Definition of square
When you multiply a whole number (not a fraction) by itself, the outcome is a square number.
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
Sn = 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)!
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.
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
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.
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
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.
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.
52 = 1+3+5+7+9 = 25
62 = 1+3+5+7+9+11 = 36