erroneous “proof” that the  of continuous functions is continuous from Cauchy’s 1821.
 * 1841 – discovers but does not publish the.
 * 1843 – discovers and presents the Laurent expansion theorem.
 * 1843 – discovers the calculus of s and deduces that they are non-commutative.
 * 1847 – formalizes  in The Mathematical Analysis of Logic, defining what is now called.
 * 1849 – shows that s can arise from a combination of periodic waves.
 * 1850 – distinguishes between poles and branch points and introduces the concept of.
 * 1850 – George Gabriel Stokes rediscovers and proves Stokes' theorem.
 * 1854 – introduces.
 * 1854 – shows that quaternions can be used to represent rotations in four-dimensional.
 * 1858 – invents the.
 * 1858 – solves the general quintic equation by means of elliptic and modular functions.
 * 1859 – Bernhard Riemann formulates the, which has strong implications about the distribution of s.
 * 1868 – demonstrates  of ’s  from the other axioms of.
 * 1870 – constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate.
 * 1872 – invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers.
 * 1873 – proves that  is.
 * 1873 – presents his method for finding series solutions to linear differential equations with s.
 * 1874 – proves that the set of all s is  but the set of all real s is .  does not use his, which he published in 1891.
 * 1882 – proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge.
 * 1882 – Felix Klein invents the.
 * 1895 – and  derive the  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 s and the.
 * 1895 – publishes paper "" which started modern topology.
 * 1896 – and  independently prove the.
 * 1896 – presents Geometry of numbers.
 * 1899 – Georg Cantor discovers a contradiction in his set theory.
 * 1899 – presents a set of self-consistent geometric axioms in Foundations of Geometry.
 * 1900 – David Hilbert states his, which show where some further mathematical work is needed.

20th century

 * 1901 – develops the.
 * 1901 – publishes on.
 * 1903 – presents a  algorithm
 * 1903 – gives considerably simpler proof of the prime number theorem.
 * 1908 – axiomizes, thus avoiding Cantor's contradictions.
 * 1908 – solves the Riemann problem about the existence of a differential equation with a given  and uses Sokhotsky – Plemelj formulae.
 * 1912 – presents the.
 * 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent n = 5.
 * 1915 – proves, which shows that every  has a corresponding.
 * 1916 – introduces . This conjecture is later generalized by.
 * 1919 – defines  B2 for s.
 * 1921 – Emmy Noether introduces the first general definition of a.
 * 1928 – begins devising the principles of  and proves the.
 * 1929 – Emmy Noether introduces the first general representation theory of groups and algebras.
 * 1930 – shows that the  has no solution.
 * 1930 – introduces.
 * 1931 – proves, which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
 * 1931 – develops theorems in  and es.
 * 1933 – and  present the.
 * 1933 – publishes his book Basic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung), which contains an  based on.
 * 1938 - introduces.
 * 1940 – Kurt Gödel shows that neither the nor the  can be disproven from the standard axioms of set theory.
 * 1942 – and  develop a  algorithm.
 * 1943 – proposes a method for nonlinear least squares fitting.
 * 1945 – introduces.
 * 1945 – and  start.
 * 1945 – and  give the  for (co-)homology.
 * 1946 – introduces the.
 * 1948 – John von Neumann mathematically studies .I
 * 1948 - and  prove independently in an elementary way the.
 * 1949 – and L.R. Smith compute π to 2,037 decimal places using.
 * 1949 – develops notion of.
 * 1950 – and John von Neumann present  dynamical systems.
 * 1953 – introduces the idea of thermodynamic  algorithms.
 * 1955 – et al. publish the complete list of.
 * 1955 –, , Stanisław Ulam, and numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior.
 * 1956 – describes a  of s.
 * 1956 – discovers the existence of an  in seven dimensions, inaugurating the field of.
 * 1957 – develops.
 * 1957 – provides the  for crease-free.
 * 1958 – 's proof of the is published.
 * 1959 – creates.
 * 1960 – invents the  algorithm.
 * 1960 – and  present the.
 * 1961 – and  compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
 * 1961 – and  independently develop the  to calculate the  and  of a matrix.
 * 1961 – Stephen Smale proves the for all dimensions greater than or equal to 5.
 * 1962 – proposes the.
 * 1962 – becomes the third African American woman to receive a PhD in mathematics.
 * 1963 – uses his technique of  to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory.
 * 1963 – and  analytically study the  in the continuum limit and find that the  governs this system.
 * 1963 – meteorologist and mathematician published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and s or  – also the.
 * 1965 – Iranian mathematician founded  theory as an extension of the classical notion of  and he founded the field of.
 * 1965 – Martin Kruskal and Norman Zabusky numerically study colliding in  and find that they do not disperse after collisions.
 * 1965 – and  present an influential fast Fourier transform algorithm.
 * 1966 – presents two methods for computing the  in terms of a polynomial in that matrix.
 * 1966 – presents.
 * 1967 – formulates the influential  of conjectures relating number theory and representation theory.
 * 1968 – and  prove the  about the index of s.
 * 1973 – founded the field of.
 * 1974 - solves the last and deepest of the, completing the program of Grothendieck.
 * 1975 – publishes Les objets fractals, forme, hasard et dimension.
 * 1976 – and  use a computer to prove the.
 * 1981 – gives an influential talk "Simulating Physics with Computers" (in 1980  proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)).
 * 1983 – proves the  and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
 * 1985 – proves the.
 * 1986 – proves.
 * 1987 –, , , and use iterative modular equation approximations to elliptic integrals and a   to compute π to 134 million decimal places.
 * 1991 – and  develop.
 * 1992 – and  develop the, one of the first examples of a  that is exponentially faster than any possible deterministic classical algorithm.
 * 1994 – proves part of the  and thereby proves.
 * 1994 – formulates, a  for.
 * 1995 – discovers  capable of finding the nth binary digit of π.
 * 1998 – (almost certainly) proves the.
 * 1999 – the full is proven.
 * 2000 – the proposes the seven  of unsolved important classic mathematical questions.

21st century

 * 2002 –, , and of  present an unconditional deterministic  algorithm to determine whether a given number is  (the ).
 * 2002 –, Y. Ushiro, , and a team of nine more compute π to 1241.1 billion digits using a  64-node.
 * 2002 – proves.
 * 2003 – proves the.
 * 2004 – the, a collaborative work involving some hundred mathematicians and spanning fifty years, is completed.
 * 2004 – and  prove the.
 * 2007 – a team of researchers throughout North America and Europe used networks of computers to map.
 * 2009 – had been  by.
 * 2010 – and  solve the.
 * 2013 – proves the first finite bound on gaps between prime numbers.
 * 2014 – Project Flyspeck announces that it completed proof of.
 * 2014 – Using Alexander Yee's y-cruncher "houkouonchi" successfully calculated π to 13.3 trillion digits.
 * 2015 – solved The
 * 2015 – found that a quasipolynomial complexity algorithm would solve the
 * 2016 – Using Alexander Yee's y-cruncher Peter Trueb successfully to 22.4 trillion digits
 * 2019 – using y-cruncher v0.7.6 calculated π to 31.4 trillion digits.