He also made important advances in mathematical notation, and was one of the first mathematicians to introduce symbolism into algebra, using an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. He was perhaps the first to recognize fractions as numbers in their own right, allowing positive rational numbers for the coefficients and solutions of his equations. Diophantus applied himself to some quite complex algebraic problems, particularly what has since become known as Diophantine Analysis, which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns. Diophantine equations can be defined as polynomial equations with integer coefficients to which only integer solutions are sought.

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Quick Info Born about probably Alexandria, Egypt Died about probably Alexandria, Egypt Summary Diophantus was a Greek mathematician sometimes known as 'the father of algebra' who is best known for his Arithmetica. This had an enormous influence on the development of number theory.

Biography Diophantus , often known as the 'father of algebra', is best known for his Arithmetica , a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. There are a few limits which can be put on the dates of Diophantus's life. On the one hand Diophantus quotes the definition of a polygonal number from the work of Hypsicles so he must have written this later than BC.

On the other hand Theon of Alexandria, the father of Hypatia , quotes one of Diophantus's definitions so this means that Diophantus wrote no later than AD.

However this leaves a span of years, so we have not narrowed down Diophantus's dates a great deal by these pieces of information. There is another piece of information which was accepted for many years as giving fairly accurate dates. Heath [ 3 ] quotes from a letter by Michael Psellus who lived in the last half of the 11 th century.

Psellus wrote Heath's translation in [ 3 ] :- Diophantus dealt with [ Egyptian arithmetic ] more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way and in the most succinct form, dedicating his work to Diophantus.

Psellus also describes in this letter the fact that Diophantus gave different names to powers of the unknown to those given by the Egyptians. This letter was first published by Paul Tannery in [ 7 ] and in that work he comments that he believes that Psellus is quoting from a commentary on Diophantus which is now lost and was probably written by Hypatia.

However, the quote given above has been used to date Diophantus using the theory that the Anatolius referred to here is the bishop of Laodicea who was a writer and teacher of mathematics and lived in the third century.

From this it was deduced that Diophantus wrote around AD and the dates we have given for him are based on this argument. Knorr in [ 16 ] criticises this interpretation, however:- But one immediately suspects something is amiss: it seems peculiar that someone would compile an abridgement of another man's work and then dedicate it to him, while the qualification "in a different way", in itself vacuous, ought to be redundant, in view of the terms "most essential" and "most succinct".

Knorr gives a different translation of the same passage showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek which has a remarkably different meaning:- Diophantus dealt with [ Egyptian arithmetic ] more accurately, but the very learned Anatolius, having collected the most essential parts of that man's doctrine, to a different Diophantus most succinctly addressed it.

The conclusion of Knorr as to Diophantus's dates is [ 16 ] The most details we have of Diophantus's life and these may be totally fictitious come from the Greek Anthology, compiled by Metrodorus around AD.

This collection of puzzles contain one about Diophantus which says So he married at the age of 26 and had a son who died at the age of 42 , four years before Diophantus himself died aged Based on this information we have given him a life span of 84 years.

The Arithmetica is a collection of problems giving numerical solutions of determinate equations those with a unique solution , and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written. There are many Arabic translations, for example by Abu'l-Wafa , but only material from these six books appeared.

Heath writes in [ 4 ] in :- The missing books were evidently lost at a very early date. Paul Tannery suggests that Hypatia 's commentary extended only to the first six books, and that she left untouched the remaining seven, which, partly as a consequence, were first forgotten and then lost. F Sezgin made this remarkable discovery in In [ 19 ] and [ 20 ] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus.

Rozenfeld, in reviewing these two articles is, however, not completely convinced:- The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators perhaps Hypatia?

It is time to take a look at this most outstanding work on algebra in Greek mathematics. The work considers the solution of many problems concerning linear and quadratic equations , but considers only positive rational solutions to these problems.

Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless. In other words how could a problem lead to the solution - 4 books? There is no evidence to suggest that Diophantus realised that a quadratic equation could have two solutions. However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realise today.

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 a, b, c a , b , c to all be positive in each of the three cases above.

There are, however, many other types of problems considered by Diophantus. He solved problems such as pairs of simultaneous quadratic equations. Diophantus would solve this by creating a single quadratic equation in x. Quotations by Diophantus Other Mathematicians born in Egypt.

References show. Biography in Encyclopaedia Britannica. J Klein, Greek mathematical thought and the origin of algebra London, P Tannery, Diophanti Alexandrini Opera omnia cum graecis commentariis 2 vols. Leipzig, - XII Wiesbaden, , - Russian , Mat. R Rashed, Les travaux perdus de Diophante. I, Rev. II, Rev. E I Slavutin, Some questions on the structure of the 'Arithmetic' of Diophantus of Alexandria Russian , in Studies in the history of mathematics 18 'Nauka' Moscow, , - , Additional Resources show.

Other pages about Diophantus: The title page from the translation by Bachet of Arithmetica and another page showing the transcription of Fermat's marginal note Herbert Jennings Rose's Greek mathematical literature.

Honours show. Cross-references show. History Topics: Arabic mathematics : forgotten brilliance?

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Its historical importance is twofold: it is the first known work to employ algebra in a modern style, and it inspired the rebirth of number theory. Credit for the first proof is given to the 17th-century French amateur mathematician Pierre de Fermat. The first published proof of the four-square theorem was…. His writing, the Arithmetica , originally in 13 books six survive in Greek, another four in medieval Arabic translation , sets out hundreds of arithmetic problems with their solutions. For example, Book II, problem 8, seeks to express a given square number as the sum of two square numbers here….

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## Arithmetica

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. It should be in the public domain obviously , so I'd thought I could find the English text on the web somewhere. Apparently not? The original Greek text is lost to us.

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## Diophantus

Quick Info Born about probably Alexandria, Egypt Died about probably Alexandria, Egypt Summary Diophantus was a Greek mathematician sometimes known as 'the father of algebra' who is best known for his Arithmetica. This had an enormous influence on the development of number theory. Biography Diophantus , often known as the 'father of algebra', is best known for his Arithmetica , a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. There are a few limits which can be put on the dates of Diophantus's life.

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## DIOPHANTUS OF ALEXANDRIA

Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations. In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers.

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