Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly between 200 and 214, d. Events By Place World Human population reaches about 257 million Events By Place Roman Empire The kingdom of Osroene becomes a Province of the Roman Empire. between 284 and 298 AD), sometimes called "the father of algebra", a title he shares with al-Khwārizmī, was an Alexandrian mathematician. Events By Place Roman Empire November 20 — Diocletian becomes Emperor. Events By Place Roman Empire Constantius Chlorus defeats the Alamanni in the territory of the Lingones (Langres and strengthens Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια 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 He is the author of a series of books called Arithmetica that deal with solving algebraic equations, many of which are now lost. Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE. In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q Pierre de Fermat studied Arithmetica and made a fateful note in the margin of his copy of the book that a certain equation similar to the Pythagorean equation considered by Diophantus has no solutions and he found "a truly marvelous proof of this proposition", the celebrated Fermat's Last Theorem. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like This led to tremendous advances in number theory, and the study of diophantine equations ("diophantine geometry") and of diophantine approximations remain important areas of mathematical research. 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 In Number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of Real numbers by Rational Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. A negative number is a Number that is less than zero, such as −2 In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In modern use, diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Diophantus also made advances in mathematical notation.
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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. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. While most scholars consider Diophantus to have been a Greek,[1][2][3][4] others speculate him to have been a non-Greek,[5] possibly either a Hellenized Babylonian,[6] an Egyptian,[7][4] a Jew, or a Chaldean. The Greeks ( Greek: Έλληνες) are a Nation and Ethnic group native to Greece, Cyprus and neighbouring regions Hellenization (or Hellenisation) is a term used to describe the spread of Greek culture. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital This article is about the contemporary North African ethnic group PLEASE TAKE NOTE************ Chaldea (from Greek grc Χαλδαία Chaldaia; Akkadian akk māt Kaldu Hebrew כשדים Kaśdim, "the Chaldees" of the [8] Much of our knowledge the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly One of the problems states:
'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life. '
This puzzle implies that Diophantus lived to be about 84 years old. However, the accuracy of the information cannot be independently confirmed. This puzzle was the Puzzle No. 142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first. is the second game in the Professor Layton series by Level-5.
Arithmetica
- See also: Arithmetica
The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. Arithmetica is an ancient Greek text on Mathematics written by the Mathematician Diophantus in the 3rd century CE. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent 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. Some Diophantine problems from Arithmetica have been found in Arabic sources.
It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus. Hermann Hankel ( February 14, 1839 - August 29, 1873) was a German Mathematician who was born in Halle,
“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”
History
During the Dark Ages, Diophantus was forgotten and like many other mathematical treatises from the classical period, Arithmetica survived through the Arab tradition. This article is about the phrase "Dark Age(s" as a characterization of the Early Middle Ages in Western Europe 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 . Johannes Müller von Königsberg ( June 6, 1436 &ndash July 6, 1476) known by his Latin Pseudonym Regiomontanus . . . ”
Arithmetica was first translated into Latin by Bombelli in 1570, but the translation was never published. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. In Classical scholarship, editio princeps is a Term of art. It means roughly the first printed edition of a work that previously had existed only in Guilielmus Xylander ( Wilhelm Holtzman, according to his own spelling ( December 26, 1532 - February 10, 1576) was a German The best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Claude Gaspard Bachet de Méziriac ( October 9, 1581 - February 26, 1638) was a French Mathematician born in Bourg-en-Bresse Pierre de Fermat owned a copy, studied it, and made notes in the margins. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the
Margin writing by Fermat and Planudes
The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy:
- “If an integer n is greater than 2, then an + bn = cn has no solutions in non-zero integers a, b, and c. Claude Gaspard Bachet de Méziriac ( October 9, 1581 - February 26, 1638) was a French Mathematician born in Bourg-en-Bresse Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like 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. Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University 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.
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 1260 &ndash 1330 was a Byzantine Greek Grammarian and theologian who lived and worked during the reigns of Michael VIII Palaeologus
Other works
Diophantus wrote several other books besides Arithmetica, but very few of them 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. In Mathematics, a lemma (plural lemmata or lemmas from the Greek λήμμα "lemma" meaning "anything which is received Some scholars think that The porisms may have actually been a section of Arithmetica that is now lost.
Although The Porisms is lost, we know three lemmas contained there since Diophantus refers to them in the Arithmetica. One lemma states 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 a and b, with a > b, there exist c and d, all positive and rational, such that
- a3 − b3 = c3 + d3.
Polygonal numbers and geometric elements
Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. In Mathematics, a polygonal number is a Number that can be arranged as a regular Polygon. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Fragments of a book dealing with polygonal numbers are extant.
A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. Hero (or Heron) of Alexandria ( Ήρων ο Αλεξανδρεύς) (c It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus [9].
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.
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 [10]. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of 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. ”
Diophantine analysis
- See also: Diophantine equation
Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only 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.
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. ”
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
References
- ^ Research Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing, 312. “Diophantus (lived c. AD 270-280) Greek mathematician who, in solving linear mathematical problems, developed an early form of algebra. ”
- ^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "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. ”
- ^ 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. ”
- ^ a b Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0321016181.
"But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C. E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [. . . ] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definatively. Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca Pappus of Alexandria ( Greek) (c 290 &ndash c 350 was one of the last great Greek mathematicians of antiquity known for his Synagoge or Collection Hypatia of Alexandria (haɪˈpeɪʃə ( Greek:; born between AD 350 and 370 – 415 was a Greek scholar from Alexandria in Egypt, considered But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [. . . ] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privaleged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist. "
- ^ 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. . . "
- ^ D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm. C. Brown Publishers).
"Diophantos was most likely a Hellenized Babylonian. "
- ^ George Sarton (1936). George Alfred Leon Sarton (1884-1956 was a Belgian -American Polymath, historian of science, and father of the writer May Sarton. "The Unity and Diversity of the Mediterranean World", Osiris 2, p. 406-463 [429].
- ^ Oswald Spengler (1923), Der Untergang des Abendlandes, 2 Bande:
- ^ Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol. Plotinus ( Greek:) (ca AD 204–270 was a major philosopher of the ancient world who is widely considered the founder of Neoplatonism (along with his PLEASE TAKE NOTE************ Chaldea (from Greek grc Χαλδαία Chaldaia; Akkadian akk māt Kaldu Hebrew כשדים Kaśdim, "the Chaldees" of the 20, No. 2, 180-192
- ^ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
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. Sir Thomas Little Heath ( October 5, 1861 &ndash March 16, 1940) was a British civil servant Mathematician, classical
- D. C. Robinson and Luke Hodgkin. History of Mathematics, King's College London, 2003. King's College London is a British Higher education institution and co-founding constituent college of the federal University of London.
- 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. Year 1982 ( MCMLXXXII) was a Common year starting on Friday (link displays the 1982 Gregorian calendar) ISBN 0-387-90690-8.
External links
- O'Connor, John J. & Robertson, Edmund F. , “Diophantus”, MacTutor History of Mathematics archive
- Diophantus's Riddle Diophantus' epitaph, by E. The MacTutor History of Mathematics archive is an award-winning website maintained by John J Weisstein
- Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.
- Tannery's edition of the Works of Diophantus, now in the public domain (Classical Greek)
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