Definition:Integer

Informal Definition
The numbers $\left\{{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}\right\}$ are called the integers.

This set is usually denoted $\Z$.

An individual element of $\Z$ is called an integer.

Also known as
The integers are also referred to as whole numbers, so as to distinguish them from fractions. However, use of this term is discouraged because it is ambiguous: it can refer to the integers, the positive integers, or the negative integers, depending on the preference of the author.

Some sources refer to the integers as rational integers, to clearly distinguish them from the algebraic integers.

Some sources use the term directed numbers or signed numbers for integers, so as to distinguish them from the natural numbers which are, by definition defined without a sign.

Variants on $\Z$ are often seen, for example $\mathbf Z$ and $\mathcal Z$, or even just $Z$.

Some sources use $I$, while others use $\mathbf J$ or a variant.

Historical Note
The use of $Z$ appears to have originated with.

Earlier, the symbol $\overline{\mathfrak Z}$ had been used by for the integers.