---
tag: 0C4JS9RF5M
kind: object
name: Integers
aliases: [integer, set-of-integers]
msc: [11A99]
sources:
  - url: https://en.wikipedia.org/wiki/Integer
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Integer
    locator: Algebraic properties
same_as:
  wikidata: Q12503
  wikipedia: Integer
---

## Description

The integers are the numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$.
We write $\mathbb{Z}$ for the set of integers.
The set $\mathbb{Z}$ contains the [[natural numbers|0MWKJ44V5A]] $0, 1, 2, \dots$ and their negatives $-1, -2, -3, \dots$.
With addition and multiplication, $\mathbb{Z}$ is a [[commutative ring|0BXA4Q0JHW]]. Each integer $a$ has an additive inverse $-a$, thus subtraction is always possible in $\mathbb{Z}$.
The integers have a [[total order|0F6FM9Y4Y6]] that extends the order of $\mathbb{N}$.

## Remarks

The letter $\mathbb{Z}$ comes from the German word "Zahlen", which means "numbers".
One [[construction of the integers|0NS58JP68E]] uses pairs of natural numbers.
The [[rational numbers|0NCJ5WXN6Z]] contain the integers as the fractions $a / 1$.

## Explanations

### L1 human

The integers are the whole numbers with a sign: positive, negative or zero.
You need them to answer questions such as $5 - 8 = ?$, which has no answer in the natural numbers.
On a number line, the negative integers are to the left of $0$.