This document discusses squares, square roots, and their properties. It includes: definitions of squares and perfect squares; methods for finding square roots like repeated subtraction, prime factorization, and long division; properties like Pythagorean triplets and patterns in square numbers; and how to find square roots of decimal numbers. Videos are provided explaining different concepts like squares of integers, perfect squares, and methods for finding square roots.

1. SQUARES & SQUARE ROOTS
BY:
ROHIT KUMAR

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

5. Table of squares
 NUMBERS(1 TO 10) MULTIPLICATION SQUARE NUMBER
 1 1*1=12 1
 2 2*2=22 4
 3 3*3=32 9
 4 4*4=42 16
 5 5*5=52 25
 6 6*6=62 36
 7 7*7=72 49
 8 8*8=82 64
 9 9*9=92 81
 10 10*10=102 100

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.