# Squares and Square Roots

Notes of chapter: Squares and Square Roots are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 8.

(1) Square Number-

In simple words, multiplication of a number by itself is known as square number of that number. If a natural number m can be expressed as n2, where n is a natural number, then m is a square number.

m = n2

Where, m and n are natural numbers.

Eg:- Represent 36 and 34 as a square number.

(i)36 is a natural number.

It can be represent as 6 × 6 or 62.

36 = 62

Hence 36 is a square of 6.

(ii)

34 is a natural number.

But, it can not be represent as n2. [Because fraction of 32 = 2, 2, 2, 2, 2]

Hence 34 is not a square number.

(2)Perfect squares-

Squares numbers are also called perfect squares.

(3) Properties of square numbers-

(i) If a number has 0, 1, 4, 5, 6, or 9 at ones place, then it must be a square number.

Eg:-

16, 81, 64, 169, 100 are square numbers.

(ii) If a number has 1 or 9 at the unit place, then its square ends in 1.

Eg:-

 Number Square 1 1 9 81 11 121 19 361 21 441

(iii) If a number has 4 or 6 at its unit place, then its square ends in 6.

Eg:-

 Number Square 4 16 6 36 14 196 16 256

(iv) Square numbers can only have even numbers of zeros at the end of the number.

Eg:-

 Number Square 10 100 20 400 100 10000 20 40000 1000 1000000 2000 2000000

(v) Even numbers have even square numbers.

Eg:-

 Even Number Square of Even Number 2 4 6 36 14 144 10 100 22 484 26 676

(vi) Odd numbers have odd square number.

Eg:-

 Odd Numbers Square of Odd Numbers 3 9 5 25 9 81 11 121 25 625 27 729

(4) Some interesting patterns

(a)Triangular numbers are those numbers whose dot patterns can be arranged as triangles.

Eg:- (b)If we combine two consecutive triangular numbers, we get a square number.

Eg:-    (ii)Numbers between square numbers

Method 1-

There are 2n non perfect square numbers between the squares of the numbers n and (n+1).

Eg:- Find how many non square numbers are there between the square of numbers 1 and 2.

Ans-

We know that there are 2n non perfect square numbers between the squares of the numbers n and (n+1).

n = 1

Non perfect square numbers between the squares of the numbers n and (n+1)

= 2n

Non perfect square numbers between the squares of the numbers 1 and 2

= 2× 1=2

Hence,non perfect square numbers between square of 1 and 2 will be 2.

Method 2-

Non squares numbers between two square numbers is one less than the difference between squares of those two consecutive numbers.

Eg:- Find how many non square numbers are there between the square of numbers 1 and 2.

Ans-

Square of 1 = 12 = 1

Square of 2 = 22 = 4

Non square numbers between 1 and 4 are 2, 3.

Difference between squares of two consecutive numbers = 4 – 1 = 3

Non squares numbers between two square numbers is one less than the difference between squares of those two consecutive numbers.  (iv)Subtracting odd numbers

If a natural number can be expressed as a subtraction of successive odd natural numbers starting with 1, then it is a perfect square.

Eg:-

(a)Check whether 25 is a perfect square.

Ans-

Subtraction of the successive odd numbers starting from 1

(a) 25 – 1 = 24

(b) 24 – 3 = 21

(c) 21 – 5 = 16

(d) 16 – 7 = 9

(e) 9 – 9 = 0

1 + 3 + 5 + 7 + 9 = 25, a perfect square.

(b)Check whether 27 is a perfect square.

Ans-

Eg:-

Subtraction of the successive odd numbers starting from 1

(a) 27 – 1 = 26

(b) 26 – 3 = 23

(c) 23 – 5 = 18

(d) 18 – 7 = 11

(e) 11 – 9 = 2

(f) 2 – 11 = – 9

Hence, 27 is not a perfect square.       (9) Finding square root

(i) Finding square root through repeated subtraction

Sum of first n numbers = n2

or

If the number is a square number, it has to be the sum of successive odd numbers starting from 1.

Eg:-Find √81

(a) 81 – 1 = 80

(b) 80 – 3 = 77

(c) 77 – 5 = 72

(d) 72 – 7 = 65

(e) 65 – 9 = 56

(f) 56 – 11 = 45

(g) 45 – 13 = 32

(h) 32 – 15 = 17

(i) 17 – 17 = 0

Total steps to subtract successive odd numbers = 9

√81 = 9      Step 4-

Double the divisor and enter it with a blank on its right.                             Helping Topics

NCERT solutions

Practice sheet