Square and square root:

 A natural number m is said to be square number or perfect square if it can be expressed in terms of n², where n is also a natural number. It is necessary to know about square and square root to solve number system related problems.
Example: 4 is a square number as 4 = 2²

Properties of perfect square:

1. All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.

2. None of these end with 2, 3, 7 or 8 at unit’s place.

3. If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.

4. when a perfect square number ends in 6, the number whose square it is will have either 4 or 6 in unit’s place.

5. There are 2n non-perfect square numbers between n² and (n + 1)²
Example: between 2² and 3² the no. of non-perfect square no. are 2x2 =4 as (5, 6, 7, 8)

6. sum of first n odd natural numbers is n² also Remember it for square and square root related terms.
Example: 1+3+5 = 9 = 3²

7. Any perfect square number can be represented as sum of odd natural numbers starting from 1.

8. Also, Any odd perfect square number can be represented as sum of two consecutive natural number.
Example: 49 = 24+25, 121 = 60 + 61

9. For any natural number (n + 1) × (n – 1) = n² – 1.
Example: 6 x 4 = 5² – 1 = 24

10. Square numbers can only have even number of 0’s at the end.

11. If a perfect square is of n-digits, then its square root will have \(\frac{n}{2}\) digits if n is even, or \(\frac{n + 1}{2}\) if n is odd?

Pythagorean triplet:

For any natural number a > 1, we have (2a)²+(a² – 1)²=(a² + 1)²

So, 2a, a² – 1 and a²+ 1 forms a Pythagorean triplet.

By this rule, if any one of the members of Pythagorean triplet is given then you can find the rest. this also necessary for getting knowledge of square and square root.

Example:
1. 3² +  4² = 5² so, 3, 4, 5 are Pythagorean triplet.

2. If 5 is one of the members of the Pythagorean triplet then find the rest.

Solution:
Try 2a = 5 then a will not be an integer so take next
a² – 1 = 5 then a² = 6 again not an integer
try a² + 1 = 5 then a² = 4 thus, a = 2

2a = 4
a² – 1 = 3

Thus, Pythagorean triplet are 3, 4, 5.

Square root:

It is simply the inverse operation of square. Square and square root are in converse of each other.
There are many ways to find the square root of any number:

1. By just inversion of square.
Example: 3² = 9 and √9 = 3

2. By repeated subtraction of odd numbers.
Example: √9,
9 – 1 = 8, 8 – 3 = 5, 5 – 5 = 0, by 3 times consecutive subtraction we got zero thus √9 = 3.

3. By prime factorization.
√36 = 2x2x3x3 = 2² x 3² thus square root of 36 is 2x3 i.e. 6

4. By division method.
Try to solve examples based on square and square root. So that it will be easy for you to solve problems related to it.