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
(i)Adding triangular numbers-
(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