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# Integer

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, $\displaystyle{ \mathbb{Z} }$), which stands for Zahlen (German for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set.

## Algebraic properties

Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.

closure: $\displaystyle{ a + b }$ is an integer $\displaystyle{ a \cdot b }$ is an integer
associativity: $\displaystyle{ a + (b + c) = (a + b) + c }$ $\displaystyle{ a \cdot (b \cdot c) = (a \cdot b) \cdot c }$
commutativity: $\displaystyle{ a + b = b + a }$ $\displaystyle{ a \cdot b = b \cdot a }$
existence of an identity element: $\displaystyle{ a + 0 = a }$ $\displaystyle{ a \cdot 1 = a }$
existence of inverse elements: $\displaystyle{ a + (-a) = 0 }$
distributivity: $\displaystyle{ a \cdot (b + c) = (a \cdot b) + (a \cdot c) }$

## Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by

$\displaystyle{ \ldots \lt -2 \lt -1 \lt 0 \lt 1 \lt 2 \lt \ldots }$

An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:

1. If $\displaystyle{ a \lt b }$ and $\displaystyle{ c \lt d }$, then $\displaystyle{ a + c \lt b + d }$
2. If $\displaystyle{ a \lt b }$ and $\displaystyle{ 0 \lt c }$, then $\displaystyle{ ac \lt bc }$

(From this fact, one can show that if $\displaystyle{ c \lt 0 }$, then $\displaystyle{ ac \gt bc }$.)