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The integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French[1]) are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers.

The set of all integers is often denoted by a boldface Z (or blackboard bold \mathbb{Z}, Unicode U+2124 ), which stands for Zahlen (German for numbers, pronounced [ˈtsaːlən]).[2] The set \mathbb{Z}_n is the finite set of integers modulo n (for example, \mathbb{Z}_2=\{0,1\}).

The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set.

In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as rational integers to distinguish them from the more broadly defined algebraic integers (but with "rational" meaning "quotient of integers", this attempt at precision suffers from circularity).


Algebraic properties

Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).

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