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This book features a host of problems, the most significant of which have come to be called Diophantine equations. 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 84may 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: The Arithmetica begins with an introduction addressed to Dionysius—arguably St. 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.
We now know that the numbering of the Greek books must be modified: 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.
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: 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.
Diophantus later gives the condition for an integer: 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:Algebra Help - Lessons, examples, practice questions and other resources in algebra for learning and teaching algebra, How to solve equations and inequalities, How to solve different types of algebra word problems, Rational expressions, examples with step by step solutions.
is the arithmetic work of Diophantus of Alexandria (c. 3rd century ce).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 kaja-net.com example, Book II, problem 8, .
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial kaja-net.comly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (). It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n, and it is given by the formula =!!(−)!.
Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Introduction to polynomials.
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