What is an Integer?

The integers (from the Latin integer, literally "untouched," hence "whole": the word entire comes from the same origin, but via French) 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, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers.

The set of all integers is often denoted by a boldface Z (or blackboard bold , Unicode U+2124 ℤ), which stands for Zahlen (German for numbers, pronounced [ˈtsaːlən]).

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.

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