DIOPHANTUS ARITHMETICA PDF

His texts deal with solving algebraic equations. 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. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Little is known about the life of Diophantus.

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His texts deal with solving algebraic equations. 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.

Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.

Little is known about the life of Diophantus. He lived in Alexandria , Egypt , during the Roman era , probably from between AD and to or It was at first found that Diophantus lived between AD by analysing the price of wine used in many of his mathematical texts and finding out the period during which wine was sold at that price.

Diophantus has variously been described by historians as either Greek , [2] [3] [4] non-Greek, [5] Hellenized Egyptian , [6] Hellenized Babylonian , [7] Jewish , or Chaldean.

One of the problems sometimes called his epitaph states:. This puzzle implies that Diophantus' age x can be expressed as. However, the accuracy of the information cannot be independently confirmed. In popular culture, this puzzle was the Puzzle No. 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 Arabic books discovered in are also by Diophantus.

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. Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the so-called Dark Ages , since the study of ancient Greek, and literacy in general, had greatly declined.

The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos — are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes — , who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.

In German mathematician Regiomontanus wrote:. Arithmetica was first translated from Greek into Latin by Bombelli in , but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in by Xylander. The best known Latin translation of Arithmetica was made by Bachet in and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.

The edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous " Last Theorem " in the margins of his copy:. 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 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 Even though the text is otherwise inferior to the edition, Fermat's annotations—including the "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 scholar John Chortasmenos — had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem. Diophantus wrote several other books besides Arithmetica , but very few of them have survived.

Diophantus himself refers [ citation needed ] to a work which consists of a collection of lemmas called The Porisms or Porismata , but this book is entirely 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.

Diophantus is also known to have written on polygonal numbers , a topic of great interest to Pythagoras and Pythagoreans. 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. It has been studied recently by Wilbur Knorr , who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus.

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 17th and 18th 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. 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.

It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. 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". 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. Diophantus made important advances in mathematical notation, becoming the first person known 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: [16]. 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. He also lacked a symbol for a general number n. Algebra still had a long way to go before very general problems could be written down and solved succinctly.

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; 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 [ It is, of course, impossible to answer this question definitively. 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 [ In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged 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.

From Wikipedia, the free encyclopedia. Alexandrian Greek mathematician. For the general, see Diophantus general. For the sophist, see Diophantus the Arab. See also: Arithmetica. See also: Diophantine equation. The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. Diophantus lived c. A History of Mathematics Second ed.

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. 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.

Hankel , 2nd ed. Katz A History of Mathematics: An Introduction , p. Burton , Brown Publishers. Sesiano Margins and Metropolis: Authority across the Byzantine Empire. Princeton University Press. Retrieved 10 April

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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?

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DIOPHANTUS OF ALEXANDRIA

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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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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