In mathematics,  integers are numbers that are written without a fractional component. 

For example, 21, 4, and −2048 are integers; 9.75, 5½, and √2 are not integers. 

The set of integers consists of the natural numbers (1, 2, 3, ...), zero (0) and the opposites of the natural numbers (−1, −2, −3, ..., that are negative).

The symbol for integers. image: wikipedia

The origin of the word integer

The name derives from the Latin integer (meaning literally "untouched," hence "whole": the word entire comes from the same origin, but via French). 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) it is a subset of the sets of rational and real numbers.

Operations and countability

The integers (with addition as operation) form the smallest group containing the additive monoid (a single operation) 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.