PROPERTIES OF PERFECT SQUARE

When a number is multiplied by itself, we say that the number is squared.

It is denoted by a number raised to the power 2.

For example, 

3 x 3 = 3² = 9

5 x 5 = 5² = 25

In the example above, 5² is read as 5 to the power of 2 or 5 raised to the power 2 or 5 squared. 25 is the square of 5.

Similarly, 49 and 81 are the squares of 7 and 9 respectively.

The numbers 1, 4, 9, 16, 25, g are called perfect squares or square numbers as 1 = 1², 4 = 2², 9 = 3², 16 = 4²  and so on.

A number is called a perfect square, if it is expressed as the square of a number.

We observe the following properties through the patterns of perfect squares.

Property 1 : 

In perfect squares, the digits at the one’s place are always 0, 1, 4, 5, 6 or 9. The numbers having 2, 3, 7 or 8 at its one' place are not perfect square numbers.

Property 2 :

If a number has 1 or 9 in the one's place then its square ends in 1.

Property 3 :

If a number has 2 or 8 in the one's place then its square ends in 4.

Property 4 :

If a number has 3 or 7 in the one's place then its square ends in 9.

Property 5 :

If a number has 4 or 6 in the one's place then its square ends in 6.

Property 6 :

If a number has 5 in the one's place then its square ends in 5.

Property 7 :

Consider the following square numbers :

From the perfect squares given above, we infer that

Property 8 :

A perfect square number followed by even number of zeros will be a perfect square and a perfect square number followed by odd number of zeros will not be a perfect square.  

Consider the following square numbers :

Therefore, 100 is a perfect square and 81000 is not a perfect square.

Property 9 :

Square of even numbers is always even. 

Property 10 :

Square of odd numbers is always odd. 

From property 11 and property 12, we infer that

Some Interesting Patterns of Square Numbers

Addition of consecutive odd numbers :

The above figure illustrates the result that the sum of the first n natural odd numbers is n².

And, square of a rational number a/b is given by

Kaprekar Numbers

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