![]() ![]() ![]() In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm Elements, Prop. (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.)Įuclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements). Theaetetus was, like Plato, a disciple of Theodorus's he worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. 1800 BC) contains a list of " Pythagorean triples", that is, integers ( a, b, c ) are irrational. The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 ( Larsa, Mesopotamia, ca. ![]() In particular, arithmetical is commonly preferred as an adjective to number-theoretic. (The word " arithmetic" is used by the general public to mean " elementary calculations" it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating-point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. By the early twentieth century, it had been superseded by "number theory". The older term for number theory is arithmetic. One may also study real numbers in relation to rational numbers, for example, as approximated by the latter ( Diophantine approximation). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion ( analytic number theory). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences-and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. ![]()
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