# PROPERTIES OF SQUARE NUMBERS

## About "Properties of square numbers"

²

Properties of square numbers :

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

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

For example,

(i)  3 x 3  =  3²  =  9

(ii)  5 x 5  =  5²  =  25

In example (ii) 5² is read as 5 to the power of 2 (or) 5 raised to the power 2 (or) 5 squared. 25 is known as the square of 5.

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

## Properties of square numbers

We observe the following properties through the patterns of square numbers.

Property 1 :

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

Property 2 :

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

Property 3 :

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

Property 4 :

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

Property 5 :

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

Property 6 :

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

Property 7 :

Consider the following square numbers :

From the square numbers given above, we infer that

Property 8 :

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.

It has been illustrated in the table given below.

Property 10 :

Square of odd numbers is always odd.

It has been illustrated in the table given below.

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

After having gone through the stuff given above, we hope that the students would have understood "Square numbers".

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