What Is An Integer? — Definition & Examples

Written by

Malcolm McKinsey

January 12, 2023

Fact-checked by

Paul Mazzola

Integer definition

An integer is a whole number from the set of negative, non-negative, and positive numbers. To be an integer, a number cannot be a decimal or a fraction.

The follow are integers:

• 130

• -9

• 0

• 25

• -7,685

Integers

Integers are all negative and positive whole numbers, and 0. Integers or integer values are part of various numbering systems.

Numbering systems are ways of counting and categorizing real and imaginary objects. Integers are one set of numbers or numbering system you use every day.

Common numbering systems you may encounter include all these:

• Real numbers

• Natural numbers

• Integers

• Imaginary numbers

• Rational numbers

• Irrational numbers

• Complex numbers

We usually write our numbers using Arabic numerals: 0, 1, 2, 3. But we can also represent them using Roman numerals (except for 0): I, II, III.

Set of integers

Mathematicians show sets of numbers using these brackets {}, and ellipsis … to show that the numbers continue without end. So we could show a set of integers like this:

We understand that the negative numbers keep going, and so do the positive whole numbers.

Below are two different sets of numbers. What do each of them mean?

1. {1, 2, 3, 4, 5...}

2. {0, 2, 4, 6, 8, 10...}

The first is a set of all positive integers. The second is a set of all non-negative, even integers.

Integers that are not whole numbers

Negative integers are not whole numbers. Whole numbers are all counting numbers and 0. A set of integers that are not whole numbers looks like this:

A negative number that is not a decimal or fraction is an integer but not a whole number.

Integer examples

Integers are positive whole numbers and their additive inverse, any non-negative whole number, and the number zero by itself. Here are examples of integers:

• Negative integers: -1, -2, -3, -4, -5 and so on, without end.

• Non-negative integers: 0 and all positive whole numbers like 6, 7, 8, 9, 10 and so on.

• Positive integers: 1, 2, 3, 4, 5 and so on, without end.

• Zero: 0 all by itself.

Non-integers examples

Non-integers are any number that is a decimal, fraction, or mixed unit. These are all not integers:

• Decimals: 3.14

• Fractions: $\frac{1}{2}$

• Mixed units: $3\frac{1}{2}$

Where do we use integers?

Integers pop up in most things you count each day:

• Temperatures: $45°C$ or $76°F$

• Grades: 80%

• Teams: 11 football players; 9 baseball players

• Altitude: Commercial planes fly at 35,000 feet

• Coins: You have three quarters and two dimes so you have 0.95 cents.

Whenever you are counting real objects such as books, pencils, shoes, hats, friends, or the number of hairs on your head, you are counting with integers.

Also, every time you round a number to a whole value, you change it from a decimal, fraction, or mixed unit to an integer:

• 5'6" becomes the integer 6'

• $\mathbf{3}\frac{\mathbf{7}}{\mathbf{8}}$ becomes the integer 4

• $98.6° F$ becomes the integer $99 °F$

• $1.97 becomes the integer $2

• 364.75 miles becomes the integer 365 miles.

Characteristics of integers

Testing to see if a number is an integer is as easy as asking two questions:

1. Is it a whole number? – Integer!

2. Is it the number 0? – Integer!

3. Is it negative of a whole number? – Integer!

4. Does it include multiple parts, such as 5'6" or $1.97 – Not an integer!

Which of the following are integers?

1. One

2. 3.14159265358979

3. -15°C

4. $\frac{11}{3}$

5. -365

Answers 1, 3, and 5 are integers.