NCERT Textbook- Squares and Square Roots | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


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5.1  Introduction
You know that the area of a square = side × side (where ‘side’  means ‘the length of
a side’). Study the following table.
Side of a square (in cm) Area of the square (in cm
2
)
1 1 × 1 = 1 = 1
2
2 2 × 2 = 4 = 2
2
3 3 × 3 = 9 = 3
2
5 5 × 5 = 25 = 5
2
8 8 × 8 = 64 = 8
2
a a × a = a
2
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2
2
, 9 can be expressed as 3 × 3 = 3
2
, all such
numbers can be expressed as the product of the number with itself.
Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers.
In general, if a natural number m can be expressed as n
2
, where n is also a natural
number, then m is a square number. Is 32 a square number?
We know that 5
2
 = 25 and 6
2
 = 36. If 32 is a square number , it must be the square of
a natural number  between 5 and 6. But there is no natural number between 5 and 6.
Therefore 32 is not a square number .
Consider the following numbers and their squares.
Number Square
1 1 × 1 = 1
2 2 × 2 = 4
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?????
? ? ? ? ? ? ?
?
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Page 2


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5.1  Introduction
You know that the area of a square = side × side (where ‘side’  means ‘the length of
a side’). Study the following table.
Side of a square (in cm) Area of the square (in cm
2
)
1 1 × 1 = 1 = 1
2
2 2 × 2 = 4 = 2
2
3 3 × 3 = 9 = 3
2
5 5 × 5 = 25 = 5
2
8 8 × 8 = 64 = 8
2
a a × a = a
2
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2
2
, 9 can be expressed as 3 × 3 = 3
2
, all such
numbers can be expressed as the product of the number with itself.
Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers.
In general, if a natural number m can be expressed as n
2
, where n is also a natural
number, then m is a square number. Is 32 a square number?
We know that 5
2
 = 25 and 6
2
 = 36. If 32 is a square number , it must be the square of
a natural number  between 5 and 6. But there is no natural number between 5 and 6.
Therefore 32 is not a square number .
Consider the following numbers and their squares.
Number Square
1 1 × 1 = 1
2 2 × 2 = 4
???????? ???? ??????
?????
? ? ? ? ? ? ?
?
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
3 3 × 3 = 9
4 4 × 4 = 16
5 5 × 5 = 25
6 -----------
7 -----------
8 -----------
9 -----------
10 -----------
From the above table, can we enlist the square numbers between 1 and 100? Are
there any natural square numbers upto 100 left out?
You will find that the rest of the numbers are not square numbers.
The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect
squares.
1. Find the perfect square numbers between    (i)  30 and 40    (ii)  50 and 60
5.2  Properties of Square Numbers
Following table shows the squares of numbers from 1 to 20.
Number Square Number Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400
Study the square numbers in the above table. What are the ending digits (that is, digits in
the units place) of the square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 at
units place. None of these end with 2, 3, 7 or 8 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? Think about it.
1. Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
(v) 1069 (vi) 2061
TRY THESE
Can you
complete it?
Reprint 2024-25
Page 3


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5.1  Introduction
You know that the area of a square = side × side (where ‘side’  means ‘the length of
a side’). Study the following table.
Side of a square (in cm) Area of the square (in cm
2
)
1 1 × 1 = 1 = 1
2
2 2 × 2 = 4 = 2
2
3 3 × 3 = 9 = 3
2
5 5 × 5 = 25 = 5
2
8 8 × 8 = 64 = 8
2
a a × a = a
2
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2
2
, 9 can be expressed as 3 × 3 = 3
2
, all such
numbers can be expressed as the product of the number with itself.
Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers.
In general, if a natural number m can be expressed as n
2
, where n is also a natural
number, then m is a square number. Is 32 a square number?
We know that 5
2
 = 25 and 6
2
 = 36. If 32 is a square number , it must be the square of
a natural number  between 5 and 6. But there is no natural number between 5 and 6.
Therefore 32 is not a square number .
Consider the following numbers and their squares.
Number Square
1 1 × 1 = 1
2 2 × 2 = 4
???????? ???? ??????
?????
? ? ? ? ? ? ?
?
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
3 3 × 3 = 9
4 4 × 4 = 16
5 5 × 5 = 25
6 -----------
7 -----------
8 -----------
9 -----------
10 -----------
From the above table, can we enlist the square numbers between 1 and 100? Are
there any natural square numbers upto 100 left out?
You will find that the rest of the numbers are not square numbers.
The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect
squares.
1. Find the perfect square numbers between    (i)  30 and 40    (ii)  50 and 60
5.2  Properties of Square Numbers
Following table shows the squares of numbers from 1 to 20.
Number Square Number Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400
Study the square numbers in the above table. What are the ending digits (that is, digits in
the units place) of the square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 at
units place. None of these end with 2, 3, 7 or 8 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? Think about it.
1. Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
(v) 1069 (vi) 2061
TRY THESE
Can you
complete it?
Reprint 2024-25
? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
Write five numbers which you can decide by looking at their units digit that they are
not square numbers.
2. Write five numbers which you cannot decide just by looking at their units digit
(or units place) whether they are square numbers or not.
• Study the following table of some numbers and their squares and observe the one’ s
place in both.
Table 1
Number Square Number Square Number Square
1 1 11 121 21 441
2 4 12 144 22 484
3 9 13 169 23 529
4 16 14 196 24 576
5 25 15 225 25 625
6 36 16 256 30 900
7 49 17 289 35 1225
8 64 18 324 40 1600
9 81 19 361 45 2025
10 100 20 400 50 2500
The following square numbers end with digit 1.
Square Number
1 1
81 9
121 11
361 19
441 21
Write the next two square numbers which end in 1 and their corresponding numbers.
You will see that if a number has 1 or 9 in the units place, then it’ s square ends in 1.
• Let us consider square numbers ending in 6.
Square Number
16 4
36 6
196 14
256 16
TRY THESE
Which of 123
2
, 77
2
, 82
2
,
161
2
, 109
2
 would end with
digit 1?
TRY THESE
Which of the following numbers would have digit
6 at unit place.
(i) 19
2
(ii) 24
2
(iii) 26
2
(iv) 36
2
(v) 34
2
Reprint 2024-25
Page 4


? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
5.1  Introduction
You know that the area of a square = side × side (where ‘side’  means ‘the length of
a side’). Study the following table.
Side of a square (in cm) Area of the square (in cm
2
)
1 1 × 1 = 1 = 1
2
2 2 × 2 = 4 = 2
2
3 3 × 3 = 9 = 3
2
5 5 × 5 = 25 = 5
2
8 8 × 8 = 64 = 8
2
a a × a = a
2
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2
2
, 9 can be expressed as 3 × 3 = 3
2
, all such
numbers can be expressed as the product of the number with itself.
Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers.
In general, if a natural number m can be expressed as n
2
, where n is also a natural
number, then m is a square number. Is 32 a square number?
We know that 5
2
 = 25 and 6
2
 = 36. If 32 is a square number , it must be the square of
a natural number  between 5 and 6. But there is no natural number between 5 and 6.
Therefore 32 is not a square number .
Consider the following numbers and their squares.
Number Square
1 1 × 1 = 1
2 2 × 2 = 4
???????? ???? ??????
?????
? ? ? ? ? ? ?
?
Reprint 2024-25
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TRY THESE
3 3 × 3 = 9
4 4 × 4 = 16
5 5 × 5 = 25
6 -----------
7 -----------
8 -----------
9 -----------
10 -----------
From the above table, can we enlist the square numbers between 1 and 100? Are
there any natural square numbers upto 100 left out?
You will find that the rest of the numbers are not square numbers.
The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect
squares.
1. Find the perfect square numbers between    (i)  30 and 40    (ii)  50 and 60
5.2  Properties of Square Numbers
Following table shows the squares of numbers from 1 to 20.
Number Square Number Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400
Study the square numbers in the above table. What are the ending digits (that is, digits in
the units place) of the square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 at
units place. None of these end with 2, 3, 7 or 8 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? Think about it.
1. Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
(v) 1069 (vi) 2061
TRY THESE
Can you
complete it?
Reprint 2024-25
? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
Write five numbers which you can decide by looking at their units digit that they are
not square numbers.
2. Write five numbers which you cannot decide just by looking at their units digit
(or units place) whether they are square numbers or not.
• Study the following table of some numbers and their squares and observe the one’ s
place in both.
Table 1
Number Square Number Square Number Square
1 1 11 121 21 441
2 4 12 144 22 484
3 9 13 169 23 529
4 16 14 196 24 576
5 25 15 225 25 625
6 36 16 256 30 900
7 49 17 289 35 1225
8 64 18 324 40 1600
9 81 19 361 45 2025
10 100 20 400 50 2500
The following square numbers end with digit 1.
Square Number
1 1
81 9
121 11
361 19
441 21
Write the next two square numbers which end in 1 and their corresponding numbers.
You will see that if a number has 1 or 9 in the units place, then it’ s square ends in 1.
• Let us consider square numbers ending in 6.
Square Number
16 4
36 6
196 14
256 16
TRY THESE
Which of 123
2
, 77
2
, 82
2
,
161
2
, 109
2
 would end with
digit 1?
TRY THESE
Which of the following numbers would have digit
6 at unit place.
(i) 19
2
(ii) 24
2
(iii) 26
2
(iv) 36
2
(v) 34
2
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
TRY THESE
We can see that when a square number ends in 6, the number whose square it is, will
have either 4 or 6 in unit’s place.
Can you find more such rules by observing the numbers and their squares (Table 1)?
What will be the “one’ s digit” in the square of the following numbers?
(i) 1234 (ii) 26387 (iii) 52698 (iv) 99880
(v) 21222 (vi) 9106
• Consider the following numbers and their squares.
10
2 
= 100
20
2
 = 400
80
2
 = 6400
100
2 
= 10000
200
2
 = 40000
700
2
 = 490000
900
2
 = 810000
If a number contains 3 zeros at the end, how many zeros will its square have ?
What do you notice about the number of zeros at the end of  the number and the
number of zeros at the end of its square?
Can we say that square numbers can only have even number of zeros at the end?
• See Table 1 with numbers and their squares.
What can you say about the squares of even numbers and squares of odd numbers?
1. The square of which of the following numbers would be an odd number/an even
number? Why?
(i) 727 (ii) 158 (iii) 269 (iv) 1980
2. What will be the number of zeros in the square of the following numbers?
(i) 60 (ii) 400
5.3  Some More Interesting Patterns
1. Adding triangular numbers.
Do you remember triangular numbers (numbers whose dot patterns can be arranged
as triangles)?
*
* * *
* ** * **
* ** *** * ***
* ** *** **** * ****
136 10 15
But we have
four zeros
But we have
two zeros
We have
one zero
We have
two zeros
Reprint 2024-25
Page 5


? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
5.1  Introduction
You know that the area of a square = side × side (where ‘side’  means ‘the length of
a side’). Study the following table.
Side of a square (in cm) Area of the square (in cm
2
)
1 1 × 1 = 1 = 1
2
2 2 × 2 = 4 = 2
2
3 3 × 3 = 9 = 3
2
5 5 × 5 = 25 = 5
2
8 8 × 8 = 64 = 8
2
a a × a = a
2
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2
2
, 9 can be expressed as 3 × 3 = 3
2
, all such
numbers can be expressed as the product of the number with itself.
Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers.
In general, if a natural number m can be expressed as n
2
, where n is also a natural
number, then m is a square number. Is 32 a square number?
We know that 5
2
 = 25 and 6
2
 = 36. If 32 is a square number , it must be the square of
a natural number  between 5 and 6. But there is no natural number between 5 and 6.
Therefore 32 is not a square number .
Consider the following numbers and their squares.
Number Square
1 1 × 1 = 1
2 2 × 2 = 4
???????? ???? ??????
?????
? ? ? ? ? ? ?
?
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
3 3 × 3 = 9
4 4 × 4 = 16
5 5 × 5 = 25
6 -----------
7 -----------
8 -----------
9 -----------
10 -----------
From the above table, can we enlist the square numbers between 1 and 100? Are
there any natural square numbers upto 100 left out?
You will find that the rest of the numbers are not square numbers.
The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect
squares.
1. Find the perfect square numbers between    (i)  30 and 40    (ii)  50 and 60
5.2  Properties of Square Numbers
Following table shows the squares of numbers from 1 to 20.
Number Square Number Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400
Study the square numbers in the above table. What are the ending digits (that is, digits in
the units place) of the square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 at
units place. None of these end with 2, 3, 7 or 8 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? Think about it.
1. Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057 (ii) 23453 (iii) 7928 (iv) 222222
(v) 1069 (vi) 2061
TRY THESE
Can you
complete it?
Reprint 2024-25
? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
Write five numbers which you can decide by looking at their units digit that they are
not square numbers.
2. Write five numbers which you cannot decide just by looking at their units digit
(or units place) whether they are square numbers or not.
• Study the following table of some numbers and their squares and observe the one’ s
place in both.
Table 1
Number Square Number Square Number Square
1 1 11 121 21 441
2 4 12 144 22 484
3 9 13 169 23 529
4 16 14 196 24 576
5 25 15 225 25 625
6 36 16 256 30 900
7 49 17 289 35 1225
8 64 18 324 40 1600
9 81 19 361 45 2025
10 100 20 400 50 2500
The following square numbers end with digit 1.
Square Number
1 1
81 9
121 11
361 19
441 21
Write the next two square numbers which end in 1 and their corresponding numbers.
You will see that if a number has 1 or 9 in the units place, then it’ s square ends in 1.
• Let us consider square numbers ending in 6.
Square Number
16 4
36 6
196 14
256 16
TRY THESE
Which of 123
2
, 77
2
, 82
2
,
161
2
, 109
2
 would end with
digit 1?
TRY THESE
Which of the following numbers would have digit
6 at unit place.
(i) 19
2
(ii) 24
2
(iii) 26
2
(iv) 36
2
(v) 34
2
Reprint 2024-25
??? ? ? ? ????? ????
TRY THESE
TRY THESE
We can see that when a square number ends in 6, the number whose square it is, will
have either 4 or 6 in unit’s place.
Can you find more such rules by observing the numbers and their squares (Table 1)?
What will be the “one’ s digit” in the square of the following numbers?
(i) 1234 (ii) 26387 (iii) 52698 (iv) 99880
(v) 21222 (vi) 9106
• Consider the following numbers and their squares.
10
2 
= 100
20
2
 = 400
80
2
 = 6400
100
2 
= 10000
200
2
 = 40000
700
2
 = 490000
900
2
 = 810000
If a number contains 3 zeros at the end, how many zeros will its square have ?
What do you notice about the number of zeros at the end of  the number and the
number of zeros at the end of its square?
Can we say that square numbers can only have even number of zeros at the end?
• See Table 1 with numbers and their squares.
What can you say about the squares of even numbers and squares of odd numbers?
1. The square of which of the following numbers would be an odd number/an even
number? Why?
(i) 727 (ii) 158 (iii) 269 (iv) 1980
2. What will be the number of zeros in the square of the following numbers?
(i) 60 (ii) 400
5.3  Some More Interesting Patterns
1. Adding triangular numbers.
Do you remember triangular numbers (numbers whose dot patterns can be arranged
as triangles)?
*
* * *
* ** * **
* ** *** * ***
* ** *** **** * ****
136 10 15
But we have
four zeros
But we have
two zeros
We have
one zero
We have
two zeros
Reprint 2024-25
? ?????? ? ???????????? ? ? ? ? ? ? ? ? ?
If we combine two consecutive triangular numbers, we get a square number, like
1 + 3 = 4 3 + 6 = 9 6 + 10 = 16
          = 2
2
= 3
2
= 4
2
2. Numbers between square numbers
Let us now see if we can find some interesting pattern between two consecutive
square numbers.
1 (= 1
2
)
2, 3, 4 (= 2
2
)
5, 6, 7, 8, 9 (= 3
2
)
10, 11, 12, 13, 14, 15, 16 (= 4
2
)
17, 18, 19, 20, 21, 22, 23, 24, 25 (= 5
2
)
Between 1
2
(=1) and 2
2
(= 4) there are two (i.e., 2 × 1) non square numbers 2, 3.
Between 2
2
(= 4) and 3
2
(= 9) there are four (i.e., 2 × 2) non square numbers 5, 6, 7, 8.
Now, 3
2
 = 9,       4
2
 = 16
Therefore, 4
2
 – 3
2
 = 16 – 9 = 7
Between 9(=3
2
) and 16(= 4
2
) the numbers are 10, 11, 12, 13, 14, 15 that is, six
non-square numbers which is 1 less than the difference of two squares.
We have 4
2
 = 16     and     5
2
 = 25
Therefore, 5
2
 – 4
2
 = 9
Between 16(= 4
2
) and 25(= 5
2
) the numbers are 17, 18, ... , 24 that is, eight non square
numbers which is 1 less than the difference of two squares.
Consider 7
2
 and 6
2
. Can you say how many numbers are there between 6
2
 and 7
2
?
If we think of any natural number n and (n + 1), then,
(n + 1)
2
 – n
2
 = (n
2
 + 2n + 1) – n
2
 = 2n + 1.
We find that between n
2
 and (n + 1)
2
 there are 2n numbers which is 1 less than the
difference of two squares.
Thus, in general we can say that there are 2n non perfect square numbers between
the squares of the numbers n and (n + 1). Check for n = 5, n = 6 etc., and verify.
Two non square numbers
between the two square
numbers 1 (=1
2
) and 4(=2
2
).
4 non square numbers
between the two square
numbers 4(=2
2
) and 9(3
2
).
8 non square
numbers between
the two square
numbers 16(= 4
2
)
and 25(=5
2
).
6 non square numbers between
the two square numbers 9(=3
2
)
and 16(= 4
2
).
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