Integer

In mathematics, integers are whole numbers (natural numbers, their negatives, and zero).^[1] Integers can also be shown on a number line like this:

${\displaystyle {...-4,-3,-2,-1,0,+1,+2,+3,+4,...}\,\!}$

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 ${\displaystyle {1,2,3,4}}$).

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.

${\displaystyle \mathbf {Z} }$ or ${\displaystyle \mathbb {Z} }$ is the name of the set of integers, and ${\displaystyle \mathbb {Z} _{+}\!}$ is the set of positive integers.^[2][3] (${\displaystyle \mathbf {Z} }$, +, 0) is an abelian group.

The number of integers is infinite. The cardinal number of ${\displaystyle \mathbf {Z} }$ is ${\displaystyle \aleph _{0}}$. The ordinal number of ${\displaystyle \mathbf {Z} }$ is ${\displaystyle ^{*}\omega +\omega }$.

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

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

Somewhat confusingly ^[4] , integers don't always correspond to exactly one type in a programming language, such as:

• char, short, int, long, long long in C
• int, bignum in Python

All seven of the above types are integers.

"Unsigned" integers are not allowed to have a negative sign, such as ${\displaystyle {0,+1,+2,+3,...}}$

"Signed" integers are allowed to have a negative sign (or positive sign or no sign), such as ${\displaystyle {-3,-2,-1,0,+1,+2,+3,...}}$

1. ↑ Negative numbers have a minus (−) in front of the number. Positive numbers have no sign or a plus (+) sign in front. Zero usually has no sign, but in rare cases such as One's Complement and Sign-magnitude, both ${\displaystyle +0}$ and ${\displaystyle -0}$ co-exist.
2. ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-11.
3. ↑ Weisstein, Eric W. "Integer". mathworld.wolfram.com. Retrieved 2020-08-11.
4. ↑ Multiple types exist for integers due to the biggest and smallest representable numbers varying among the types.

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