ARITHMETICA DIOFANTO PDF

Biography[ edit ] 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. After consoling his fate by the science of numbers for four years, he ended his life.

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Of later Greek mathematicians, especially noteworthy is Diophantus of Alexandria flourished c. An arithmetic epigram from the Anthologia Graeca of late antiquity, purported to retrace some landmarks of his life marriage at 33, birth of his son at 38, death of his son four years before his own at 84 , may well be contrived.

Two works have come down to us under his name, both incomplete. The first is a small fragment on polygonal numbers a number is polygonal if that same number of dots can be arranged in the form of a regular polygon. The second, a large and extremely influential treatise upon which all the ancient and modern fame of Diophantus reposes, is his Arithmetica. 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.

The Arithmetica begins with an introduction addressed to Dionysius—arguably St. Dionysius of Alexandria. After some generalities about numbers, Diophantus explains his symbolism—he uses symbols for the unknown corresponding to our x and its powers, positive or negative, as well as for some arithmetic operations—most of these symbols are clearly scribal abbreviations.

This is the first and only occurrence of algebraic symbolism before the 15th century. After teaching multiplication of the powers of the unknown, Diophantus explains the multiplication of positive and negative terms and then how to reduce an equation to one with only positive terms the standard form preferred in antiquity.

With these preliminaries out of the way, Diophantus proceeds to the problems. Indeed, the Arithmetica is essentially a collection of problems with solutions, about in the part still extant. The introduction also states that the work is divided into 13 books. However, the Arabic text lacks mathematical symbolism, and it appears to be based on a later Greek commentary—perhaps that of Hypatia c.

Further renumbering is unlikely; it is fairly certain that the Byzantines only knew the six books they transmitted and the Arabs no more than Books I to VII in the commented version. Get exclusive access to content from our First Edition with your subscription.

Subscribe today The problems of Book I are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. In three problems of Book II it is explained how to represent: 1 any given square number as a sum of the squares of two rational numbers; 2 any given non-square number, which is the sum of two known squares, as a sum of two other squares; and 3 any given rational number as the difference of two squares.

While the first and third problems are stated generally, the assumed knowledge of one solution in the second problem suggests that not every rational number is the sum of two squares.

Such examples motivated the rebirth of number theory. Although Diophantus is typically satisfied to obtain one solution to a problem, he occasionally mentions in problems that an infinite number of solutions exists.

In Books IV to VII Diophantus extends basic methods such as those outlined above to problems of higher degrees that can be reduced to a binomial equation of the first- or second-degree. For instance, one problem involves decomposing a given integer into the sum of two squares that are arbitrarily close to one another. Book X presumably Greek Book VI deals with right-angled triangles with rational sides and subject to various further conditions.

Indeterminate equations restricted to integral solutions have come to be known, though inappropriately, as Diophantine equations. Learn More in these related Britannica articles:.

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