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For the general, see Diophantus.

Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς circa 200/214 – circa 284/298) was a Greek mathematician of the Hellenistic era. Little is known of his life except that he lived in Alexandria, Egypt and worked in the Greek tradition of mathematics, principally on number theory.

He was known for his study of equations with variables which take on rational values and these Diophantine equations are named after him. Diophantus is considered the "father of Algebra" because Arithmetica contains the earliest know use of syncopated notation. Contrary to popular belief, al-Khwarizmi did not develop a fully symbolic algebra, infact "the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even the numbers were written out in words rather than symbols!"<ref>Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228</ref> His most famous work is the Arithmetica — originally thirteen Greek books, of which only six survive today in extant Greek manuscripts. Some Diophantine problems from these books have been found in Arabic sources. An additional four books of the Arithmetica, apparently from the lost Greek books, were discovered in an Arabic manuscript in 1968. Diophantus also wrote a treatise on polygonal numbers, of which part survives.

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.

In 1637, while reviewing his copy of Diophantus' Arithmetica Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy of Bachet's 1621 edition of the Arithmetica. Although this original copy is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Although 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 (scholia) to Diophantus on the same problem (II.8), the Byzantine mathematician Maximus Planudes had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems, and of this one in particular".

Little is known about the life of Diophantus. Some biographical information can be computed from a 5th and 6th century math puzzle involving Diophantus' age and styled as his epitaph (see links below).

"This tomb holds 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."

Diophantus, who is often known as ‘the father of algebra’, is primarily famous for his work the Arithmetica, which is a collection of problems with solutions, and for his contribution to algebraic notation. Diophantus worked with equations which we now call Diophantine equations, and 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, which will be covered briefly later.

First of all, what little information is known about his life will be revealed. We know that he lived in Alexandria, Egypt, and that is important, as we know what resources he had access to and how his influence spread. Almost everything we know about Diophantus comes from a single 5th century Greek anthology, which is a collection of number games and strategy puzzles. Here is one puzzle:

"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. We only know he lived between 150 BC and 350 AD which leaves a good span of 500 years.

The most famous work of Diophantus, Arithmetica, has great importance as the first work known to contain algebraic symbolism and it is one of the greatest contributions to number theory ever.

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 ax2 + bx = c, ax2 = bx + c, and ax2 + 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 hi problems. Diophantus would not deal with negative solutions. He 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 meaningless answer. In other words, how could a problem lead to the solution -4 books? One solution was all he required to a quadratic equation. He also considered simultaneous quadratic equations. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. 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 realize today. Diophantus is actually the first Greek mathematician to even study algebra.

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. He also noticed that numbers of the form 4n + 3 cannot be the sum of 2 squares. Diophantus also appears to know that every number can be written as the sum of 4 squares. If he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Lagrange proved it using results due to Euler.

Diophantus did not just write Arithmetica. He had 2 other books that we know of, but the whole of them does not survive either. Fragments of one of Diophantus's other books on polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived.

Diophantus himself refers to another work which consists of a collection of lemmas (or subsidiary propositions assumed to be valid and used to demonstrate a principal proposition) called The Porisms (or Porismata), but this book is entirely 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, d such that a3 - b3= c3 + d3. We are not sure but 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. They are quite sure though, that the work on polygonal numbers was a separate work of Diophantus. Another extant work Preliminaries to the Geometric Elements, which has been attributed to Heron, has been studied recently and it is suggested that the attribution to Heron is incorrect, and that the work is due to Diophantus.

Diophantus was the first person ever to use algebraic notation and symbolism. Before him everyone had to completely write out an equation. Although Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown. As Kurt Vogel a currently living mathematician 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.”

Diophantus has laid a groundwork plan for future mathematicians whom study algebra and in fact many of the other advanced maths as algebra is a groundwork plan for them itself. This may be one of or in fact the largest testimony of the greatness of Diophantus as a mathematician. Diophantus used an abridged notation for frequently occurring operations, and a special symbol for the unknown. Thus for the unknown he wrote , if it occurred once. For our 3x, he wrote , where is the plural  of the unknown and  represents the coefficient 3. Addition was denoted by simply placing the summands next to each other, and subtraction was indicated by the symbol  . Instead of our sign for equality, he wrote  . Also, terms which were not tied to the unknown were preceded by the symbol . Even with this ineffective symbolism Diophantus was able to solve a great many problems quicker than he would have had he had to completely write each equation out in words. Even so he was much slower than today’s mathematicians. Diophantus was concerned with particular problems more often than with general methods. The reason for this is that although he made important advances in symbolism, he still lacked the necessary notation to express more general methods. For instance 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)/(n2 -3), Diophantus has to write in words like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.

Despite the improved notation that Diophantus introduced, algebra had a long way to go before really general problems could be written down and solved succinctly. Though it was his writings that inspired other later Arab mathematicians to devise on the symbolism he had already created.

Diophantus was the first Greek mathematician whom frankly recognized frations as numbers. In spite of his ineffective symbolism, and in spite of the inelegance of his methods, he is the person who most deserves to be awarded the title ‘Father of Algebra’ as he was the one who invented notation, contributed to the study of equations, and practically gave birth to number theory. Sadly soon after the Diophantus came the Dark Ages which spread a shadow upon math and science and veiled his greatness in Europe for about 1500 years. Perhaps the only reason his work may have survived could be because the Arabs took a keen interest in his works. Many Arab scholars studied the works of Diophantus and preserved this knowledge for later generations. 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 great debate and personal opinion.

Diophantus had not even been heard of until the 15th century in Europe and wasn’t even studied until the 16th . European mathematicians did not learn of the wonders and knowledge in Diophantus's Arithmetica until Regiomontanus wrote in 1463: “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...”

The first Latin translation of Arithmetica was by Bombelli whom 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 most famous Latin translation of the Diophantus's Arithmetica is due to Bachet in 1621 whom was the first person to widely publish an edition of Arithmetica available to the public.

It was that very 1621 edition which Fermat studied and caused a spark in him to grow. Certainly Fermat was inspired by this work which has become famous in recent years due to its connection with Fermat's Last Theorem.

In 1637, while reviewing his copy of Diophantus' Arithmetica Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy of Bachet's 1621 edition of Diophantus’s Arithmetica. Although this original copy 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. All the hype is about this small single equation -

an+bn=cn

about which Fermat says:

“If an integer n is greater than 2, then an + bn = cn 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."

Most do not believe that Fermat ever had a proof. It is this theorem that had the mathematical world community stumped for 356 years. It has stumped many of the greatest mathematicians after that time including Euler and LaGrange. In 1994 Andrew Wiles who had been working on the proof for seven years finally finished it.

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.”

Although Diophantus was sometimes given the title “Father of Algebra”,it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to Babylonian mathematics. For this reason 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.”

Essentially nothing is known of Diophantus' life, and there has been much debate regarding the date at which he lived. The most details we have of Diophantus's life come from the Greek Anthology which is a large collection of mathematical games and puzzles, compiled by Metrodorus around 500 and gives away that Diophantus lived to be about 84 years old.

Diophantus is best known for his Arithmetica, the most outstanding work on algebra in Greek mathematics. It is a collection of 130 problems giving numerical solutions of both determinate and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only 6 of the original 13 books have survived, though there are some who believe that 4 other Arab books discovered in 1968 are also due to Diophantus.

This work contains the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems. Diophantus looked at 3 different types of quadratic equations, because he did not have any notion of zero and he avoided negative coefficients. He also considered simultaneous quadratic equations.

Fragments of another book by Diophantus, On polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. Diophantus himself refers to another work called The Porisms which consists of a collection of lemmas, but this book is entirely lost. We do know three lemmas contained in it since Diophantus refers to them in the Arithmetica. One such lemma is that given any numbers a, b then there exist numbers c, d such that a3 - b3 = c3 + d3.

Although Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown. Since an abbreviation is also employed for the word "equals", he took a fundamental step from verbal algebra towards symbolic algebra.

Diophantus was concerned with particular problems more than with general methods. The reason for this is that although he made important advances in symbolism, he still lacked the necessary notation to express more general methods.

Soon after Diophantus had made his forever standing testimony to his ingeniousness Europe was plunged into the Dark Ages and did not even hear about Diophantus until the late 15th century and wasn’t even seriously studied until the late 16th century. Diophantus and his works have always influenced the Arab mathematicians and were of great fame among them. When Arithmetica was finally translated and published in Europe in 1621 its value as mathematical gem was finally realized. After that point on it had great influence upon mathematicians. So in spite of his ineffective notation and inelegance of his methods, Diophantus must be regarded as the precursor of modern algebra.

[edit] References

[edit] Sources

• A. Allard, "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl. 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.

[edit] External links

• O'Connor, John J., and Edmund F. Robertson. "Diophantus". MacTutor History of Mathematics archive.
• Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
• Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.da:Diofant

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