If a natural number m can be expressed as n2 (where n is a natural number), then m is the square root or perfect square.

i.e. if m = n^{2} (m, n – natural numbers)

E.g. 81 = 3 × 3 × 3 × 3

= 3^{2} × 3^{2} = (3 × 3)^{2} = 9^{2}

Hence, 9 is the square root of 81.

Below is the table that has squares of numbers from 1 to 10.

If we see the above results carefully, we can conclude that numbers ending with 0, 1, 4, 5, 6, or 9 at units place are perfect squares None of these end with 2, 3, 7 or 8, So numbers that end with 2, 3, 7, 8 are not perfect squares.

Thus, numbers like 122, 457, 183, 928 are not perfect squares.

(1). The ones digit in the square of number can be determined if the ones digit of the number is known.

(2). The number of zeros at the end of a perfect square is always even and double the number of zeros at the end of the number

E.g.

Double zero {700^{2} = 490000} four zero (even)

(3). The square of an even number is always an even number and square of an odd number is always an odd number.

E.g.

Number between square numbers

There are ‘2a’ non perfect square numbers between the square of two Consecutive natural numbers n + (n + 1)

Between 22 = 4 & 32 = 9 → 5, 6, 7, 8

2 × 2 = 4 non square numbers

Between

3^{2} = 9 & 4^{2} = 16 → 10, 11, 12, 13, 14, 15

2 × 3 = 6 non square numbers.

So, we can conclude that the sum of first in odd natural numbers in n^{2} or, we can say if the number is a square number, it has to be the sum of successive odd numbers.

E.g. 36

Successively substract 1, 3, 5, 7 ... from 36

36 – 1 = 35

35 – 3 = 32

32 – 5 = 27

27 – 7 = 20

20 – 9 = 11

11 – 11 = 0

∴ 36 is a perfect square

We can use the below to methods for Calculating square of any natural number

(a + b)^{2} = a^{2} + b^{2} + 2ab

(a � b)^{2} = a^{2} + b^{2} � 2ab

The logic behind using these formulae is to convert square of unknow number into square of a known number.

E.g.

(53)^{2} = (50 + 3)^{2}

= 50^{2} + 3^{2} + 2.50.3

= 250 + 9 + 300

= 2809

99^{2} = (100 – 1)^{2}

= 100^{2} + 1^{2} – 2.100.1

= 10000 + 1 – 200

= 9801

Finding the Square root is inverse operation of squaring.

If 9 × 9 = 81 i.e, 9^{2} = 81

Then the square root of 81 = 9

In other words, if p = q^{2}, the q is called the square root of p

The Square root of a number is the number which when multiplied with itself gives the number as the product. The square root of a number is denoted by the symbol

Some Properties of square root

Finding square root through repeated subtraction Recall the pattern formed while adding Consecutive odd numbers.

1 + 3 + 5 + 7 + 9 = 5^{2} = 25

1 + 3 + 5 + 7 + 9 + 11 = 6^{2} = 36

Sum of first n odd numbers = n^{2}

The above Pattern can be used to find the square root of the given number.

Finding square root through Prime factorization. In order to find the square root of a perfect square by prime factorization.

We follow the following steps.

E.g.

Find the square root of 400

Finding the square root by division Method when the numbers are large even the method of finding square root by prime factorisation becomes slightly difficult. To over come this problem we use long division method.

Step 1: Place a bar over every pair of digits starting from the digit at one�s place. If the number of digits is odd, then the left most single digit too will have a bar. Thus we have

Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar (2^{2} < 7 < 3^{2}). Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend (here 7) Divide and get the remainder (5 in this case)

Step 3: Bring down the number under the next bar (i.e, 84 in this case) to the right of the remainder. So the new divided is 384.

Step 4: Double the divisor and enter it with a blank on its right

Step 5: Gives a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.

In this case 47 × 7 = 329

As 48 × 8 = 384 so we choose

the new digit as 8. Get the remainder

Step 6: Since the remainder is 0 and no digits are left in the given number, therefore