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| A timeline of key algebraic developments are as follows:
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| ! style="width:15%" | Year || Event
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| | Circa 1800 BC || The [[First Babylonian Dynasty|Old Babylonian]] [[Strassburg tablet]] seeks the solution of a quadratic elliptic equation.{{Citation needed|date=April 2007}}
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| | Circa 1800 BC || The ''[[Plimpton 322]]'' tablet gives a table of [[Pythagorean triples]] in [[Babylonia]]n [[Cuneiform script]].{{Citation needed|date=July 2007}}
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| | 1800 BC || [[Berlin papyrus 6619]] (19th dynasty) contains a quadratic equation and its solution. [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin]
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| | 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 places
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| | Circa 300 BC || In Book II of his Elements, [[Euclid]] gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.{{Citation needed|date=April 2007}}
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| | 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]].{{Citation needed|date=April 2007}}
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| | 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 equations, and [[permutations]] and [[combinations]]
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| | Circa 100 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.{{Citation needed|date=April 2007}}
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| | 1st century || [[Heron of Alexandria]], the earliest fleeting reference to square roots of negative numbers.
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| | Circa 150 || Greek mathematician [[Hero of Alexandria]], treats algebraic equations in three volumes of mathematics.{{Citation needed|date=April 2007}}
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| | Circa 200 || Hellenistic mathematician [[Diophantus]] 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.{{Citation needed|date=April 2007}}
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| | 499 || Indian mathematician [[Aryabhata]], in his treatise ''Aryabhatiya'', obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a [[differential equation]].{{Citation needed|date=April 2007}}
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| | Circa 625 || Chinese mathematician [[Wang Xiaotong]] finds numerical solutions to certain cubic equations.<ref>{{MacTutor|id=Wang_Xiaotong |title=Wang Xiaotong}}</ref>
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| | Circa 7th century<br>Dates vary from the 3rd to the 12th centuries.<ref>{{Harv|Hayashi|2005|p=371}} Quote:"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."</ref> || 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 equations]] 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}}
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| | 7th century || [[Brahmagupta]] 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
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| | 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 roots]], methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]], and the [[Brahmagupta theorem]]
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| | 700s || [[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]] and knows its laws
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| | Circa 800 || The [[Abbasid]] patrons of learning, [[al-Mansur]], [[Haroun al-Raschid]], and [[al-Mamun]], had Greek, Babylonian, and Indian mathematical and scientific works translated into Arabic and began a cultural, scientific and mathematical awakening after a century devoid of mathematical achievements.<ref name="Boyer Intro Islamic Algebra">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|page=227|quote=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.}}</ref>
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| | 820 || The word ''algebra'' is derived from operations described in the treatise written by the [[Mathematics in medieval Islam|Persian mathematician]], {{Unicode|[[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.<ref name=Boyer-229>{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 229}} "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."</ref> His algebra was also no longer concerned "with a series of [[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=harv}}</ref>
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| | Circa 850 || [[Persian people|Persian]] mathematician [[al-Mahani]] conceived the idea of reducing geometrical problems such as [[Doubling the cube|duplicating the cube]] to problems in algebra.{{Citation needed|date=April 2007}}
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| | 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 discovered the first numerical solution to equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c.<ref name="Boyer al-Karkhi ax2n">{{Harv|Boyer|1991|loc="The Arabic Hegemony" p. 239}} "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),"</ref> 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>
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| | 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 number]]s can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
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| | 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 was 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 started a school of algebra which flourished for several hundreds of years”. He also discovered the [[binomial theorem]] for [[integer]] [[exponent]]s, which “was a major factor in the development of [[numerical analysis]] based on the decimal system.”
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| | ca. 1000 || [[Abū Sahl al-Qūhī]] (Kuhi) solves [[equation]]s higher than the [[Quadratic equation|second degree]].
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| | Circa 1050 || Chinese mathematician [[Jia Xian]] finds numerical solutions of polynomial equations of arbitrary degree.<ref>{{MacTutor|id=Jia_Xian|title=Jia Xian}}</ref>
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| | 1070 || [[Omar Khayyám]] begins to write ''Treatise on Demonstration of Problems of Algebra'' and classifies cubic equations.
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| | 1072 || Persian mathematician [[Omar Khayyam]] 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.<ref name="Boyer Omar Khayyam positive roots">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=241–242|quote=Omar Khayyam (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 Khayyam 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 Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots).}}</ref>
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| | 1100s || [[Bhaskara II|Bhaskara Acharya]] writes the “[[Bijaganita]]” (“[[Algebra]]”), which is the first text that recognizes that a positive number has two square roots
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| | 1130 || [[Al-Samawal]] 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.”<ref name=MacTutor/>
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| | 1135 || [[Sharafeddin Tusi]] followed al-Khayyam's application of algebra to geometry, and wrote a treatise on [[cubic equation]]s which “represents an essential contribution to another [[algebra]] which aimed to study [[curve]]s by means of [[equation]]s, thus inaugurating the beginning of [[algebraic geometry]].”<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>
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| | Circa 1200 || [[Sharaf al-Dīn al-Tūsī]] (1135–1213) wrote the ''Al-Mu'adalat'' (''Treatise on Equations''), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "[[Ruffini's rule|Ruffini]]-[[Horner scheme|Horner]] method" to [[Numerical analysis|numerically]] approximate the [[root of a function|root]] of a cubic equation. He also developed 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 understood the importance of the [[discriminant]] of the cubic equation and used 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=harv}}</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=harv}}</ref>
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| | 1202 || Algebra is introduced to [[Europe]] largely through the work of [[Leonardo Fibonacci]] of [[Pisa]] in his work ''[[Liber Abaci]]''.{{Citation needed|date=April 2007}}
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| | 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.{{Citation needed|date=April 2007}}
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| | Circa 1400 || [[Jamshīd al-Kāshī]] developed an early form of [[Newton's method]] to numerically solve the equation <math>x^P - N = 0</math> to find roots of ''N''.<ref>Tjalling J. Ypma (1995), "Historical development of the Newton-Raphson method", ''SIAM Review'' '''37''' (4): 531–51, {{doi|10.1137/1037125}}</ref>
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| | Circa 1400 || Indian mathematician [[Madhava of Sangamagrama]] finds the solution of [[Transcendental function|transcendental equations]] by [[iteration]], [[iterative method]]s for the solution of non-linear equations, and solutions of differential equations.{{Citation needed|date=April 2007}}
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| | 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
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| | 1412–1482 || Arab mathematician [[Abū al-Hasan ibn Alī al-Qalasādī]] took "the first steps toward the introduction of [[Mathematical notation|algebraic symbolism]]." He used "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>
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| | 1535 || [[Niccolò Fontana Tartaglia]] and others mathematicians in Italy independently solved the general cubic equation.<ref name="Stewart">{{cite book|first=Ian|last=Stewart|title=Galois Theory|edition=Third|publisher=Chapman & Hall/CRC Mathematics|year=2004}}</ref>
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| | 1545 || Girolamo [[Cardano]] publishes ''Ars magna'' -''The great art'' which gives Fontana's solution to the general quartic equation.<ref name="Stewart" />
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| | 1572 || [[Rafael Bombelli]] recognizes the complex roots of the cubic and improves current notation.{{Citation needed|date=April 2007}}
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| | 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}}
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| | 1619 || [[René Descartes]] discovers [[analytic geometry]] ([[Pierre de Fermat]] claimed that he also discovered it independently),
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| | 1631 || [[Thomas Harriot]] in a posthumous publication is the first to use symbols < and > to indicate "less than" and "greater than", respectively.<ref>{{cite book |first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second |publisher=John Wiley & Sons, Inc. |year=1991 |chapter=Prelude to Modern Mathematics |page=306 |isbn=0-471-54397-7 |quote=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."}}</ref>
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| | 1637 || Pierre de Fermat claims to have proven [[Fermat's Last Theorem]] in his copy of [[Diophantus]]' ''Arithmetica'',
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| | 1637 || First use of the term [[imaginary number]] by [[René Descartes]]; it was meant to be derogatory.
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| | 1682 || [[Gottfried Leibniz|Gottfried Wilhelm Leibniz]] develops his notion of symbolic manipulation with formal rules which he calls ''characteristica generalis''.{{Citation needed|date=April 2007}}
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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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| | 1685 || Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations.{{Citation needed|date=April 2007}}
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| | 1693 || [[Gottfried Wilhelm Leibniz|Leibniz]] solves systems of simultaneous linear equations using matrices and determinants.{{Citation needed|date=April 2007}}
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| | 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,
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| | 1799 || Carl Friedrich Gauss proves the [[fundamental theorem of algebra]] (every polynomial equation has a solution among the complex numbers),
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| | 1799 || [[Paolo Ruffini]] partially proves the [[Abel–Ruffini theorem]] that [[Quintic equation|quintic]] or higher equations cannot be solved by a general formula,
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| | 1806 || [[Jean-Robert Argand]] publishes proof of the [[Fundamental theorem of algebra]] and the [[Argand diagram]],
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| | 1824 || [[Niels Henrik Abel]] proved that the general quintic equation is insoluble by radicals.<ref name="Stewart" />
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| | 1832 || [[Galois theory]] is developed by [[Évariste Galois]] in his work on abstract algebra.<ref name="Stewart" />
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| | 1847 || [[George Boole]] formalizes [[symbolic logic]] in ''The Mathematical Analysis of Logic'', defining what now is called [[Boolean algebra (logic)|Boolean algebra]],
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| | 1873 || [[Charles Hermite]] proves that [[e (mathematical constant)|e]] is transcendental,
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| | 1878 || Charles Hermite solves the general quintic equation by means of elliptic and modular functions
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| | 1981 || [[Mikhail Gromov (mathematician)|Mikhail Gromov]] develops the theory of [[hyperbolic group]]s, revolutionizing both infinite group theory and global differential geometry,
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| | 2007 || a team of researches throughout North America and Europe used networks of computers to map [[E8 (mathematics)]].<ref>Elizabeth A. Thompson, MIT News Office, ''Math research team maps E8'' http://www.huliq.com/15695/mathematicians-map-e8</ref>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Timeline Of Algebra}}
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| [[Category:History of algebra]]
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| [[Category:Mathematics timelines|Algebra]]
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