# integer

Any positive or negative whole number or zero: ...-3, -2, -1, 0, 1, 2, 3, ... "Integer" is Latin for "whole" or "intact." The set of all integers is denoted by Z, which stands for Zahlen (German for "number"). The integers are an extension of the natural numbers to include negative numbers and so make possible the solution of all equations of the form a + x = b, where a and b are natural numbers. Integers can be added and subtracted, multiplied, and compared. Like the natural numbers, the integers form a countably infinite set. However, the integers don't form a field since, for instance, there is no integer x such that 2x = 1; the smallest field containing the integers is that of the rational numbers. An important property of the integers is division with remainder: given two integers a and b with b not equal to 0, it is always possible to find integers q and r such that a = bq + r, and such that 0 r < |b|. q is called the quotient and r is called the remainder resulting from division of a by b. The numbers q and r are uniquely determined by a and b. From this follows the fundamental theorem of arithmetic, which states that integers can be written as products of prime numbers in an essentially unique way.

## Related category

• TYPES OF NUMBERS