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What Is an Integer? (Mathematics Glossary)

What Is an Integer?

An integer is a whole number. An integer may be positive, negative, or zero:

Mathematicians use the symbol to represent the set of integers.

Dictionary Definition

The Oxford English Dictionary defines an integer as "a whole number or undivided quantity. Opposed to fraction."

Examples of Integers

An integer is a whole number. Some examples of integers are given below:

An integer does not contain a fractional part. They cannot be fractions or decimals. Some examples of numbers that are not integers are shown below:

Integers on a Number Line

The image below shows the integers on the number line.

  • The integers on the number line go on forever to the left and right.

  • Integers do not exist in between the whole numbers labelled.

  • The number line shows the ordering of the integers. The further right on the number line an integer is, the greater it is.

    Any integer is greater than an integer to its left. Any integer is less than any integer to its right.

    This means that 3 is greater than 2, but -3 is less than -2.

Integers, Natural Number and Whole Numbers

The integers contain:
  • The counting numbers (called the natural numbers): {1, 2, 3 ...}
  • 0
  • The negatives of the natural numbers: {-1, -2, -3 ...}
If we include 0 in with the natural numbers, we get the whole numbers:
{0, 1, 2, 3 ...}

Putting all this together, we see that the integers contain the whole numbers and the natural numbers:

Read more about the types of whole numbers

Properties of Integers

The slider below shows some of the properties of integers:
Interactive Test

Here's a second test on integers.
Here's a third test on integers.

What's in a Name?

The word integer comes from the Latin word 'integer', meaning "whole".

The Latin word 'integer' itself is derived from the Latin 'in' ('not') + 'tangere' ('to touch').

If a number was untouched, it was unbroken, whole and entire.

This should be contrasted with the Latin root for fraction, 'fractus' meaning 'broken'.

What Does the Symbol for Integers Mean?

Mathematicians use the symbol to represent the set of integers.

This symbol is a Z, from the German word 'Zahlen', meaning 'numbers'.