Square Numbers
(KS2, Year 6)

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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: squares_square

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. first_10_square_numbers

Interactive Game on Square Numbers

Here is an interactive game to help you learn about square numbers.

You are the fish.
Pop the bubbles with square numbers to collect red hearts. You need five to win.
If a bubble with a square number reaches the top, you will lose a red heart.
Avoid the bubbles with non-square numbers. If you pop one, you will lose one of your three lives.
Good luck!
The fish moves towards your clicks. (Hint: The farther the distance, the faster it moves.)

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").