for class 8th the basics of squares n square roots

1. KENDRIYA VIDYALAYA
IFFCO AONLA

2. MATHS
PROJECT
WORK

3. SQUARES
&
SQUARE ROOTS

4. INTRODUCTION
We know that the area of a square =side*side(where side means length of the side)
Side of a square Area of square
1 1×1=1=1²
2 2×2=4=2²
5 5×5=25=5²
8 8×8=64=8²
a a×a=a²

5. What is special about the numbers4,9,25,64 and other such
numbers?
Since, 4 can be expressed as 2×2 = 2², 9can be expressed as
3×3=3², all such numbers can be expressed as product of the
number with itself.
Such numbers like1,4,9,16,25,… are known as square
numbers.
In general, if a natural number m can be expressed as n²,
where n is also a natural number, then m s a square number.
is 32 a square number.
We know that 5²=25and 6²=36.If 32is a square number, it
must be the square of a natural number between 5&6.but
there is no natural number between 5&6.therefore 32 is not a
square number.

6. Consider following numbers and their squares.
From the above table, can we enlist the square numbers between
1&100 ?Are there any natural numbers upto 100 left out?
We will find that rest of the numbers are not square numbers.
The numbers 1,4,9,16…are square numbers. So these are called
perfect squares.
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100

7. FOLLOWING TABLE SHOWS THE SQUARES OF NUMBERS FROM 1 TO 20.
PROPERTIES OF SQUARE NUMBERS
NUMBER SQUARE NUMBER SQUARE
1
2
3
4
5
6
7
8
9
10
1
4
9
16
25
36
49
64
81
100
11
12
13
14
15
16
17
18
19
20
121
144
169
196
225
256
289
324
361
400

8. Study the square numbers in table in the
last slide .What are ending digits (that is,
digits in the one᾿s place)of the square
numbers? All these numbers end
with0,1,4,5,6 or9 at unit ᾿s place. None of
these end with 2,3,7 or8 at unit ᾿s place.
Can we say that if a number ends in
0,1,4,5,6 or 9,then it must be a square
number? yes of course

9. • Study the following table of some numbers and their
squares and observe the one᾿s place in both.
NUMBER SQUARE NUMBER SQUARE
1
2
3
4
5
6
7
8
9
10
1
4
9
16
25
36
49
64
81
100
11
12
13
14
15
16
17
18
19
20
121
144
169
196
225
256
289
324
361
400
NUMBE
R
SQUARE
21
22
23
24
25
30
35
40
45
50
441
484
529
576
625
900
1225
1600
2025
2500

10. The following square numbers end with digit 1.
If a number contains one zero at the end, its square
have two zeros and if a number contains two zero at
the end, its square have four zeros. For egg:-
10²=100 100²=10000
20²=400 200²=40000
80²=6400 900²=810000
SQUARE NUMBER
1
81
121
361
441
1
9
11
19
21

11. SOME MORE INTERESTING PATTERNS
ADDING TRIANGULAR NUMBERS:-If we combine two consecutive
triangular nos. ,we get a square no.. Like:- 1+3=4=2²,
3+6=9=3²
NUMBERS BETWEEN SQUARE NUMBERS:-Let us now see if we can
find some interesting pattern between two consecutive
square numbers.
Between 1²& 2² there are two non-square numbers2,3
If we think of any natural no. n &(n+1),then,
(n+1)²-n²=(n²+2n+1)-n²=2n+1.
We can say that there are 2n non-perfect square numbers
between the squares of the number n&(n+1)

12. Adding odd numbers:-Consider the following:-
1+3+5(Sum of first 3 odd numbers)=9=3²
So, we can say that sum of first n odd natural no. is
n².
NOTE:-If a natural no. can’t be expressed as a sum of
successive odd natural nos. starting with 1,then it is
not a perfect square.

13. A sum of consecutive natural number:-Consider
the following-
3²=9=4=5
5²=25=12+13
Product of two consecutive even/odd nos.:-
11×13=143=12²-1
11×13=(12-1)×(12+1)=12²-1
So in general we can say that(a+1) ×(a-
1)=a²-1

14. Finding the square of a number:-We can find
the square of a number quickly by
knowing the method shown below:-
To find the square of 23:
23=20+3
23²=(20+3)²=20(20+3)+3(20+3)
=20²+20×3+3×20+3²
=400+60+60+9=529

15. PYTHAGOREAN TRIPLETS
Consider the following
3²+4²=9+16=25=5²
The collection of nos. 3,4,5 is known as Pythagorean
triplet.
FORMULA:-
(2m)² + (m²-1) ²=(m²+1) ²

16. SQUARE ROOTS
Square root is the inverse operation of squaring
and its symbol is √.
FOR EG:-
1²=1,therefore square root of 1 is 1.

17. Statement Inference Statement Inference
1²=1
2²=4
3²=9
4²=16
5²=25
√2
√98
√98/√2
√14
√25=5
6²=36
7²=49
8²=64
9²=81
10²=100
√36=6
√49=7
√64=8
√81=9
√100=10

18. Finding square root through prime factorisation:-Consider the
prime factorisation of the following numbers and their
squares .
Each prime factor in prime factorisation of the square of a
number,occurs twice the number of times it occurs in the
prime factorisation of the number itself.
Prime factorisation of a
number
Prime factorisation of its
squares
6=2×3
8=2×2×2
12=2×2×3
36=2×2×3×3
64=2×2×2×2×2×2
144=2×2×2×2×3×3

19. Findingsquarerootby divisionmethod:- When
the numbers are large, even the
method of finding square root by
prime factorisation becomes lengthy
and difficult. To overcome this problem
we use Long Division Method.

20. Square Roots of Decimals
Consider √17.64
Step 1:-To find the square root of a decimal number we put
bars on the integral part of the number in the usual
manner. And place bars on the decimal part on every pair of
digits beginning with the first decimal place proceed as
usual.We get 17.64
Step 2:-Now proceed in a similar manner.The left most bar
is on 17&4<17<5.Take this number as the divisor and the
number under the left-most bar as the dividend.Divide and
get the remainder.
Step 3:-the remainder is 1.Write the number under the
next bar on the right of this remainder, to get 164.

21. Step 4:-Double the divisor and enter it
with a blank on its right. Since is the
decimal part so put a decimal point in
the quotient.
Step5:-We know 82×2=164,therefore
the new digit is 2. Divide and get the
remainder.
Step6:-Since the remainder is 0 and no
bar left, therefore √17.64=4.2

22. EXAMPLES
Example 1:- Find a Pythagorean triplet in which one number
is 12.
Solution:- If we take m²-1=12
Then, m²=12+1=13
Then the value of m will not be an integer.
So, we try to take m² +1=12.again m²=11 will not give an
integer value for m.
So, let us take 2m=12
Then m=6
Thus, m²-1=36-1=35 and m²+1=36+1=37
Therefore, the required triplet is 12,35,37.

23. Example 2 :- Is 90 a perfect square?
Solution:- We have 90=2×3×3×5
The prime factors 2&5 do not occur in
pairs.
Therefore,90is not a perfect square. That
90 is not a perfect square can also be seen
from the fact that it has only one zero.

24. MAIN POINTS
1. If a natural number m can be expressed as n²,
where n is also a natural number, then m is a
square number.
2. All square numbers end with 0, 1,4,5,6 or 9 at units
place.
3. Square numbers can only have even number of
zeros at the end.
4. Square root is the inverse proportion of square.

25. SUBMITTED TO:-
SHRI P.K. SHRIVASTAVA SIR

26. THE END
THANKS