Perfect Squares

A perfect square number is obtained by the product of two same integers. More specifically, if we multiply an integer with itself then the resultant number will be a perfect square number. So the general form of a perfect square is r2 for some integer r. As 16=42, the number 16 is an example of a perfect square. In this section, we will learn about perfect square numbers.

Definition of Perfect Square

Perfect square definition: A number is called a perfect square if its square root is a whole number. Note that the square root of a perfect square is an integer.

For example, the square root of 4 is 2. Since 2 is an integer, we can say that 4 is a perfect square. More examples of perfect squares: $9, 16, 25, 36, 49, 64, 81$ etc.

 

Properties of Perfect Squares

From the definition, the square root of a perfect square is an integer.

If $X$ is a perfect square, then we can express $X$ as $X=n^2$ for some natural number $n.$

The square root of a perfect square must have unit digit $0, 1, 4, 5$ or $9.$

The numbers ending with $2, 3, 6, 7$ or $8$ cannot be a perfect square number. Those numbers are called non-perfect square numbers.

The square root of a non-perfect square is a quadratic surd. As $\sqrt{8}=2\sqrt{2}$ is not an integer, $8$ is an example of non-perfect square numbers, and the square root $\sqrt{8}$ is a quadratic surd.

 

List of Perfect Squares

Integer (n) n ×n Perfect square
0 0×0=0 0
1 1×1=1 1
2 2×2=4 4
3 3×3=9 9
4 4×4=16 16
5 5×5=25 25
6 6×6=36 36
7 7×7=49 49
8 8×8=64 64
9 9×9=81 81
10 10×10=100 100

From the above table, we see that the perfect squares between 1 to 100 are the numbers $1, 4, 9, 16, 25, 36, 49, 64, 81$ and $100.$ So there are 10 perfect squares from 1 to 100.