2. CONTENT
SQUARES
PERFECT SQUARES
TABLE OF SQUARES
PROPERTIES OF SQUARES
PROPERTIES OF PERFECT SQUARES
PYTHAGOREAN TRIPLET
SQUARES OF INTEGERS
SQUARE ROOTS
REPEATED SUBTRACTION
PRIME FACTORISATION
LONG-DIVISION METHOD
SQUARE ROOTS OF NUMBERS IN DECIMAL FORM
PATTERN OF SQUARE NUMBER
QUICK NOTES
3. SQUARES
In mathematics square of a number is obtained by
multiplying the number by itself.
The usual notation for the formula for the square of a
number n is not the product n × n, but the
equivalent exponentiation n2
FOR EXAMPLE:
6 2 =6*6=36
On the next slide there is a video clipping by Adhithan
who explains about SQUARES.
4. Perfect squares
A Perfect square is a natural number which is the
square of another natural number .
For Example consider two number 84 and 36. The
factors of 84 are 2*2*3*7
Factors of 36 are 2*2*3*3. The Factor of 84 cannot be
grouped into pairs of identical factors. So, 84 is not a
perfect. But the factor of 36 can be grouped into pairs
of identical factors , like
36 = 2*2 *3*3 =62
6. PROPERTIES OF SQUARES
The number m is a square number if and only if one
can compose a square of m equal (lesser) squares:
m = 12 = 1 =
m = 22 = 4 =
m = 32 = 9 =
m = 42 = 16 =
m = 52 = 25 =
7. PROPERTIES OF PERFECT
SQUARES
A number ending in 2,3,7or 8 is never a perfect square.
A number ending in an odd number of zeros is never a
perfect square.
The square of even number is even.
The square of odd number is odd.
The square of a proper fraction is smaller than the
fraction.
The square of a natural number ‘n’ is equal to the sum
of the first ‘n’ odd numbers .
For example : n is equal to the sum of the first ‘n’ odd
numbers.
8. Pythagorean triplet
Consider the following:-
32+42=9+16=25=52
The collection of numbers 3,4 and 5 are known as
Pythagorean triplet
For any natural number m>1, we have
(2m)2+(m2-1)2 = (m2+1)2
9. SQUARES OF INTEGERS
Squares of negative integers:-
The square of a negative integer is always a positive
integer. For example :- -m*-m=m2
-5*-5= 52 = 25
Squares of positive integers:-
The square of a positive integer is always a positive
integer. For example :- m*m= m2
5* 5= 52 = 25
On the next slide there is a video clipping by
Maharajan who explains about SQUARES OF
INTEGERS
10. Square Roots
In mathematics, a square root of a number x is a
number y such that y2 = x ( symbol - ). For
example :
There are 3 methods to find square roots ,
they are :-
REPEATED SUBTRACTION
( for small squares)
PRIME FACTORIZATION
LONG DIVISION
On the next slide there is a video clipping by Adhithan
who explains about Square Roots
11. Repeated subtraction
Repeated subtraction method e.g.,- √81
Sol.:- 81-1=80
(2) 80-3=77
(3) 77-5=72
(4) 72-7=65
(5) 65-9=56
(6) 56-11=45
(7) 45-13=32
(8)32-15=17
(9) 17-17=0
Result=9
On the next slide there is a video clipping by Tarun Prasad who
explains about Repeated subtraction
12. PRIME FACTORISATION
PRIME FACTORIZATION METHOD In order to find the
square root of a perfect square , resolve it into prime
factors; make pairs of similar factors , and take the product
of prime factors , choosing one out of every pair.
On the next slide there is a video clipping by Tarun Prasad
who explains about PRIME FACTORISATION
13. LONG-DIVISION METHOD
When numbers are very large , the method of finding
their square roots by factorization becomes lengthy
and difficult .So, we use long-division method.
For example :
On the next slide there is a video clipping by Rohit
Kumar who explains about LONG-DIVISION
METHOD
14. SQUARE ROOTS OF NUMBERS
IN DECIMAL FORM
For finding the square root of a decimal fraction ,
make the number of decimal places even by affixing a
zero , if necessary; mark the periods , and find out the
square root, putting the decimal point in the square
root as soon as the integral part is exhausted.
For example :
On the next slide there is a video clipping by Rohit Kumar
who explains about SQUARE ROOTS OF
NUMBERS IN DECIMAL FORM
15. Pattern of square number
Pattern of square number
12 =1
112 =121
1112 =12321
11112=1234321
111112 =123454321
1111112 =12345654321
11111112 =1234567654321
111111112 =123456787654321
1111111112 =12345678987654321
16. QUICK NOTES
If p=m 2 , where m is a natural number, then p is a
perfect square. When the sum of odd numbers is even
it is a perfect square of even number and when the
sum of odd numbers is odd it is a perfect square of odd
numbers. To find a square root of a decimal number
correct up to “n” places , we find the square root up to
(n+1) places and round it off to “n” places.