The document discusses squares and square roots. It defines a square number as a number multiplied by itself. It provides examples of perfect square numbers and their properties, such as being even or odd and having an even number of zeros at the end. The document also discusses finding the square root as the inverse of squaring a number using prime factorization or long division. Pythagorean triplets are introduced as sets of numbers that satisfy the Pythagorean theorem for a right triangle. In conclusion, the key properties of square numbers are that they end in 0, 1, 4, 5, 6, or 9 and have an even number of zeros at the end.

1. SQUARES AND
SQUARE ROOTS
Powerpoint presentation
by RISIKA PANWAR

2. INTRODUCTION TO
SQUARES
Square of a number is a × a = a(square) 1 ×
1 = 1 2× 2 = 4
3 × 3 = 9 5 × 5 =
25 8 × 8 = 64
What is special about these numbers?
The above numbers can are product of a
number multiplied by itself. These are known
as square numbers.
If a number m can be expressed as n*n . Here
m is the SQUARE NUMBER and n is a
NATURAL NUMBER.

3. Important square to
remember
1*1= 1
2*2=4
3*3=9
4*4=16
5*5=25
6*6=36
7*7=49
8*8=64
9*9=81
10*10= 100
THESE NUMBERS ARE ALSO KNOWN AS PERFECT
SQUARES.

4. Properties of square numbers
O The numbers ending with either 0, 1, 4, 5, 6, 9 must be a
perfect square. Though it isn’t always necessary that
numbers with unit digits 0,1,4,5,6,9 are always a square
number.
O The unit’s digit of the square of a natural number is the
unit’s digit of the square of the digit at unit’s place of the
given natural number.
Example :
1) Unit digit of square of 146.
Solution : Unit digit of 6 2 = 36 and the unit digit of 36 is
6, so the unit digit of square of 146 is 6.
C) Squares of even numbers are always even numbers and
square of odd numbers are always odd.
Example : 12 2 = 12 x 12 = 144. (both are even numbers)
19 2 = 19 x 19 = 361 (both are odd numbers)

5. Properties continued…
O The number of zeros at the end of a perfect square is always
even. In other words, a number ending in an odd number of
zeros is never a perfect square.
Example : 2500 is a perfect square as number of zeros are
2(even) and 25000 is not a perfect square as the number of
zeros are 3 (odd).
O PYTHAGOREAN TRIPLETS: This is a topic in maths that
helps us understand the relation between the sides of a
right-angled triangle.
O For any natural number m greater than 1,
(2m, m 2 - 1, m 2 + 1) is a Pythagorean triplet.
O : The square of a number n is equal to the sum of first n odd
natural numbers.
1 2 = 1
2 2 = 1 + 3
3 2 = 1 + 3 + 5
4 2 = 1 + 3 + 5 + 7

6. Square of a number
Finding squares of big numbers like 23 is
not easy we will have to multiply 23 by
itself.
23 = 20 + 3
23 = (20 + 3)2 = 20(20 + 3) + 3(20 + 3) =
202 + 20 × 3 + 3 × 20 + 32 = 400 + 60 + 60
+ 9 = 529
PYTHAGOREAN TRIPLETS:
The collection of numbers 3,
4
and 5 is known as
Pythagorean
triplet.
MEMBERS OF
PYTHAGOREAN TRIPLETS
ARE : m+1 , m-1 and 2m

7. Square Roots
The inverse (opposite) operation of addition is
subtraction and the inverse operation of
multiplication is division. Similarly, finding the
square root is the inverse operation of
squares.
1. PRIME FACTORISATION : Let’s consider
finding square root of 256.
Prime factorisation of 256 is 256 = 2 × 2 × 2 ×
2 × 2 × 2 × 2 × 2
By pairing the prime factors we get, 256 = 2 ×
2 × 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2 × 2 × 2)
Therefore, 256 = 2 × 2 × 2 × 2 = 16

8. Square root by long division
method
By long division method we can find the
square root of not only natural numbers also
of decimals.
Let us try to find root 5607 by long division
method.
We get the remainder 131
. It shows that 742 is less than 5607 by 131
. This means if we subtract the remainder
from the number, we get a perfect square.
Therefore, the required perfect square is
5607 – 131 = 5476. And, 5476 = 74.
Same procedure can be used for decimals as
well.

9. LET’S SUMMARISE
1. If a natural number m can be
expressed as n 2 , 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 unit’s place.
3. Square numbers can only have even
number of zeros at the end.
4. Square root is the inverse operation of
square.

10. Thankyou everyone for paying attention at my presentation. I
hope y0u liked it . I tried and covered each and every topic I
could, still anything left I’M SORRY.