Key Points - Squares and Square Roots

- Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 22 = 2 × 2 = 4 (square of 2 is 4).
- If a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number.
- All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.
- Square numbers can only have even number of zeros at the end.
- Square root is the inverse operation of square.
- There are two integral square roots of a perfect square number.
- Positive square root of a number is denoted by the symbol
For example, 3 ^ 2 =9 gives 9 =3
- Perfect Square or Square number: It is the square of some natural number. If m = n 2 , then m is a perfect square number where m and n are natural numbers. Example: 1 = 1× 1 = 12 ,
4 = 2 × 2 = 22 .

- Properties of Square number: 
(i) A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
(ii) A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number.
Example: 20, 31, 24, etc.
(iii) Square of even numbers are even. Example: 22 = 4 , 4 2 = 16 , etc.
(iv) Square of odd numbers are odd. Example: 52 = 25 , 92 = 81 , etc.
(v) A number ending in an odd number of zeroes cannot be a perferct square. 
Example: 10, 1000, 900000, etc.
(vi) The difference of squares of two consecutive natural number is equal to their sum.
 ( n + 1) (vii)
2
− n 2 = n + 1 + n . Example: 42 − 32 = 4 + 3 = 7 , 12 2 − 112 = 12 + 11 = 23 , etc.

A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean triplet, if m 2 + n 2 = p 2 ;32 + 42 = 25 = 52