Download 1. Squares Description Reflect and Review Teasers

1. Squares
Description

Square Numbers or Perfect
Squares:
Reflect and Review

121 = 11 × 11, so it is
a perfect square.

To verify the
properties for
squares ending with
different digits, check
the examples below:
The numbers, which can be
expressed as the product of two
identical numbers, are known as
square numbers or perfect
squares.

-

-
Squares of numbers ending
with different digits:
If a number ends with 1 or 9, its
square ends with the digit 1.
If a number ends with 2 or 8, its
square ends with the digit 4.
If a number ends with 3 or 7, its
square ends with the digit 9.
If a number ends with 4 or 6, its
square ends with the digit 6.
If a number ends with 5, its
square also ends with the digit 5.
If a number ends with 0, its
square also ends with the digit 0.
Even and Odd Square
Numbers:
The square of an even number is
always an even number.
The square of an odd number is
always an odd number.
The square of any number can
never end with odd number of
zeroes.
Teasers
1) Find the
square of the
following
number:
122, 37, 781,
304.
2) What will be
the ones digit
in the square
of the
following
numbers?
29, 42, 313,
44, 800.
Answers
1) 14884,
1369,
609961,
92416
2) 1,
4,
9,
6,
0
212 = 441
382 = 1444
172 = 289
562 = 3136
452 = 2025
303 = 900

The square of 24
(which is an even
number):
242 = 576 (an even
number)
The square of 39
(which is an odd
number):
392 = 1521 (an odd
number)
The squares of 20
and 300 are 400
1
(ending with 2
zeroes)and 90000
(ending with 4
zeroes).
2. Interesting Patterns Involving Square Numbers
Description
Reflect and Review

Finding Natural Numbers
between two Consecutive
Square Numbers:

To find the number of natural
numbers between 32 and 42:
We have, 42 = (3 + 1)2.
Between two consecutive
square numbers n2 and (n+1)2,
we have 2n non-square
numbers.

Square of an Odd Number as
the Sum of Two Consecutive
Positive Integers:
So, number of natural
numbers between 32 and 42
= 2 × 3 = 6.

Teasers
1) Find how
many nonsquares are
there
between
1052 and
1062.
Answers
1) 210
To represent 112 as the sum
of two consecutive positive
integers:
If n is any odd number, then
= 60 + 61 ( = 121).
Here, 60 and 61 are two
consecutive positive integers.

Square of a Number as the Sum 
of Odd Numbers:
To express 62 as the sum of
first 6 odd natural numbers:
The sum of the first ‘n’ odd
natural numbers = n2.
62 = 1 + 3 + 5 + 7 + 9 + 11.
3. Finding the Square of a Number Without Actual Multiplication
Description
Reflect and Review
Teasers
The square of any number can be
easily calculated using its nearest 10
multiple values.
4. Pythagorean Triplets
2
2
To calculate 52 , without actual 1) Find the square
multiplication:
of 54, without
actual
522
multiplication.
2
= (50 + 2)
= (50 + 2) (50 + 2)
= (50)2 + (2 × 50 × 2) + 22
= 2500 + 200 + 4
= 2704
Answers
2916
Description

Reflect and Review
If the sum of the squares of
two numbers is equal to the
square of a third number, then
the three numbers form a
Pythagorean triplet.

For any natural number m > 1:
2m, m2 – 1 and m2 + 1 form a
Pythagorean triplet.
2
2
6 +8
=6×6+8×8
= 36 + 64
= 100 = 102

The Square Root
of a number is a
value that, when
multiplied by
itself, gives the
number, i.e., the
square root of
any number is
one of its equal
factors.
The symbol for
square root is √.
Answers
2) Find a
2) 22, 120
Pythagorean
and 122
triplet whose
smallest
member is 22.
So, 6, 8 and 10 form a
Pythagorean triplet.
5. Square Roots of a Number
Description
Reflect and Review

Teasers
To find the square root of
36:
We know, 6 × 6 = 36.
So, 6 is the square root of
36, i.e., √
Teasers
Find the least number by
which 55296 should be
divided so as to get a
perfect square. Also, find
the square root of the
resulting number.
Answers
6; 96
6. Finding the square root of a number by Repeated Subtraction method
Description

The square root of a
number can be obtained
by repeated subtraction
of odd numbers 1, 3, 5,
7, 9, 11… till we get 0.

The number of times
subtraction is done to
get zero gives the square
root.

This method is not
suitable for finding the
square root of large
numbers as it is very
tedious and tiresome.
Reflect and Review
To find the square root of
144 by repeated subtraction:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
144 – 1 = 143
143 – 3 = 140
143 – 5 = 135
135 – 7 = 128
128 – 9 = 119
119 – 11 = 108
108 – 13 = 95
95 – 15 = 80
95 – 17 = 63
63 – 19 = 44
44 – 21 = 23
23 – 23 = 0
Teasers
Find the square
root of 529 by
repeated
subtraction.
Answers
23
3
So, √
7. Finding the square root of a number by prime factorisation method
Description
Reflect and Review
Teasers

Square root of a number
can be obtained by prime
factorisation.
To obtain the square root
of 1225 by prime
factorisation:
Find the square
root of 3136 by
prime factorisation.

First, we find the prime
factors of the square
number. Then by pairing
the prime factors, we get
the square root.
We find the prime factors
of 1225 as follows,
5
5
7
7
Answers
56
1225
245
49
7
1
1225 = 5 × 5 × 7 × 7
= 52 × 72
Now, by pairing the prime
factors, we obtain the
square root,
√
√
8. Finding the square root of a number by division method
Description
Reflect and Review
Teasers


4
Square root
of numbers
can be
calculated by
division
method
irrespective
of the
numbers
being perfect
squares or
non-perfect
squares.
This method
is usually

To find the square root of 676 by
division method:
Step 1: Place a bar over every pair
of digits starting from the right
hand side. If there is odd number
of digits, the extreme left digit is
without a bar.
So, we have ̅̅̅̅.
Step 2: The number in the
extreme left is 6. We have to find
the greatest number whose
square is less than or equal to 6.
We take this number both as the
1) Find the square
root of 11881
using division
method
2) What is the least
number that
should be added
to 6082 in order
to obtain a
perfect square?
Also, find the
square root of
the resulting
number.
Answers
1) 109
2) 2; 78
adopted
when the
numbers are
very large.
divisor and the quotient.
We have
and 32.
} 6 lying between 22
So, 2 is the quotient and 2 is also
the divisor. 2 multiplied by 2 gives
4.
Write 4 below 6 and subtract. The
remainder is 2.
2
̅̅̅̅
2
4
2
Step 3: Bring down 76 to the right
of 2. The new dividend is 276.
2
̅̅̅̅
2
-4
̅̅̅̅
4_
Step 4: For the new divisor, double
the quotient and write it leaving a
blank space next to it.
Step 5: The new divisor is 2
followed by a digit. This digit will
also be the new quotient such that
the new quotient multiplied by the
new divisor will be less than or
equal to 276.
Step 6: Clearly that digit is 6
because 46 × 6 = 276.
Write 276 below 276 and subtract.
The remainder is 0.
26
̅̅̅̅
2
-4
46
276
-276
0
√
5
9. Square Root of Decimal Numbers
Description

Square root
of decimal
numbers can
also be
calculated by
division
method.
Reflect and Review

To find the square root of 65.61
by division method:
Step 1: A decimal number has an
integral part and a decimal part.
(For the integral part, place a bar
over every pair of digits starting
from the right hand side.)
Step 2: For the decimal part,
place a bar over every pair of
digits starting from the first
decimal place. If the number of
digits in the decimal part is not
even, we add a zero to the
extreme right and then pair up
the digits in the decimal part.
Step 3: So, we have̅̅̅̅ ̅̅̅̅. The
pair in the extreme left is 65. Find
the greatest number whose
square is less than or equal to 65.
Take the number as the divisor
and the quotient.
We have
} 64 lying
2
between 8 and 92.
So, 8 is the quotient and 8 is also
the divisor. 8 multiplied by 8
gives 64.
We write 64 below 65 and
subtract. The remainder is 1.
8
8 ̅̅̅̅ ̅̅̅̅
-16
1
Step 4: Since the next pair of digit
is of the decimal part, place
decimal after 8 in the quotient
and bring down the pair 61 next
6
Teasers
Find the square root
of 207.36 using
division method
Answers
14.4
to 1.
The new dividend is 161.
8
̅̅̅̅ ̅̅̅̅
8
-16
161
Step 4: For the new divisor,
double the quotient and write it
leaving a blank space next to it.
8.
̅̅̅̅ ̅̅̅̅
8
-64
16_
161
Step 5: The new divisor is 16
followed by a digit. This digit will
also be the new quotient such
that the new quotient multiplied
by the new divisor will be less
than or equal to 161.
Step 6: Clearly that digit is 1
because 161 × 1 = 161.
Write 161 below 161 and
subtract. The remainder is 0.
8.1
̅̅̅̅ ̅̅̅̅
8
-64
161
161
-161
0
√
10. Estimating the Square Root
Description
Reflect and Review
To find a
number whose
square is close
to a given
number.
To find the estimate value of square root
of 240:
240 lies between 225 and 256,
i.e.,
√
Also, 240 is closer to 225 than 256.
is closer to √
than √
.
√
Hence, √
Teasers
Estimate the value
of the following
square roots to the
nearest whole
number:
1) √
2) √
Answers
1) 36
2) 30
is approximately equal to 15.
7
11.Square Root of a Non-Perfect Square Number
Description
Reflect and Review
Square root of
non-perfect
squares can be
calculated in the
same way as that
of perfect squares,
using division
method.
To find the square root of 5, up to two decimal places:
We follow the same method as we did in the procedure for
finding the square root by division method.
Since, we are asked to find the value of √ up to two
decimal places, we will calculate the value up to three
decimal places and then take approximation.
2.236
2 ̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅
-4
42 100
-84
443
1600
-1329
4466
27100
-26796
304
√
8
(up to 2 decimal places)
Teasers
Answers
Find the
1) 2.92
square root of 2) 6.71
the following
up to 2 places
of decimal.
1) 8.526
2) 45