Integer

In mathematics, integers are whole numbers (natural numbers, their negatives, and zero). Integers can also be shown on a number line like this:

[math]\displaystyle{ {...-4, -3, -2, -1, 0, +1, +2, +3, +4,...}\,\! }[/math]

In particular, zero or 0 is an integer that is neither positive nor negative. An integer is a whole number with no fractional or decimal part.

Integers have no smallest or largest value because for any given integer, there is always a larger and smaller integer. Any integer is either greater than or smaller than any other integer. Consecutive integers are integers that come after each other (as in [math]\displaystyle{ {1, 2, 3, 4} }[/math]).

The sum, difference and product of integers is always an integer.(for example, (12 + 2345 × (67 - 8)) × 9 is an integer). An integer divided by an integer is sometimes not an integer (for example, 1 ÷ 2 = 0.5).

[math]\displaystyle{ \mathbf{Z} }[/math] or [math]\displaystyle{ \mathbb{Z} }[/math] is the name of the set of integers, and [math]\displaystyle{ \mathbb{Z}_+\! }[/math] is the set of positive integers. ([math]\displaystyle{ \mathbf{Z} }[/math], +, 0) is an abelian group.

The number of integers is infinite. The cardinal number of [math]\displaystyle{ \mathbf{Z} }[/math] is [math]\displaystyle{ \aleph_0 }[/math]. The ordinal number of [math]\displaystyle{ \mathbf{Z} }[/math] is [math]\displaystyle{ ^*\omega + \omega }[/math].

To sum up, an integer is a whole number that has no decimals.

In programming

In some programming languages, like C, there are types called "int" or "integer."

Integer Media

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  The integers arranged on a number line

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  Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in blue and negative integers in red.

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  A figure representing the equivalence classes of the relative numbers constructed as a pair of natural numbers. Any relative number (eg. -5) has an infinity of equivalent possible representation, eg. (2,7), (0,5), … The equivalent representations (in red) are on the same blue dotted line, and the number in blue at the end on the line is the corresponding relative number