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Encyclopedia > Diophantus
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.

Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. between 200 and 214, d. between 284 and 298 AD), sometimes called "the father of algebra", was a Hellenistic mathematician. He is the author of a series of classical mathematical books called Arithmetica and worked with equations which we now call Diophantine equations; the method to solve those problems is now called Diophantine analysis. The study of Diophantine equations is one of the central areas of number theory. The findings and works of Diophantus have influenced mathematics greatly and caused many other questions to arise. The most famous of these is Fermat's Last Theorem. Diophantus also made advances in mathematical notation and was the first Hellenistic mathematician who frankly recognized fractions as numbers. Image File history File links Download high resolution version (668x1081, 368 KB) Summary Work by Diophantus (died in about 280 B.C.), translated from Greek into Latin by Claude Gaspard Bachet de Méziriac. ... Image File history File links Download high resolution version (668x1081, 368 KB) Summary Work by Diophantus (died in about 280 B.C.), translated from Greek into Latin by Claude Gaspard Bachet de Méziriac. ... Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... Claude Gaspard Bachet de Méziriac (October 9, 1581 - February 26, 1638) was a French mathematician born in Bourg-en-Bresse. ... For other uses, see number 200. ... Events The kingdom of Osroene becomes a province of the Roman Empire. ... For other uses, see number 284. ... Events Constantius Chlorus overthrows the Alamanni in the territory of the Lingones (Langres) and strengthened the Rhine frontier Christians are expelled from the Roman army Baths of Diocletian built in Rome Births Athanasius of Alexandria, bishop and opponent of Arianism Deaths Diophantus, mathematician (approximate date) King Chaekgye of Baekje Categories... The term Hellenistic (derived from HéllÄ“n, the Greeks traditional self-described ethnic name) was established by the German historian Johann Gustav Droysen to refer to the spreading of Greek culture over the non-Greek people that were conquered by Alexander the Great. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Pierre de Fermats informal conjecture written in the margin of his book proved to be one of the most intriguing and enigmatic math problems ever devised. ...

Contents

Biography

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between 200 and 214 to 284 or 298 AD. There is no agreement on Diophantus' ethnic origins.[1] While some consider Diophantus to have been a Greek,[2][3] others consider him to have been a Hellenized Babylonian,[4][5] and still others suggest that he may have been non-Greek,[6] possibly Jewish or Chaldean.[5] Almost everything we know about Diophantus comes from a single fifth century Greek anthology, which is a collection of number games and strategy puzzles. Here is one of the puzzles: Alexandria (Greek: , Coptic: , Arabic: , Egyptian Arabic: Iskindireyya), (population of 3. ... An ethnic group or ethnicity is a population of human beings whose members identify with each other, usually on the basis of a presumed common genealogy or ancestry (Smith 1987). ... Hellenization (or Hellenisation) is a term used to describe a cultural change in which something non-Greek becomes Greek (Hellenistic civilization). ... Babylonia was a state in the south part of Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. ... For other uses, see Jew (disambiguation). ... For other uses, see Chaldean. ...


“This tomb hold Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.”


This puzzle reveals that Diophantus lived to be about 84 years old. We cannot be sure if this puzzle is accurate or not.


Arithmetica

See also: Arithmetica Arithmetica, an ancient text on mathematics written by classical period Greek mathematician Diophantus in the second century AD is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. ...


The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus[citation needed]. Some Diophantine problems from Arithmetica have been found in Arabic sources. An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...


History

Soon after Diophantus's death, the Dark Ages began, spreading a shadow on math and science, and causing knowledge of Diophantus and the Arithmetica to be lost in Europe for about 1500 years. Possibly the only reason that some of his work has survived is that many Arab scholars studied his works and preserved this knowledge for later generations. In 1463 German mathematician Regiomontanus wrote: “No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hidden . . . .” Petrarch, who conceived the idea of a European Dark Age. From Cycle of Famous Men and Women, Andrea di Bartolo di Bargillac, c. ... Johannes Müller von Königsberg (June 6, 1436 – July 6, 1476), known by his Latin pseudonym Regiomontanus, was an important German mathematician, astronomer and astrologer. ...


The first Latin translation of Arithmetica was by Bombelli who translated much of the work in 1570 but it was never published. Bombelli did however borrow many of Diophantus's problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The most famous Latin translation of Arithmetica was by Bachet in 1621 which was the first translation of Arithmetica available to the public. Latin is an ancient Indo-European language originally spoken in Latium, the region immediately surrounding Rome. ... Rafael Bombelli (1526-1573) was an Italian mathematician. ... In classical scholarship, editio princeps is a term of art. ... Guilielmus Xylander (Wilhelm Holtzman, according to his own spelling) (December 26, 1532 - February 10, 1576) was a German classical scholar. ... Cover of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ...


Margin writing by Fermat and Planudes

Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's last theorem.
Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's last theorem.

The 1621 edition of Arithmetica by Bombelli gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy: Image File history File links Download high resolution version (978x1520, 271 KB) Summary Work by Diophantus (died in about 280 B.C.), with additions by Pierre de Fermat (died in 1665). ... Image File history File links Download high resolution version (978x1520, 271 KB) Summary Work by Diophantus (died in about 280 B.C.), with additions by Pierre de Fermat (died in 1665). ... Pierre de Fermats informal conjecture written in the margin of his book proved to be one of the most intriguing and enigmatic math problems ever devised. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Pierre de Fermats informal conjecture written in the margin of his book proved to be one of the most intriguing and enigmatic math problems ever devised. ...


“If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”


Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations --- including his famous "Last Theorem" --- were printed in this version. Andrew Wiles should not be confused with André Weil, another famous mathematician who, like Wiles, did important work in the area of elliptic curves. ...


Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine mathematician Maximus Planudes had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem. Maximus Planudes (c. ...


Other works

Diophantus did not just write Arithmetica, but very few of his other works have survived.


The Porisms

Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost. Many scholars and researchers believe that The Porisms may have actually been a section included inside Arithmetica or indeed may have been the rest of Arithmetica.[citation needed] In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. ...


Although the Porisms is lost we do know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such lemma is that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any numbers a, b then there exist numbers c and d such that a3b3 = c3 + d3.


On polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers. Fragments of one of Diophantus' books on polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. An extant work called Preliminaries to the Geometric Elements, which has been attributed to Hero of Alexandria, has been studied recently and it is suggested that the attribution to Hero is incorrect, and that the work is actually by Diophantus [7]. In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ... Pythagoras of Samos (Greek: ; circa 580 BC – circa 500 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... Heros aeolipile Hero (or Heron) of Alexandria (Greek: Ήρων ο Αλεξανδρεύς) (c. ...


Influence

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the seventeenth and eighteenth centuries. Diophantus and his works have also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how much he affected India is a matter of debate. In the history of mathematics, Islamic mathematics refers to the mathematics developed by mathematicians of the Islamic culture, from the beginning of Islam until the 17th century — mostly including Arab and Persian mathematicians, as well as other Muslims and non-Muslims that were a part of the Islamic culture. ...


The father of algebra?

Diophantus is often called “the father of algebra" because he contributed greatly to number theory, mathematical notation, and because Arithmetica contains the earliest known use of syncopated notation [8]. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this reason mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.” According to some historians of mathematics, like Florian Cajori, Diophantus got the first knowledge of algebra from India, [9][10], although other historians disagree.[11] Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since... Florian Cajori at Colorado College Florian Cajori was born February 28, 1859 in St Aignan (near Thusis), Graubünden, Switzerland. ...


Diophantine analysis

See also: Diophantine equation

Today Diophantine analysis is the area of study where integral (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integral coefficients to which only integral solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: ax2 + bx = c, ax2 = bx + c, and ax2 + c = bx. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a,b,c to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a negative value for x. One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...


Mathematical notation

Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:


“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”[citation needed]


Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write (12 + 6n) / (n2 − 3), Diophantus has to resort to constructions like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.


Algebra still had a long way to go before very general problems could be written down and solved succinctly.


See also

  • Muhammad ibn Mūsā al-Khwārizmī

(Arabic: ) was a Persian[1] mathematician, astronomer, astrologer and geographer. ...

References

  1. ^ Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History. Louis Pasteur University.
  2. ^ Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 178. ISBN 0471543977. “At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.” 
  3. ^ Cooke, Roger (1997). "The Nature of Mathematics", The History of Mathematics: A Brief Course. Wiley-Interscience, 7. ISBN 0471180823. “Some enlargement in the sphere in which symbols were used occurred in the writings of the third-century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.” 
  4. ^ D. M. Burton (1991, 1995), History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers):

    "Diophantos was most likely a Hellenized Babylonian." The Université Louis Pasteur, also known as Strasbourg I or ULP is a large university in Strasbourg, Alsace, France. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Hellenization (or Hellenisation) is a term used to describe a cultural change in which something non-Greek becomes Greek (Hellenistic civilization). ... Babylonia was a state in the south part of Mesopotamia (in modern Iraq), combining the territories of Sumer and Akkad. ...

  5. ^ a b Oswald Spengler (1923), Der Untergang des Abendlandes, 2 Bande:

    "Were Plotin and Diophantus maybe of Jewish or Chaldaic origins?" Plotinus Plotinus (Greek: ) (ca. ... For other uses, see Jew (disambiguation). ... For other uses, see Chaldean. ...

  6. ^ H. Hankel (1874, 2nd ed. 1965), Zur Geschichte der Mathematik im Altertum und Mittelalter, Leipzig:

    "Here, in the midst of this sad and barren landscape of the Greek accomplishments in arithmetic, suddenly springs up a man with youthful energy: Diophantus. Where does he come from, where does he go to? Who were his predecessors, who his successors? We do not know. It is all one big riddle. He lived in Alexandria. If a conjecture were permitted, I would say he was not Greek; ... if his writings were not in Greek, no-one would ever think that they were an outgrowth of Greek culture..."

  7. ^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
  8. ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  9. ^ Florian Cajori, A History of Elementary Mathematics, 1898
  10. ^ Saradakanta Ganguli, Notes on Indian Mathematics. A Criticism of George Rusby Kaye's Interpretation, Isis, Vol. 12, No. 1 (Feb., 1929), pp. 132-145
  11. ^ "Heeffer, Albrecht, The Reception of Ancient Indian Mathematics by Western Historians, Ghent University, Belgium."

Bibliography

  • A. Allard, "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibo. Nat. 4678 et les Vaticani gr. 191 et 304," Byzantion 53. Brussels, 1983: 682-710.
  • P. Ver Eecke, Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1926.
  • T. L. Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
  • D. C. Robinson and Luke Hodgkin. History of Mathematics, King's College London, 2003.
  • P. L. Tannery, Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
  • Jacques Sesiano, Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā, Heidelberg: Springer-Verlag, 1982. ISBN 0-387-90690-8.

Kings College London is the largest college of the University of London and one of a number of university institutions founded in England in the early 19th century: only the Universities of Oxford and Cambridge have royal charters predating that of Kings. ... 1982 (MCMLXXXII) was a common year starting on Friday of the Gregorian calendar. ...

External links


  Results from FactBites:
 
Diophantus - Wikipedia, the free encyclopedia (623 words)
Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς circa 200/214 – circa 284/298) was a Hellenistic mathematician.
Diophantus is sometimes known as the "father of Algebra" perhaps because his unusual syncopated notation seems reminiscient of the fully symbolic algebra that would develop much later.
The editio princeps of Diophantus was published in 1575 by Xylander, and editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the seventeenth and eighteenth centuries.
History of Algebra - Diophantus of Alexandria (fl. 360 A.D.) (464 words)
About the middle of the 4th century of the Christian era, a period when the mathematical sciences were on the decline, and their cultivators, instead of producing original works of genius contented themselves with commentaries on the works of their more illustrious predecessors, there was a valuable addition made to the fabric of ancient learning.
Diophantus may have been the inventor of the Greek algebra, but it is more likely that its principles were not unknown before his time; and that, taking the science in the state in which he found it as the basis of his labours, he enriched it with new applications.
The elegant solutions of Diophantus show that he possessed great address in the particular branch of which he treated, and that he was able to resolve determinate equations of the second degree.
  More results at FactBites »

 
 

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