An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by a boldface Z ("Z") or blackboard bold
Z
{\displaystyle \mathbb {Z} }
(Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite.
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, the (rational) integers are the algebraic integers that are also rational numbers.
The symbol Z can be annotated to denote various sets, with varying usage amongst differentauthors: Z+, Z+ or Z> for the positive integers, Z≥ for non-negative integers, Z≠ for non-zero integers. Some authors use Z* for non-zero integers, others use it for non-negative integers, or for {–1, 1}. Additionally, Zp is used to denote either the set of integers modulo p, i.e., a set of congruence classes of integers, or the set of p-adic integers.