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Timeline of algebra

From Wikipedia, the free encyclopedia

The following is a timeline of key developments of algebra:

Year Event
c. 1800 BC The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.[citation needed]
c. 1800 BC The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.[1]
1800 BC Berlin Papyrus 6619 (19th dynasty) contains a quadratic equation and its solution.[2][3]
800 BC Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places
c. 300 BC Euclid's Elements gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots.[4] The construction is due to the Pythagorean School of geometry.[citation needed]
c. 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 Jain mathematicians in 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.
250 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.[5]
1st century AD Hero of Alexandria gives the earliest fleeting reference to square roots of negative numbers.
c. 150 Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.
c. 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]
c. 625 Chinese mathematician Wang Xiaotong finds numerical solutions to certain cubic equations.[6]
c. 7th century
Dates vary from the 3rd to the 12th centuries.[7]		 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]
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 Brahmasphuta-siddhanta, where zero is clearly explained, and where the modern place-value 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 and quadratic equations, and rules for summing 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 infinite procedure, and also deals with the logarithm to base 2
c. 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.[8]
820 The word algebra is derived from operations described in the treatise written by the Persian mathematician, Muḥammad ibn Mūsā al-Ḵhwārizmī, 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.[9] 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."[10]
c. 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.[11] He also discovers the first numerical solution to equations of the form ax2n + bxn = c.[12] 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.[11]
895 Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations. He also generalized the Pythagorean theorem, and discovered the 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 monomials , , , … and , , , … and to give rules for 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 exponents, which “was a major factor in the development of numerical analysis based on the decimal system.”
c. 1000 Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree.
c. 1050 Chinese mathematician Jia Xian finds numerical solutions of polynomial equations of arbitrary degree.[13]
1070 Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations.
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.[14]
12th century 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.”[15]
c. 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-Horner method" to numerically approximate the 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.[16] He understands the importance of the discriminant of the cubic equation and uses an early version of Cardano's formula[17] 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.[18]
1202 Leonardo Fibonacci of Pisa publishes his Liber Abaci, a work on algebra that introduces Arabic numerals to Europe.[19]
c. 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.[20]
c. 1400 Indian mathematician Madhava of Sangamagrama documents infinite series approximations of trigonometry functions.[21]
15th century Nilakantha Somayaji, a Kerala school mathematician, writes the “Aryabhatiya Bhasya”, which contains work on infinite-series expansions, problems of algebra, and spherical geometry
1412–1482 Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī takes "the first steps toward the introduction of algebraic symbolism." He uses "short Arabic words, or just their initial letters, as mathematical symbols."[22]
1535 Scipione del Ferro and Niccolò Fontana Tartaglia, in Italy, independently solve the general cubic equation.[23]
1545 Girolamo Cardano publishes Ars magna -The great art which gives del Ferro's solution to the cubic equation[23] and Lodovico Ferrari's solution to the quartic equation.
1572 Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.[24]
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]
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.[25]
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.[26][27]
1637 The term imaginary number is first used by René Descartes; it is meant to be derogatory.
1682 Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.[28]
1683 Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant,[29] discriminant,[citation needed] and Bernoulli numbers.[29]
1693 Leibniz solves systems of simultaneous linear equations using matrices and determinants.[citation needed]
1722 Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers,
1750 Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.[30]
1797 Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,
1799 Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers),
1799 Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula,
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.[23]
1832 Galois theory is developed by Évariste Galois in his work on abstract algebra.[23]
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.
1873 Charles Hermite proves that 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 algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions, providing the foundation for modern algebra.
1981 Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry,

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Transcription

What I want to do in this video is think about the origins of algebra. The origins of algebra, and the word, especially in association with the ideas that algebra now represents, comes from this book, or actually this is a page of the book right over there. The English translation for the title of this book is the "Compendious Book on Calculation by Completion and Balancing." And it was written by a Persian mathematician who lived in Baghdad in, I believe, it was in the eighth or ninth century. I believe it was actually 820 AD when he wrote this book. AD. And algebra is the Arabic word, that here is the actual title that he gave to it, which is the Arabic title. Algebra means restoration or completion. Restoration or completion. And he associated it in his book with a very specific operation, really taking something from one side of an equation to another side of an equation. But we can actually see it right over here, and I don't know Arabic, but I actually do know some languages that seems to have borrowed a little bit from Arabic, or maybe it went the other way around. But this says Al-kitab, and I know just enough Urdu and Hindi to understand a good India movie, but Al-kitab, kitab means book. So this part is book. Book. Al-mukhtasar, well, I think that means compendious, because I don't know the word for compendious and that seems like that. Fihisab, hisab means calculation in Hindi or Urdu, so this is calculation. Calculation. Al-gabr, this is the root. This is the famous algebra, this is where it shows up. So this is for completion, you could view that as completion. Completion. And then wa'l-muqabala, and that means essentially balancing. Balancing. Completion and balancing. So if we wanted to translate it-- I know this isn't a video on translating Arabic, but the book, I guess this is saying compendious on calculation by completion and balancing is the rough translation right over there. But that is the source of the word algebra, and this is a very, very, very important book. Not just because it was the first use of the word algebra, but many people viewed this book as the first time that algebra took a lot of its modern-- took on many of its modern ideas. Ideas of balancing an equation. The abstract problem itself, not trying to do one off problems here or there. But al-Khwarizmi was not the first person, and just to get an idea of where all this is happening. So he was hanging out in Baghdad, and this part of the world shows up a lot in the history of algebra. But he was hanging out right there in around the eighth or ninth century. So let me draw a time line here, just so we can appreciate everything. So that is timeline, and then whether or not you are religious, most of our modern dates are dependent on the birth of Jesus, so that is right there. Maybe I'll put a cross over there to signify that. When we want to be non-religious, we say the common era. Before the common era, when we want to be religious we say AD, which means in the year of our lord. I don't know the Latin, Anno Domini, I believe, year of our lord. And then when we want-- in the religious context, instead of saying before common era, we say before Christ, BC. But either way, so this is 1000 in the common era. This is 2000 in the common era. And obviously, we are sitting-- at least when I'm making this video, I'm sitting right about there. And then this is 1000 before the common era, and this is 2000 before the common era. So the first traces-- and I'm skipping out, and really, it's just what we can find. I'm sure if we were able to dig more, we might be able to find other evidence of different civilizations and different people stumbling on many of the ideas in algebra. But our first records of people really exploring the ideas that are hit upon in algebra come from ancient Babylon around 2000 years before the common era, before Christ. So right around there there are stone tablets where it looks like people were exploring some of the fundamental ideas of algebra. They weren't using the same symbols. They weren't using the same ways of representing the numbers, but it was algebra that they were working on. And that was, once again, in this part of the world. Babylon was right about there. And Babylon, it's kind of kept the tradition of Sumeria. This whole region was called Mesopotamia, Greek for between two rivers. But that's the first traces of people that we know of that where people were starting to do what we would call real, real algebra. And then you fast forward. And I'm sure we're missing-- and I'm sure even our historians don't know all of the different instances of people using algebra, but the major contributions to algebra, we saw it here in Babylon 2000 years ago. And then if we fast forward to about 200 to 300 AD, so right over there, you have a Greek gentleman who lived in Alexandria. So this is Greece right over here, but he lived in Alexandria, which at the time was part of the Roman Empire. So Alexandria is right over here, and he was a gentleman by the name of Diophantus, or Diophantus. I don't know how to pronounce it, Diophantus. And he is sometimes credited with being the father of algebra, and it's debatable whether it's Diophantus or al-Khwarizmi. al-Khwarizmi, who kind of started using these terms of balancing equations and talking about math in a purer way, while Diophantus was more focused on particular problems. And both of them were kind of beat to the punch by the Babylonians, although they all did contribute in their own way. It's not like they were just copying what the Babylonians did. They had their own unique contributions to what we now consider algebra. But many, especially Western historians, associate Diophantus as the father of algebra. And now, al-Khwarizmi is sometimes what other people would argue as the father of algebra, so he made significant contributions. And if you go to 600 AD-- so if you go to about 600 AD, another famous mathematician in the history of algebra was Brahmagupta, in India. Brahmagupta, in India. So obviously, and actually, I don't know where in India he lived. I should look that up, but roughly in that part of the world. And he also made significant contributions. And then you have al-Khwarizmi, who shows up right there, al-Khwarizmi. And he is the gentleman that definitely we credit with the name algebra, comes from Arabic for restoration, and some people also consider him to be, if not the father of algebra, although some people say he is the father, he is one of the fathers of algebra because he really started to think about algebra in the abstract sense, devoid of some specific problems and a lot of the way a modern mathematician would start to think about the field.

See also

References

  1. ^ Anglin, W.S (1994). Mathematics: A Concise History and Philosophy. Springer. p. 8. ISBN 978-0-387-94280-3.
  2. ^ Smith, David Eugene Smith (1958). History of Mathematics. Courier Dover Publications. p. 443. ISBN 978-0-486-20430-7.
  3. ^ "Egyptian Mathematics Papyri". Mathematicians and Scientists of the African Diaspora. The State University of New York at Buffalo.
  4. ^ Euclid (January 1956). Euclid's Elements. Courier Dover Publications. p. 258. ISBN 978-0-486-60089-5.
  5. ^ Crossley, John; W.-C. Lun, Anthony (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. p. 349. ISBN 978-0-19-853936-0.
  6. ^ O'Connor, John J.; Robertson, Edmund F., "Wang Xiaotong", MacTutor History of Mathematics Archive, University of St Andrews
  7. ^ Hayashi (2005), p. 371. "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."
  8. ^ Boyer (1991), "The Arabic Hegemony" p. 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."
  9. ^ 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."
  10. ^ Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–2. ISBN 0-7923-2565-6. OCLC 29181926.
  11. ^ a b O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics Archive, University of St Andrews
  12. ^ 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)."
  13. ^ O'Connor, John J.; Robertson, Edmund F., "Jia Xian", MacTutor History of Mathematics Archive, University of St Andrews
  14. ^ 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)."
  15. ^ Arabic mathematics, MacTutor History of Mathematics archive, University of St Andrews, Scotland
  16. ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics Archive, University of St Andrews
  17. ^ Rashed, Roshdi; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 342–3. ISBN 0-7923-2565-6.
  18. ^ Berggren, J. L.; Al-Tūsī, Sharaf Al-Dīn; Rashed, Roshdi; Al-Tusi, Sharaf Al-Din (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. JSTOR 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.
  19. ^ Ball, W. W. Rouse (1960). A Short Account of the History of Mathematics. Courier Dover Publications. p. 167. ISBN 978-0-486-15784-9.
  20. ^ Grattan-Guinness, Ivor (1997). The Norton History of the Mathematical Sciences. W.W. Norton. p. 108. ISBN 978-0-393-04650-2.
  21. ^ Katz, Victor J. "Ideas of Calculus in Islam and India". Mathematics Magazine.
  22. ^ O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics Archive, University of St Andrews
  23. ^ a b c d Stewart, Ian (2004). Galois Theory (Third ed.). Chapman & Hall/CRC Mathematics. ISBN 9781584883937.
  24. ^ Cooke, Roger (16 May 2008). Classical Algebra: Its Nature, Origins, and Uses. John Wiley & Sons. p. 70. ISBN 978-0-470-27797-3.
  25. ^ Boyer (1991), "Prelude to Modern Mathematics" p. 306. "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.'"
  26. ^ Cajori, Florian (1919). "How x Came to Stand for Unknown Quantity". School Science and Mathematics. 19 (8): 698–699. doi:10.1111/j.1949-8594.1919.tb07713.x.
  27. ^ Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. Chicago: Open Court Publishing. p. 381. ISBN 9780486677668.
  28. ^ Struik, D. J. A Source Book in Mathematics, 1200-1800. Harvard University Press. p. 123. ISBN 978-0-674-82355-6.
  29. ^ a b O'Connor, John J.; Robertson, Edmund F., "Takakazu Shinsuke Seki", MacTutor History of Mathematics Archive, University of St Andrews
  30. ^ O'Connor, John J.; Robertson, Edmund F., "Gabriel Cramer", MacTutor History of Mathematics Archive, University of St Andrews
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