# What Is a Square Number?

A square number is a number that results from a number being multiplied by itself.

For example,
• 1 is a square number. It is 1 × 1.

• 4 is a square number. It is 2 × 2.

• 9 is a square number. It is 3 × 3.
Note: 2 × 2 can be written 22 ("2 squared"),

#### Dictionary Definition

The Oxford English Dictionary defines a square number as "the product of a number multiplied by itself."

# The Square Numbers

The square numbers are:

In a number square, the square numbers are shaded below:

These are just the square numbers up to 100. There are infinitely many square numbers, they go on forever.

# How to Find Square Numbers

Finding square numbers is easy. Just multiply a number by itself.

The first 10 square numbers are found by multiplying 1 by itself, then 2 by itself, as shown below.

# Square Numbers in Algebra

In algebra, letters are used instead of numbers.

Just as we can square a number...
22 = 2 × 2 = 4
...we can square a letter:
x2 = x × x
Note: We would say x2 as "x squared", just like we would say 22 as "2 squared".

Here are some examples of squared letters in algebra:
• Quadratic equations have an x2 term:

• Pythagoras theorem relates the square of the lengths of the sides of a right triangle to each other.

The image below shows the formula for Pythagoras theorem and the lengths of the sides a, b and c of a right triangle:

• The area of a circle is related to the square of its radius.

The image below shows the formula for the area of a circle and the circle's radius, r

##### Interactive Test
show

Here's a second test on square numbers.
Here's a third test on square numbers.
Here's a dynamic test on square numbers.

# Square Numbers Come From Integers

Square numbers are the result of integers (whole numbers, not fractions) being multiplied by themselves.

A square number will always be an integer.

# Square Numbers Are Always Positive

A square number is always a positive number. They can never be a negative number.

Negative numbers can be multiplied by themselves to give a positive square number:
-2 × -2 = (-2)2 = 4

# "Squaring" a Number

The verb which describes the operation of multiplying a number by itself (e.g. 6 × 6) is "to square".

When we multiply a number be itself , we call this "squaring" a number.

6 × 6 = 62 (called "6 squared").

# Why "Square" Numbers?

A square number is a number times itself.
12 = 1 × 1 = 1
22 = 2 × 2 = 4
32 = 3 × 3 = 9
The squares go 1, 4, 9...

A square with sides of length 1 has an area of 1.

A square with sides of length 2 has an area of 4.

A square with sides of length 3 has an area of 9.

The areas go 1, 4, 9...

The square of a number and the area of a square with side lengths of that number are the same.

# Square Roots

If 2² = 4, then 4 is the square of 2.

The inverse of the square is the square root, denoted by the √ symbol.

2 is the square root of 4, or √4 = 2.