Points to Remember- Squares and Square Roots Class 8 Notes | EduRev

Class 8 Mathematics by Full Circle

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Class 8 : Points to Remember- Squares and Square Roots Class 8 Notes | EduRev

The document Points to Remember- Squares and Square Roots Class 8 Notes | EduRev is a part of the Class 8 Course Class 8 Mathematics by Full Circle.
All you need of Class 8 at this link: Class 8

Facts That Matter

  • A natural number ‘n’ is a perfect square, if m= n for a natural number m.
  • A number ending in 2, 3, 7 or 8 is never a perfect square.
  • The squares of even numbers are even.
  • The squares of odd numbers are odd.
  • A number ending in an odd number of zeros is never a perfect square.
  • There are 2n non-perfect square numbers between the squares of the numbers n and n + 1.
  • For any natural number ‘n’ greater than 1, 2n, (n– 1) and (n+ 1) form a Pythagorean triplet.
  • Finding a square root is the inverse operation of squaring a number.
  • If ‘n’ be the number of digits of a square number then the number of digits in its square root are given by n/2 (for ‘n’ is even) and  n+1/2  (for ‘n’ is odd).

We Know That
If a whole number is multiplied by itself, the product is called the square of that number. For example,
3 * 3 = 9 = 32
i.e. the square of 3 is 9.
5 * 5 = 25 = 52
i.e. the square of 5 is 25.
We also know that: A natural number is called a perfect square or a square number, if it is the square of some natural number. For example,
16 is the square of 4, therefore, 16 is a perfect square.
289 is the square of 17, therefore, 289 is a perfect square.

Remember
All natural numbers are not perfect-squares or square numbers e.g. 32 is not a square-number In general, if a natural number ‘m’ can be expressed as n2, where n is also a natural number then ‘m’ is perfect-square. The numbers like 1, 4, 9, 16, 25, 36, … are called square numbers. Square numbers between 1 and 100 are:

Number

Square

Number

Squares

1

1

6

36

2

4

7

49

3

9

8

64

4

16

9

81

5

25

10

100

 

Properties of A Square Number
Let us considers square of all natural numbers from 1 to 20. 

Number

Square

Number

Square

1

1

11

121

2

4

12

144

3

9

13

169

4

16

14

196

5

25

15

225

6

36

16

256

7

49

17

289

8

64

18

324

9

81

19

361

10

100

20

400

From the table we conclude that:

Property 1: “The ending digits (the digits in the one’s place) of a square number is 0, 1, 4, 5, 6 or 9 only.” 

Some Interesting Patterns
1. Triangular numbers are: 1, 3, 6, 10, 15, 21, etc. If we combine two consecutive triangular numbers, we get a square number.
i.e 1 + 3 = 4, ‘4’ is a square number.
3 + 6 = 9, ‘9’ is a square number
6 + 10 = 16, ‘16’ is a square number
and so on.

2.
1=1
112 = 121
1112 = 12321
11112 = 1234321

3. We have:
7= 49
672 = 4489
6672 = 444889
66672 = 44448889 and so on.

Solved Examples:
Problem: What will be the unit’s digit in the square of the following numbers?
1. 12487
2. 1324
3. 91478
4. 1251
Solution: 
The unit’s digit in the square of the following is:
1. 12487 is 9 (as 72 = 49. 9 in the unit’s place).
2. 1324 is 6 (as 42 = 16. 6 in the unit’s place).
3. 91478 is 4 (as 82 = 64. 4 in the unit’s place).
4. 1251 is 1 (as 12 = 1. 1 in the unit’s place).

Problem: Comment on the square of an even number and of an odd number?
Solution: 
The square of an even number is always an even number and the square of an odd number is always an odd number. The square of an even number will always have 4, 6, or even number of zeros in its unit’s place. And the square of an odd number will always have 1, 5 or 9 in its unit’s place.

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