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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.

Formal Definition

Let $\left ({\N, +}\right)$ be the commutative semigroup of natural numbers under addition.

From Inverse Completion of Natural Numbers, we can create $\left({\N', +'}\right)$, an inverse completion of $\left ({\N, +}\right)$.

From Construction of Inverse Completion, this is done as follows:

Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

From Cross-Relation is Congruence Relation, $\boxtimes$ is a congruence relation.

Let $\left({\N \times \N, \oplus}\right)$ be the external direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$:

$\left({x_1, y_1}\right) \oplus \left({x_2, y_2}\right) = \left({x_1 + x_2, y_1 + y_2}\right)$

Let the quotient structure defined by $\boxtimes$ be $\left({\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}\right)$

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$.

Let us use $\N'$ to denote the quotient set $\dfrac {\N \times \N} \boxtimes$.

Let us use $+'$ to denote the operation $\oplus_\boxtimes$.

Thus $\left({\N', +'}\right)$ is the Inverse Completion of Natural Numbers.

As the Inverse Completion is Unique up to isomorphism, it follows that we can define the structure $\left({\Z, +}\right)$ which is isomorphic to $\left({\N', +'}\right)$.

An element of $\N'$ is therefore an equivalence class of the congruence relation $\boxtimes$.

So an element of $\Z$ is the isomorphic image of an element $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$ of $\dfrac {\N \times \N} \boxtimes$.

The set of elements $\Z$ is called the integers.

Natural Number Difference

In the context of the natural numbers, the difference is defined as:

$n - m = p \iff m + p = n$

from which it can be seen that the above congruence can be understood as:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1 \iff x_1 - y_1 = x_2 - y_2$

Thus this congruence defines an equivalence between pairs of elements which have the same difference.

Linguistic Note

The word integer is pronounced with the stress on the first syllable, and the g is soft (i.e. sounds like j): in-te-jer.

This is inconsistent with the pronunciation of the related term integral where the g is hard (as in get): in-te-gral.

Also note the use of the word integral as an adjective, meaning necessary or inherent, usually encountered in rhetoric. For further confusion, this is pronounced in-teg-ral, the stress being on the second syllable.

The symbol $\Z$ is for Zahlen, which is German for whole numbers, with overtones of unbroken.

This is reflected in English in the word integrity, which means wholeness in the sense of unbrokenness. The word also has a similarly applicable definition in the context of moral philosophy.

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 Nicolas Bourbaki.

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

Also see

  • Results about integers, in an abstract algebraic context, can be found here.