What is Square of a Number?

Definition of square

When you multiply a whole number (not a fraction) by itself, the outcome is a square number.

For example:

4 x 4 = 16

“Sixteen” is the square of “four” multiplied by itself.

Another Definition of Square

A square number also called a perfect square, is a figurate number of the form

S_n = n²

where n is an integer.

The square numbers for:

n= 0, 1, 2, 3, 4, 5, 6, 7, …

are 0, 1, 4, 9, 16, 25, 36, 49, …

Example of Squares From Geometry

Area of square is best example of square of a number.

Area of a square = Side × Side

We can say that;

Square number = s ×s = s²

Odd and Even square numbers

• Squares of even values are even numbers,

i.e., (2n)² = 4n²

• Squares of odd values are odd numbers,

i.e., (2n + 1)² = 4(n² + n) + 1.

• Since every odd square is of the form 4n + 1, the odd values that are of the form 4n + 3 are not square numbers.

Squaring Negative Numbers

As you might understand already, if you multiply a negative number by another negative number, it ends up being a positive.

An example of negative square number

-4 x -4 will become 16 just exactly same as it will, if both the 4’s were positive!

However, if you are multiplying a negative number with a positive number, like -4 x 4 it would become negative number -16 and then, of course, it wouldn’t be a square number (because -4 is a different number to 4)!

Squaring Decimals

Same as whole numbers (integers), it’s very easy to take square of a decimal number too!

An example of a decimal square

Square of 14.55

14.55^2 = 14.55 * 14.55 = 211.7025

Properties of Square Numbers

• A number with 2, 3, 7 or 8 at unit’s place shall never be a complete square. Moreover, none of the square numbers ends in 2, 3, 7 or 8.

E.g.

4² = 16, 5² = 25, 6² = 36

• If the total number of zeros at the end is even, then number is a complete square number. Else, we can say that number ending in an odd number of zeros is never a complete square

E.g.

100 = 10², 1000  n², 10000 = 100²

• If the natural numbers other than 1 is squared, it shall be either a multiple of 3 or exceeds a multiple of 3 by 1.

E.g.

4² = 16 it exceeds by 1. 6² = 36 it is multiple of 3

• If the natural numbers other than 1 is squared, it shall be either a multiple of 4 or exceeds a multiple of 4 by 1

E.g.

5² = 25 it exceeds by 1, 8²= 64 it is a multiple of 4

• It is kept in mind that the unit’s digit of the square of a number is equal to the unit’s digit of the square of the digit at unit’s place of the given number.

E.g.

42² = 1764 square of 2 is 4.

• If a number n is squared, it is equal to the addition of first n odd natural numbers.

E.g.

5² = 1+3+5+7+9 = 25

6² = 1+3+5+7+9+11 = 36