Definition:Integer

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


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 congruence relation defined on $\N \times \N$ by:

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

The fact that this is a congruence relation is proved in Equivalence Relation on Semigroup Product with Cancellable Elements.


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 $\displaystyle \frac {\N \times \N} \boxtimes$.

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 $\displaystyle \frac {\N \times \N} \boxtimes$.


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


From the comment in the proof of Construction of Inverse Completion: This Equivalence Relation is a Congruence, it can be seen that the equivalence classes which are the elements of $\Z$ can be characterized by identifying each class with the difference.


Notation

Note that $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\left[\!\left[{a, b}\right]\!\right]_\boxminus$, or $\left[\!\left[{a, b}\right]\!\right]$.

This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\left[\!\left[{a, b}\right]\!\right]$ at the start of any exposition.


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.


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.


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.


Also see

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


Sources