View source for Timeline of algebra

{{Short description|Notable events in the history of algebra}}
The following is a timeline of key developments of [[algebra]]:

{| class="wikitable" style="width:100%;"
|-
! style="width:15%" | Year || Event
|-
| {{circa}} 1800 BC || An [[First Babylonian dynasty|Old Babylonian]] tablet (Strasbourg 363) seeks the solution of a [[quadratic equation]].<ref>{{cite journal
 | last = Archibald | first = Raymond Clare
 | date = December 1936
 | doi = 10.1086/347127
 | issue = 1
 | journal = Isis
 | jstor = 225054
 | pages = 63–81
 | publisher = University of Chicago Press
 | title = Babylonian Mathematics
 | volume = 26}} See also {{cite web|url=https://cdli.mpiwg-berlin.mpg.de/artifacts/414660|title=StrKT 07 (P414660)|work=Cuneiform Digital Library Initiative}}, Mathematical tablet excavated in Uruk (mod. Warka), dated to the Old Babylonian (ca. 1900–1600&nbsp;BC) period and now kept in Bibliothèque Nationale et Universitaire de Strasbourg, Strasbourg, France, museum number BNUS 363.</ref>
|-
| {{circa}} 1800 BC || The ''[[Plimpton 322]]'' tablet gives a table of [[Pythagorean triple]]s in [[Babylonia]]n [[Cuneiform script]].<ref>{{cite book|last=Anglin|first=W.S|title=Mathematics: A Concise History and Philosophy|url=https://books.google.com/books?id=13dKav77vGsC&pg=PA8|year=1994|publisher=Springer|isbn=978-0-387-94280-3|page=8}}</ref>
|-
| 1800 BC || [[Berlin Papyrus 6619]] (19th dynasty) contains a [[quadratic equation]] and its solution.<ref>{{cite book|last=Smith|first=David Eugene Smith|title=History of Mathematics|year=1958|publisher=Courier Dover Publications |isbn=978-0-486-20430-7 |page=443 |url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA443}}</ref><ref>{{cite web |title=Egyptian Mathematics Papyri |work=Mathematicians and Scientists of the African Diaspora |publisher=The State University of New York at Buffalo |url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin}}</ref>
|-
| 800 BC || [[Baudhayana]], author of the Baudhayana [[Sulba Sutras|Sulba Sutra]], a [[Vedic Sanskrit]] geometric text, contains quadratic equations, and calculates the [[square root of 2]] correct to five [[decimal place]]s
|-
| {{circa}} 300 BC || [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots.<ref>{{cite book|author=Euclid|title=Euclid's Elements|date=January 1956|url=https://books.google.com/books?id=tXWKAAAAQBAJ&pg=PA435|publisher=Courier Dover Publications|isbn=978-0-486-60089-5|page=258}}</ref>
|-
| {{circa}} 300 BC || A geometric construction for the solution of the cubic is sought (doubling the cube problem).  It is now well known that the general cubic has no such solution using [[Euclidean tools]].
|-
| 150 BC || [[Jainism|Jain]] mathematicians in [[History of India|India]] write the “Sthananga Sutra”, which contains work on the [[theory of numbers]], arithmetical operations, [[geometry]], operations with [[fractions]], simple equations, [[cubic equations]], [[quartic equation]]s, and [[permutation]]s and [[combination]]s.
|-
| 250 BC || Algebraic equations are treated in the Chinese mathematics book ''[[The Nine Chapters on the Mathematical Art|Jiuzhang suanshu]]'' (''The Nine Chapters on the Mathematical Art''), which contains solutions of linear equations solved using the [[False position method|rule of double false position]], geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of [[simultaneous linear equations]].<ref name="oxf">{{cite book|last1=Crossley|first1=John|last2=W.-C. Lun|first2=Anthony|title=The Nine Chapters on the Mathematical Art: Companion and Commentary|url=https://books.google.com/books?id=eiTJHRGTG6YC&pg=PA349|year=1999|publisher=Oxford University Press|isbn=978-0-19-853936-0|page=349}}</ref>
|-
| 1st century AD || [[Hero of Alexandria]] gives the earliest fleeting reference to [[imaginary unit|square roots of negative numbers]].
|-
| {{circa}} 150 || Greek mathematician [[Hero of Alexandria]], treats algebraic equations in three volumes of mathematics.
|-
| {{circa}} 200 || Hellenistic mathematician [[Diophantus]], who lived in Alexandria and is often considered to be the "father of algebra", writes his famous ''[[Arithmetica]]'', a work featuring solutions of algebraic equations and on the theory of numbers.
|-
| 499 || Indian mathematician [[Aryabhata]], in his treatise ''Aryabhatiya'', obtains whole-number solutions to linear equations, describes the general solution of the indeterminate linear equation.{{Citation needed|date=April 2007}}
|-
| {{circa}} 625 || Chinese mathematician [[Wang Xiaotong]] finds numerical solutions to certain cubic equations.<ref>{{MacTutor|id=Wang_Xiaotong |title=Wang Xiaotong}}</ref>
|-
| {{circa}} 7th century<br>Dates vary from the 3rd to the 12th centuries.{{sfnp|Hayashi|2005|p=371|ps=. "The dates so far proposed for the Bakhshali work vary from the third to the twelfth centuries AD, but a recently made comparative study has shown many similarities, particularly in the style of exposition and terminology, between Bakhshalī work and Bhāskara I's commentary on the ''Āryabhatīya''.  This seems to indicate that both works belong to nearly the same period, although this does not deny the possibility that some of the rules and examples in the Bakhshālī work date from anterior periods."}} || The ''[[Indian mathematics#Bakhshali Manuscript|Bakhshali Manuscript]]'' written in [[Middle kingdoms of India|ancient India]] uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of [[linear equation]]s with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.{{Citation needed|date=April 2007}}
|-
| 7th century || [[Brahmagupta]] invents the method of solving indeterminate equations of the second degree . 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 ''[[Brahmasphutasiddhanta|Brahmasphuta-siddhanta]]'', where zero is clearly explained, and where the modern [[place-value]] [[Indian numerals|Indian numeral]] system is fully developed. It also gives rules for manipulating both [[negative and positive numbers]], methods for computing [[square root]]s, methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]], and the [[Brahmagupta theorem]]
|-
| 8th century || [[Virasena]] gives explicit rules for the [[Fibonacci sequence]], gives the derivation of the [[volume]] of a [[frustum]] using an [[Infinity|infinite]] procedure, and also deals with the [[logarithm]] to [[base 2]] 
|-
| {{circa}} 800 || The [[Abbasid]] patrons of learning, [[al-Mansur]], [[Haroun al-Raschid]], and [[al-Mamun]], has Greek, Babylonian, and Indian mathematical and scientific works translated into Arabic and begins a cultural, scientific and mathematical awakening after a century devoid of mathematical achievements.{{sfnp|Boyer|1991|loc="The Arabic Hegemony" p. 227|ps=. "The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. To Baghdad at that time were called scholars from Syria, Iran, and Mesopotamia, including Jews and Nestorian Christians; under three great Abbasid patrons of learning – al-Mansur, Haroun al-Raschid, and al-Mamun – The city became a new Alexandria. It was during the caliphate of al-Mamun (809–833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's ''Almagest'' and a complete version of Euclid's ''Elements''. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria."}}
|-
| 820 || The word ''algebra'' is derived from operations described in the treatise written by the [[Mathematics in medieval Islam|Persian mathematician]], [[Muhammad ibn Musa al-Khwarizmi|Muḥammad ibn Mūsā al-Ḵhwārizmī]], titled ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Kitab al-Jabr wa-l-Muqabala]]'' (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of [[linear equation|linear]] and quadratic equations. Al-Khwarizmi is often considered the "father of algebra", for founding algebra as an independent discipline and for introducing the methods of "[[Reduction (mathematics)|reduction]]" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which was what he originally used the term ''al-jabr'' to refer to.{{sfnp|Boyer|1991|loc="The Arabic Hegemony" p. 229|ps=. "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation."}} His algebra was also no longer concerned "with a series of [[Mathematical problem|problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."<ref name=Rashed-Armstrong>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-7923-2565-6 | oclc=29181926 | pages=11–2 }}</ref>
|-
| {{circa}} 990 || Persian mathematician [[Al-Karaji]] (also known as al-Karkhi), in his treatise ''Al-Fakhri'', further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the [[monomial]]s x, x<sup>2</sup>, x<sup>3</sup>, .. and 1/x, 1/x<sup>2</sup>, 1/x<sup>3</sup>, .. and gives rules for the products of any two of these.<ref name=Al-Karaji/> He also discovers the first numerical solution to equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c.{{sfnp|Boyer|1991|loc="The Arabic Hegemony" p. 239|ps=. "Abu'l Wefa was a capable algebraist aws well as a trionometer. [..] His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! [..] In particular, to al-Karaji is attributed the first numerical solution of equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c (only equations with positive roots were considered)."}} Al-Karaji is also regarded as the first person to free algebra from [[Geometry|geometrical]] operations and replace them with the type of [[arithmetic]] operations which are at the core of algebra today. His work on algebra and [[polynomial]]s, gave the rules for arithmetic operations to manipulate polynomials. The [[History of mathematics|historian of mathematics]] F. Woepcke, in ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi'' ([[Paris]], 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic [[calculus]]". Stemming from this, Al-Karaji investigated [[binomial coefficients]] and [[Pascal's triangle]].<ref name=Al-Karaji>{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}</ref>
|-
| 895 || [[Thabit ibn Qurra]]: the only surviving fragment of his original work contains a chapter on the solution and properties of [[cubic equation]]s. He also generalized the [[Pythagorean theorem]], and discovered the [[Thabit number|theorem]] by which pairs of [[amicable numbers]] can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
|-
| 953 || [[Al-Karaji]] 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 [is] first to define the [[monomial]]s <math>x</math>, <math>x^2</math>, <math>x^3</math>, … and <math>1/x</math>, <math>1/x^2</math>, <math>1/x^3</math>, … and to give rules for [[product (mathematics)|products]] of any two of these. He start[s] a school of algebra which flourished for several hundreds of years”. He also discovers the [[binomial theorem]] for [[integer]] [[exponent]]s, which “was a major factor in the development of [[numerical analysis]] based on the decimal system.”
|-
| {{circa}} 1000 || [[Abū Sahl al-Qūhī]] (Kuhi) solves [[equation]]s higher than the [[Quadratic equation|second degree]].
|-
| {{circa}} 1050 || Chinese mathematician [[Jia Xian]] finds numerical solutions of polynomial equations of arbitrary degree.<ref>{{MacTutor|id=Jia_Xian|title=Jia Xian}}</ref>
|-
| 1070 || [[Omar Khayyam|Omar Khayyám]] begins to write ''Treatise on Demonstration of Problems of Algebra'' and classifies cubic equations.
|-
| 1072 || Persian mathematician [[Omar Khayyam|Omar Khayyám]] gives a complete classification of cubic equations with positive roots and gives general geometric solutions to these equations found by means of intersecting conic sections.{{sfnp|Boyer|1991|loc="The Arabic Hegemony" pp. 241–242|ps=. "Omar Khayyám (ca. 1050–1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyám provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyám took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots)."}}
|-
| 12th century || [[Bhaskara II|Bhaskara Acharya]] writes the “[[Bijaganita]]” (“[[Algebra]]”), which is the first text that recognizes that a positive number has two square roots
|-
| 1130 || [[Al-Samawal]] gives 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.”<ref name=MacTutor>[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics], ''[[MacTutor History of Mathematics archive]]'', [[University of St Andrews]], Scotland</ref>
|-
| {{circa}} 1200 || [[Sharaf al-Dīn al-Tūsī]] (1135–1213) writes the ''Al-Mu'adalat'' (''Treatise on Equations''), which deals with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He uses what would later be known as the "[[Ruffini's rule|Ruffini]]–[[Horner's method|Horner]] method" to [[Numerical analysis|numerically]] approximate the [[root of a function|root]] of a cubic equation. He also develops the concepts of the [[maxima and minima]] of curves in order to solve cubic equations which may not have positive solutions.<ref>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> He understands the importance of the [[discriminant]] of the cubic equation and uses an early version of [[Gerolamo Cardano|Cardano]]'s formula<ref>{{Cite book | last1=Rashed | first1=Roshdi | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=Springer | isbn=0-7923-2565-6 | pages=342–3 }}</ref> to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the [[derivative]] of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes.<ref name=Berggren>{{Cite journal|first=J. L.|last=Berggren|year=1990|title=Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat|journal=Journal of the American Oriental Society|volume=110|issue=2|pages=304–9|quote=Rashed has argued that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance for investigating conditions under which cubic equations were solvable; however, other scholars have suggested quite difference explanations of Sharaf al-Din's thinking, which connect it with mathematics found in Euclid or Archimedes.|doi=10.2307/604533|jstor=604533|author2=Al-Tūsī, Sharaf Al-Dīn|last3=Rashed|first3=Roshdi|last4=Al-Tusi|first4=Sharaf Al-Din}}</ref>
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| 1202 || [[Leonardo Fibonacci]] of [[Pisa]] publishes his ''[[Liber Abaci]]'', a work on algebra that introduces Arabic numerals to Europe.<ref>{{cite book|last=Ball|first=W. W. Rouse|title=A Short Account of the History of Mathematics|url=https://books.google.com/books?id=_sT_psl3uYkC&pg=PA167|year=1960|publisher=Courier Dover Publications|isbn=978-0-486-15784-9|pages=167}}</ref>
|-
| {{circa}} 1300 || Chinese mathematician [[Zhu Shijie]] deals with [[polynomial algebra]], solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, [[Quintic equation|quintic]] and higher-order polynomial equations.<ref>{{cite book|last=Grattan-Guinness|first=Ivor|title=The Norton History of the Mathematical Sciences|url=https://books.google.com/books?id=3hwQQhcquBAC&pg=PA108|year=1997|publisher=W.W. Norton|isbn=978-0-393-04650-2|page=108}}</ref>
|-
| {{circa}} 1400 || Indian mathematician [[Madhava of Sangamagrama]] documents infinite series approximations of trigonometry functions.<ref>{{cite journal|journal=Mathematics Magazine|title=Ideas of Calculus in Islam and India|last=Katz|first=Victor J.|url=https://doi.org/10.2307/2691411}}</ref>
|-
| 15th century || [[Nilakantha Somayaji]], a [[Kerala school of astronomy and mathematics|Kerala school]] mathematician, writes the “Aryabhatiya Bhasya”, which contains work on infinite-series expansions, problems of algebra, and spherical geometry
|-
| 1412–1482 || Arab mathematician [[Abū al-Hasan ibn Alī al-Qalasādī]] takes "the first steps toward the introduction of [[Mathematical notation|algebraic symbolism]]." He uses "short Arabic words, or just their initial letters, as mathematical symbols."<ref name=Qalasadi>{{MacTutor Biography|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>
|-
| 1535 || [[Scipione del Ferro]] and [[Niccolò Fontana Tartaglia]], in Italy, independently solve the general cubic equation.<ref name="Stewart">{{cite book|first=Ian|last=Stewart|title=Galois Theory|url=https://archive.org/details/galoistheory0074stew|url-access=registration|edition=Third|publisher=Chapman & Hall/CRC Mathematics|year=2004|isbn=9781584883937 }}</ref>
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| 1545 || Girolamo [[Gerolamo Cardano|Cardano]] publishes ''Ars magna'' -''The great art'' which gives del Ferro's solution to the cubic equation<ref name="Stewart" /> and [[Lodovico Ferrari]]'s solution to the quartic equation.
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| 1572 || [[Rafael Bombelli]] recognizes the complex roots of the cubic and improves current notation.<ref>{{cite book|last=Cooke|first=Roger|title=Classical Algebra: Its Nature, Origins, and Uses|url=https://books.google.com/books?id=JG-skeT1eWAC&pg=PA70|date=16 May 2008|publisher=John Wiley & Sons|isbn=978-0-470-27797-3|page=70}}</ref>
|-
| 1591 || [[Franciscus Vieta]] develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in ''In artem analyticam isagoge''.{{Citation needed|date=April 2007}}
|-
| 1608 || [[Christopher Clavius]] publishes his ''Algebra''
|-
| 1619 || [[René Descartes]] discovers [[analytic geometry]]. ([[Pierre de Fermat]] claimed that he also discovered it independently.)
|-
| 1631 || [[Thomas Harriot]] in a posthumous publication is the first to use symbols < and > to indicate "less than" and "greater than", respectively.{{sfnp|Boyer|1991|loc="Prelude to Modern Mathematics" p. [https://archive.org/details/historyofmathema00boye/page/306 306]|ps=. "Harriot knew of relationships between roots and coefficients and between roots and factors, but like Viète he was hampered by failure to take note of negative and imaginary roots. In notation, however, he advanced the use of symbolism, being responsible for the signs > and < for 'greater than' and 'less than.'"}}
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| 1637 || Pierre de Fermat claims to have proven [[Fermat's Last Theorem]] in his copy of [[Diophantus]]' ''Arithmetica'',
|-
| 1637 || René Descartes introduces the use of the letters ''z'', ''y'', and ''x'' for unknown quantities.<ref>{{cite journal
 | last = Cajori
 | first = Florian 
 | author-link = Florian Cajori
 | title = How x Came to Stand for Unknown Quantity
 | journal = School Science and Mathematics
 | volume = 19
 | pages = 698–699
 | date = 1919
 | issue = 8 
 | url = https://books.google.com/books?id=X11LAAAAMAAJ&q=webster+%22prevailingly+transcribed+as+xei.%22&pg=PA698
 | doi=10.1111/j.1949-8594.1919.tb07713.x
 }}</ref><ref>{{cite book
 | last = Cajori
 | first = Florian 
 | title = A History of Mathematical Notations
 | publisher = Open Court Publishing 
 | volume = 1
 | date = 1928
 | location = Chicago
 | pages = 381
 | isbn = 9780486677668 
 | url = https://books.google.com/books?id=7juWmvQSTvwC&q=Descartes
 }}</ref>
|-
| 1637 || The term [[imaginary number]] is first used by [[René Descartes]]; it is meant to be derogatory.
|-
| 1682 || [[Gottfried Leibniz|Gottfried Wilhelm Leibniz]] develops his notion of symbolic manipulation with formal rules which he calls ''characteristica generalis''.<ref>{{cite book|last=Struik|first=D. J.|title=A Source Book in Mathematics, 1200–1800|url=https://books.google.com/books?id=XmRsZhJZGhEC&pg=PA123|publisher=Harvard University Press|isbn=978-0-674-82355-6|page=123}}</ref>
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| 1683 || Japanese mathematician [[Kowa Seki]], in his ''Method of solving the dissimulated problems'', discovers the [[determinant]],<ref name=MacTutorSeki/> discriminant,{{Citation needed|date=July 2008}} and [[Bernoulli number]]s.<ref name=MacTutorSeki>{{MacTutor|id=Seki|title=Takakazu Shinsuke Seki}}</ref>
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| 1693 || [[Gottfried Wilhelm Leibniz|Leibniz]] solves systems of simultaneous linear equations using matrices and determinants.{{Citation needed|date=April 2007}}
|-
| 1722 || [[Abraham de Moivre]] states [[de Moivre's formula]] connecting [[trigonometric function]]s and [[complex number]]s,
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| 1750 || [[Gabriel Cramer]], in his treatise ''Introduction to the analysis of algebraic curves'', states [[Cramer's rule]] and studies [[algebraic curves]], matrices and determinants.<ref>{{MacTutor|id=Cramer|title=Gabriel Cramer}}</ref>
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| 1797 || [[Caspar Wessel]] associates vectors with [[complex number]]s 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 (mathematician)|Paolo Ruffini]] partially proves the [[Abel–Ruffini theorem]] that [[Quintic equation|quintic]] or higher equations cannot be solved by a general formula,
|-
| 1806 || [[Jean-Robert Argand]] publishes proof of the [[Fundamental theorem of algebra]] and the [[Argand diagram]],
|-
| 1824 || [[Niels Henrik Abel]] proves that the general quintic equation is insoluble by radicals.<ref name="Stewart" />
|-
| 1832 || [[Galois theory]] is developed by [[Évariste Galois]] in his work on abstract algebra.<ref name="Stewart" />
|-
| 1843 || [[William Rowan Hamilton]] discovers [[quaternions]].
|-
| 1853 || [[Arthur Cayley]] provides a modern definition of groups.
|-
| 1847 || [[George Boole]] formalizes [[symbolic logic]] in  ''The Mathematical Analysis of Logic'', defining what now is called [[Boolean algebra (logic)|Boolean algebra]].
|-
| 1873 || [[Charles Hermite]] proves that [[e (mathematical constant)|e]] is transcendental.
|-
| 1878 || Charles Hermite solves the general quintic equation by means of elliptic and modular functions.
|-
| 1926 || [[Emmy Noether]] extends Hilbert's theorem on the finite basis problem to representations of a finite group over any field.
|-
| 1929 || [[Emmy Noether]] combines work on structure theory of [[associative algebra]]s and the representation theory of groups into a single arithmetic theory of [[module (mathematics)|modules]] and [[ideal (ring theory)|ideals]] in [[ring (mathematics)|rings]] satisfying [[ascending chain condition]]s, providing the foundation for modern algebra.
|-
| 1981 || [[Mikhail Gromov (mathematician)|Mikhail Gromov]] develops the theory of [[hyperbolic group]]s, revolutionizing both infinite group theory and global differential geometry,
|}

==See also==
* {{portal-inline|Mathematics}}
* [[History of algebra]] – Historical development of algebra

==References==
{{Reflist}}
*{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |year=1991 |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |isbn=0-471-54397-7 |quote= |url-access=registration |url=https://archive.org/details/historyofmathema00boye}}
*{{cite book
 | last1=Hayashi
 | first1=Takao
 | year=2005
 | chapter=Indian Mathematics
 | pages=360–375
 | editor1-last=Flood
 | editor1-first=Gavin
 | title=The Blackwell Companion to Hinduism
 | location=Oxford |publisher=Basil Blackwell
 | isbn=978-1-4051-3251-0
 }}
{{History of mathematics}}

{{DEFAULTSORT:Timeline Of Algebra}}
[[Category:History of algebra]]
[[Category:Mathematics timelines|Algebra]]