Integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French[1]) are the set of numbers consisting of the natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). 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. In other terms, integers are the numbers one can count with items such as apples or fingers, and their negatives, including 0.

More formally, the integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition.[citation needed] Like the natural numbers, the integers form a countably infinite set. The set of all integers is often denoted by a boldface Z (which stands for Zahlen (German for numbers, pronounced "tsAH-len").

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.