Square numbers – definition, properties and examples

A square number is a result of multiplying a whole number with itself. In other words, when we take a number and multiply that number with itself then the result that we get is called a square number or a perfect square.

For example, if we take 2 and then multiply 2 x 2 we get 4. We will then call the result 4 as the square number because it is the result of squaring the number or multiplying the same number with itself.

Multiplying a number with itself is called squaring the number (denoted by n2).

100 is a square number because it is a product of 10 x 10.

Technical definition: The product of an integer with itself results in another integer that is called a square number.

(If it is too confusing to remember the technical definition then don’t worry about it because we are interested in understanding the concept and not learn by rote.)

What are the properties of a square number?

The square numbers are whole numbers that are non-negative integers and whose square root will result in an integer.

If n is an integer the n2 is the square number and the square root of n2 should give us back n.

Interesting fact: The square number for n is equal to the sum of the first n odd numbers in the sequence.

For example,
If n = 3, then the sum of first 3 odd numbers 1 + 3 + 5 is equal to 9.
n2 = 32 = 9

Examples of square numbers

The square numbers from 1 to 100 are: 1, 4, 9, 25, 36, 49, 64, 81, 100
The even square numbers from 1 to 100 are: 4, 36, 64, 100
The odd square numbers from 1 to 100 are: 1, 9, 25, 49, 81

Square numbers 0-100

02 = 0
12 = 1
22 = 4 = sum of 1 + 3
32 = 9 = sum of 1 + 3 + 5
42 = 16 = sum of 1 + 3 + 5 + 7
52 = 25 = sum of 1 + 3 + 5 + 7 + 9
62 = 36 = sum of 1 + 3 + 5 + 7 + 9 + 11 + 13
72 = 49 = sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
82 = 64 = sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17
92 = 81 = sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
102 = 100 = sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

Square numbers 100-400

112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400 