The Lesson

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:

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.