الجبر و المقابلة

Timeline of algebra

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A timeline of key algebraic developments are as follows:

Year *** Event

Circa 1800 BC *** The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.[citation needed]

Circa 1800 BC: *** The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.[citation needed]

Circa 800 BC *** Indian mathematician Baudhayana, in his Baudhayana Sulba Sutra, discovers Pythagorean triples algebraically, finds geometric solutions of linear equations and quadratic equations of the forms ax2 = c and ax2 + bx = c, and finds two sets of positive integral solutions to a set of simultaneous Diophantine equations.[citation needed]

Circa 600 BC *** Indian mathematician Apastamba, in his Apastamba Sulba Sutra, solves the general linear equation and uses simultaneous Diophantine equations with up to five unknowns.[citation needed]

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]

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]

Circa 100 BC *** Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the 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]

Circa 150 AD *** Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.[citation needed]

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]

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]

Circa 625 *** Chinese mathematician Wang Xiaotong finds numerical solutions to certain cubic equations.[1]

628 *** Indian mathematician Brahmagupta, in his treatise Brahma Sputa Siddhanta, invents the chakravala method of solving indeterminate quadratic equations, including Pell's equation, and gives rules for solving linear and quadratic equations. He discovers that quadratic equations have two roots, including both negative as well as irrational roots.[citation needed]

Circa 7th century

Dates vary from the third to the twelfth centuries A.D.[2] *** The Bakhshali Manuscript written in 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]

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.[3]

820 *** The word algebra is derived from operations described in the treatise written by the Persian mathematician, Mu?ammad ibn M?s? al-?w?rizm?, titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of 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" 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.[4] His algebra was also no longer concerned "with a series of problems to be resolved, but an 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."[5]

Circa 850 *** Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[citation needed]

Circa 850 *** Indian mathematician Mahavira solves various quadratic, cubic, quartic, quintic and higher-order equations, as well as indeterminate quadratic, cubic and higher-order equations.[citation needed]

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 monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and gives rules for the products of any two of these.[6] He also discovered the first numerical solution to equations of the form ax2n + bxn = c.[7] Al-Karaji is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The 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.[6]

Circa 1050 *** Chinese mathematician Jia Xian finds numerical solutions of polynomial equations of arbitrary degree.[8]

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.[9]

1114 *** Indian mathematician Bhaskara, in his Bijaganita (Algebra), recognizes that a positive number has both a positive and negative square root, and solves quadratic equations with more than one unknown, various cubic, quartic and higher-order polynomial equations, Pell's equation, the general indeterminate quadratic equation, as well as indeterminate cubic, quartic and higher-order equations.[citation needed]

Circa 1200 *** Sharaf al-D?n al-T?s? (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-Horner method" to numerically approximate the 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.[10] He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula[11] 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.[12]

1202 *** Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci.[citation needed]

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 and higher-order polynomial equations.[citation needed]

Circa 1400 *** Jamsh?d al-K?sh? developed an early form of Newton's method to numerically solve the equation xP ? N = 0 to find roots of N.[13]

Circa 1400 *** Indian mathematician Madhava of Sangamagramma finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations.[citation needed]

1412-1482 *** Arab mathematician Ab? al-Hasan ibn Al? al-Qalas?d? took "the first steps toward the introduction of algebraic symbolism." He used "short Arabic words, or just their initial letters, as mathematical symbols."[14]

1535 *** Nicolo Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation.[15]

1545 *** Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation.[15]

1572 *** Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.[citation needed]

1591 *** Francois Viete 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]

1631 *** Thomas Harriot in a posthumous publication is the first to use symbols < and > to indicate "less than" and "greater than", respectively.[16]

1682 *** Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.[citation needed]

1683 *** Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant,[17] discriminant,[citation needed] and Bernoulli numbers.[17]

1685 *** Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations.[citation needed]

1693 *** Leibniz solves systems of simultaneous linear equations using matrices and determinants.[citation needed]

1750 *** Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.[18]

1824 *** Niels Henrik Abel proved that the general quintic equation is insoluble by radicals.[15]

1832 *** Galois theory is developed by ?variste Galois in his work on abstract algebra.[15]

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References

1. ^ O'Connor, John J.; Robertson, Edmund F., "Wang Xiaotong", MacTutor History of Mathematics archive.

2. ^ (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 Bakhshal? work and Bh?skara I's commentary on the ?ryabhat?ya. 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 Bakhsh?l? work date from anterior periods."

3. ^ Boyer (1991). "The Arabic Hegemony". pp. 227. "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."

4. ^ (Boyer 1991, "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."

5. ^ Rashed, R.; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 11-2, ISBN 0792325656, OCLC 29181926

6. ^ a b O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive.

7. ^ (Boyer 1991, "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 ax2n + bxn = c (only equations with positive roots were considered),"

8. ^ O'Connor, John J.; Robertson, Edmund F., "Jia Xian", MacTutor History of Mathematics archive.

9. ^ Boyer (1991). "The Arabic Hegemony". pp. 241-242. "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)."

10. ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics archive.

11. ^ Rashed, Roshdi; Armstrong, Angela (1994), The Development of Arabic Mathematics, Springer, pp. 342-3, ISBN 0792325656

12. ^ Berggren, J. L. (1990), "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2): 304-9, doi:10.2307/604533, "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."

13. ^ Tjalling J. Ypma (1995), "Historical development of the Newton-Raphson method", SIAM Review 37 (4): 531-51, doi:10.1137/1037125

14. ^ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics archive.

15. ^ a b c d Stewart, Ian (2004). Galois Theory (Third ed.). Chapman & Hall/CRC Mathematics.

16. ^ Boyer, Carl B. (1991). "Prelude to Modern Mathematics". A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. pp. 306. ISBN 0471543977. "Harriot knew of relationships between roots and coefficients and between roots and factors, but like Viete 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.""

17. ^ a b O'Connor, John J.; Robertson, Edmund F., "Takakazu Shinsuke Seki", MacTutor History of Mathematics archive.

18. ^ O'Connor, John J.; Robertson, Edmund F., "Gabriel Cramer", MacTutor History of Mathematics archive.

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