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A What is a integer

integer, whole-valued positive or negative number or 0. The integers are generated from the set of counting numbers 1, 2, 3,… and the operation of subtraction. When a counting number is subtracted from itself, the result is zero; for example, 4 − 4 = 0. When a larger number is subtracted from a smaller number, the result is a negative whole number; for example, 2 − 3 = −1. In this way, every integer can be derived from the counting numbers, resulting in a set of numbers closed under the operation of subtraction (see group theory).In mathematical equations, unknown or unspecified integers are represented by lowercase, italicized letters from the "late middle" of the alphabet. The most common are p, q, r, and s.

The application of positive and negative numbers in the real world is different. They are generally used to represent two contradicting situations. One common real-life application of integers is temperature measurement. The negative and positive numbers and zero in the scale denote different temperature readings. Bank credit and debit statements also use integers to represent the negative or positive values of amount. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

is not a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes, the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. (Source: en.wikipedia.org)