NCERT Textbook: The Other Side of Zero | Mathematics (Maths) Class 6 PDF Download

 Page 1


Integers
THE OTHER SIDE OF 
ZERO
10
More and More numbers!
Recall that the very ??rst numbers we learned about in the study of 
mathematics were the counting numbers 1, 2, 3, 4, …  
Then we learned that there are even more numbers! For example, 
there is the number 0 (zero), representing nothing, which comes before 
1.  The number 0 has a very important history in India and now in the 
world. For example, around the world we learn to write numbers in 
the Indian number system using the digits 0 to 9, allowing us to write 
numbers however large or however small using just these 10 digits. 
We then learned about more numbers that exist between the 
numbers 0, 1, 2, 3, 4, … , such as 
1
2
, 
3
2
, and 
13
6
. These are called 
fractions.
But are there still more numbers? Well, 0 is an additional number 
that we didn’t know about earlier, and it comes before 1 and is less 
than 1. Are there perhaps more numbers that come before 0 and are 
less than 0?
Phrased another way, we have seen the number line:
0 1 2 3 4 5 6 7 8 9 10
However, this is actually only a number ‘ray’, in the language we 
learned earlier in geometry; this ray starts at 0 and goes forever to 
the right. Do there exist numbers to the left of 0, so that this number 
ray can be completed to a true number line?
That is what we will investigate in this chapter!
Chapter 10_The Other Side of Zero.indd   242 13-08-2024   17:32:05
Reprint 2025-26
Page 2


Integers
THE OTHER SIDE OF 
ZERO
10
More and More numbers!
Recall that the very ??rst numbers we learned about in the study of 
mathematics were the counting numbers 1, 2, 3, 4, …  
Then we learned that there are even more numbers! For example, 
there is the number 0 (zero), representing nothing, which comes before 
1.  The number 0 has a very important history in India and now in the 
world. For example, around the world we learn to write numbers in 
the Indian number system using the digits 0 to 9, allowing us to write 
numbers however large or however small using just these 10 digits. 
We then learned about more numbers that exist between the 
numbers 0, 1, 2, 3, 4, … , such as 
1
2
, 
3
2
, and 
13
6
. These are called 
fractions.
But are there still more numbers? Well, 0 is an additional number 
that we didn’t know about earlier, and it comes before 1 and is less 
than 1. Are there perhaps more numbers that come before 0 and are 
less than 0?
Phrased another way, we have seen the number line:
0 1 2 3 4 5 6 7 8 9 10
However, this is actually only a number ‘ray’, in the language we 
learned earlier in geometry; this ray starts at 0 and goes forever to 
the right. Do there exist numbers to the left of 0, so that this number 
ray can be completed to a true number line?
That is what we will investigate in this chapter!
Chapter 10_The Other Side of Zero.indd   242 13-08-2024   17:32:05
Reprint 2025-26
The Other Side of Zero
243
 Can there be a number less than 0? Can you think of any ways to 
have less than 0 of something?
10.1 Bela’s Building of Fun
Children ??ock to Bela’s ice cream factory to see 
and taste her tasty ice cream. To make it even 
more fun for them, Bela purchased a multi-
storied building and ??lled it with attractions. 
She named it Bela’s Building of Fun. 
But this was no ordinary building! 
Observe that some of the ??oors in the 
‘Building of Fun’ are below the ground. What are the shops that you 
??nd on these ??oors? What is there on the ground ??oor?
A lift is used to go up and down between the ??oors. It has two 
buttons: ‘+’ to go up and ‘–’ to go down. Can you spot the lift ?
To go to the Art Centre from the ‘Welcome Hall’, you 
must press the ‘+’ button twice. 
We say that the button press is + + or + 2.
To go down two ??oors, you must press the ‘–’ 
button twice, which we write as – – or – 2.
So if you press + 1 (i.e., if you press the ‘+’ 
button once), then you will go up one ??oor and if 
you press – 1 (i.e., if you press the ‘–’ button 
once), then you will go down 1 ??oor.
Lift button presses and numbers:
+++ is written as + 3
 – – – – is written as –  4
 What do you press to go four ??oors up? 
What do you press to go three ??oors down?
Chapter 10_The Other Side of Zero.indd   243 13-08-2024   17:32:06
Reprint 2025-26
Page 3


Integers
THE OTHER SIDE OF 
ZERO
10
More and More numbers!
Recall that the very ??rst numbers we learned about in the study of 
mathematics were the counting numbers 1, 2, 3, 4, …  
Then we learned that there are even more numbers! For example, 
there is the number 0 (zero), representing nothing, which comes before 
1.  The number 0 has a very important history in India and now in the 
world. For example, around the world we learn to write numbers in 
the Indian number system using the digits 0 to 9, allowing us to write 
numbers however large or however small using just these 10 digits. 
We then learned about more numbers that exist between the 
numbers 0, 1, 2, 3, 4, … , such as 
1
2
, 
3
2
, and 
13
6
. These are called 
fractions.
But are there still more numbers? Well, 0 is an additional number 
that we didn’t know about earlier, and it comes before 1 and is less 
than 1. Are there perhaps more numbers that come before 0 and are 
less than 0?
Phrased another way, we have seen the number line:
0 1 2 3 4 5 6 7 8 9 10
However, this is actually only a number ‘ray’, in the language we 
learned earlier in geometry; this ray starts at 0 and goes forever to 
the right. Do there exist numbers to the left of 0, so that this number 
ray can be completed to a true number line?
That is what we will investigate in this chapter!
Chapter 10_The Other Side of Zero.indd   242 13-08-2024   17:32:05
Reprint 2025-26
The Other Side of Zero
243
 Can there be a number less than 0? Can you think of any ways to 
have less than 0 of something?
10.1 Bela’s Building of Fun
Children ??ock to Bela’s ice cream factory to see 
and taste her tasty ice cream. To make it even 
more fun for them, Bela purchased a multi-
storied building and ??lled it with attractions. 
She named it Bela’s Building of Fun. 
But this was no ordinary building! 
Observe that some of the ??oors in the 
‘Building of Fun’ are below the ground. What are the shops that you 
??nd on these ??oors? What is there on the ground ??oor?
A lift is used to go up and down between the ??oors. It has two 
buttons: ‘+’ to go up and ‘–’ to go down. Can you spot the lift ?
To go to the Art Centre from the ‘Welcome Hall’, you 
must press the ‘+’ button twice. 
We say that the button press is + + or + 2.
To go down two ??oors, you must press the ‘–’ 
button twice, which we write as – – or – 2.
So if you press + 1 (i.e., if you press the ‘+’ 
button once), then you will go up one ??oor and if 
you press – 1 (i.e., if you press the ‘–’ button 
once), then you will go down 1 ??oor.
Lift button presses and numbers:
+++ is written as + 3
 – – – – is written as –  4
 What do you press to go four ??oors up? 
What do you press to go three ??oors down?
Chapter 10_The Other Side of Zero.indd   243 13-08-2024   17:32:06
Reprint 2025-26
Ganita Prakash | Grade 6
244
Numbering the ??oors in the building of fun
Entry to the ‘Building of Fun’ is at the ground ??oor level and is called 
the ‘Welcome Hall’. Starting from the ground ??oor, you can reach the 
Food Court by pressing + 1 and can reach the Art Centre by pressing 
+ 2. So, we can say that the Food Court is on Floor + 1 and that the Art 
Centre is on Floor + 2.
Starting from the ground ??oor, you must press – 1 to reach the Toy 
Store. So, the Toy Store is on Floor – 1 similarly starting from the ground 
??oor, you must press – 2 to reach the Video Games shop. So, the Video 
Games shop is on Floor – 2.
The ground ??oor is called Floor 0. Can you see why?
 Number all the ??oors in the Building of Fun.
Did you notice that + 3 is the ??oor number of the Book 
Store, but it is also the number of ??oors you move 
when you press + 3? Similarly,  – 3 is the ??oor number 
but it is also the number of ??oors you go down when 
you press – 3, i.e., when you press – – – .
A number with a ‘+’ sign in front is called a positive 
number. A number with a ‘–’ sign in front is called a 
negative number. 
In the ‘Building of Fun’, the ??oors are numbered 
using the ground ??oor, Floor 0, as a reference or starting 
point. The ??oors above the ground ??oor are numbered 
with positive numbers. To get to them from the ground 
??oor, one must press the ‘+’ button some number of 
times. The ??oors below the ground are numbered with 
negative numbers. To get to them from the ground ??oor, 
one must press the ‘–’ button some number of times.
Zero is neither a positive nor a negative number. 
We do not put a ‘+’ or ‘–’ sign in front of it.
Chapter 10_The Other Side of Zero.indd   244 13-08-2024   17:32:07
Reprint 2025-26
Page 4


Integers
THE OTHER SIDE OF 
ZERO
10
More and More numbers!
Recall that the very ??rst numbers we learned about in the study of 
mathematics were the counting numbers 1, 2, 3, 4, …  
Then we learned that there are even more numbers! For example, 
there is the number 0 (zero), representing nothing, which comes before 
1.  The number 0 has a very important history in India and now in the 
world. For example, around the world we learn to write numbers in 
the Indian number system using the digits 0 to 9, allowing us to write 
numbers however large or however small using just these 10 digits. 
We then learned about more numbers that exist between the 
numbers 0, 1, 2, 3, 4, … , such as 
1
2
, 
3
2
, and 
13
6
. These are called 
fractions.
But are there still more numbers? Well, 0 is an additional number 
that we didn’t know about earlier, and it comes before 1 and is less 
than 1. Are there perhaps more numbers that come before 0 and are 
less than 0?
Phrased another way, we have seen the number line:
0 1 2 3 4 5 6 7 8 9 10
However, this is actually only a number ‘ray’, in the language we 
learned earlier in geometry; this ray starts at 0 and goes forever to 
the right. Do there exist numbers to the left of 0, so that this number 
ray can be completed to a true number line?
That is what we will investigate in this chapter!
Chapter 10_The Other Side of Zero.indd   242 13-08-2024   17:32:05
Reprint 2025-26
The Other Side of Zero
243
 Can there be a number less than 0? Can you think of any ways to 
have less than 0 of something?
10.1 Bela’s Building of Fun
Children ??ock to Bela’s ice cream factory to see 
and taste her tasty ice cream. To make it even 
more fun for them, Bela purchased a multi-
storied building and ??lled it with attractions. 
She named it Bela’s Building of Fun. 
But this was no ordinary building! 
Observe that some of the ??oors in the 
‘Building of Fun’ are below the ground. What are the shops that you 
??nd on these ??oors? What is there on the ground ??oor?
A lift is used to go up and down between the ??oors. It has two 
buttons: ‘+’ to go up and ‘–’ to go down. Can you spot the lift ?
To go to the Art Centre from the ‘Welcome Hall’, you 
must press the ‘+’ button twice. 
We say that the button press is + + or + 2.
To go down two ??oors, you must press the ‘–’ 
button twice, which we write as – – or – 2.
So if you press + 1 (i.e., if you press the ‘+’ 
button once), then you will go up one ??oor and if 
you press – 1 (i.e., if you press the ‘–’ button 
once), then you will go down 1 ??oor.
Lift button presses and numbers:
+++ is written as + 3
 – – – – is written as –  4
 What do you press to go four ??oors up? 
What do you press to go three ??oors down?
Chapter 10_The Other Side of Zero.indd   243 13-08-2024   17:32:06
Reprint 2025-26
Ganita Prakash | Grade 6
244
Numbering the ??oors in the building of fun
Entry to the ‘Building of Fun’ is at the ground ??oor level and is called 
the ‘Welcome Hall’. Starting from the ground ??oor, you can reach the 
Food Court by pressing + 1 and can reach the Art Centre by pressing 
+ 2. So, we can say that the Food Court is on Floor + 1 and that the Art 
Centre is on Floor + 2.
Starting from the ground ??oor, you must press – 1 to reach the Toy 
Store. So, the Toy Store is on Floor – 1 similarly starting from the ground 
??oor, you must press – 2 to reach the Video Games shop. So, the Video 
Games shop is on Floor – 2.
The ground ??oor is called Floor 0. Can you see why?
 Number all the ??oors in the Building of Fun.
Did you notice that + 3 is the ??oor number of the Book 
Store, but it is also the number of ??oors you move 
when you press + 3? Similarly,  – 3 is the ??oor number 
but it is also the number of ??oors you go down when 
you press – 3, i.e., when you press – – – .
A number with a ‘+’ sign in front is called a positive 
number. A number with a ‘–’ sign in front is called a 
negative number. 
In the ‘Building of Fun’, the ??oors are numbered 
using the ground ??oor, Floor 0, as a reference or starting 
point. The ??oors above the ground ??oor are numbered 
with positive numbers. To get to them from the ground 
??oor, one must press the ‘+’ button some number of 
times. The ??oors below the ground are numbered with 
negative numbers. To get to them from the ground ??oor, 
one must press the ‘–’ button some number of times.
Zero is neither a positive nor a negative number. 
We do not put a ‘+’ or ‘–’ sign in front of it.
Chapter 10_The Other Side of Zero.indd   244 13-08-2024   17:32:07
Reprint 2025-26
The Other Side of Zero
245
Addition to keep track of movement
Start from the Food Court and press + 2 in the lift. Where will you 
reach? ____________
We can describe this using an expression:
Starting Floor + Movement = Target Floor.
The starting ??oor is + 1 (Food Court) and the number of button 
presses is + 2. Therefore,  you reach the target ??oor (+?1)?+?(+?2)?=?+?3 
(Book Store).
 Figure it Out
1.  You start from Floor + 2 and press – 3 in the lift. Where will you 
reach? Write an expression for this movement.
2. Evaluate these expressions (you may think of them as Starting 
Floor + Movement by referring to the Building of Fun).
a. (+ 1) + (+ 4) = _______ b. (+ 4) + (+ 1) = _______
c. (+ 4) + (–  3) = _______ d. (– 1) + (+ 2) = _______
e. (– 1) + (+ 1) = _______ f.      0 + (+ 2) = _________
g.      0 + (– 2) = _________
3.  Starting from different ??oors, ??nd the movements required to 
reach Floor – 5. For example, if I start at Floor + 2, I must press – 7 
to reach Floor – 5. The expression is (+ 2) + (– 7) = – 5.
  Find more such starting positions and the movements needed to 
reach Floor – 5 and write the expressions.
Combining button presses is also addition 
Gurmit was in the Toy Store and wanted to go down two ??oors. 
But by mistake he pressed the ‘+’ button two times. He realised his 
mistake and quickly pressed the ‘–’ button three times. How many 
??oors below or above the Toy Store will Gurmit reach?
Gurmit will go one ??oor down. We can show the movement 
resulting from combining button presses as an expression:  
(+ 2) + (– 3) = – 1.
Chapter 10_The Other Side of Zero.indd   245 13-08-2024   17:32:07
Reprint 2025-26
Page 5


Integers
THE OTHER SIDE OF 
ZERO
10
More and More numbers!
Recall that the very ??rst numbers we learned about in the study of 
mathematics were the counting numbers 1, 2, 3, 4, …  
Then we learned that there are even more numbers! For example, 
there is the number 0 (zero), representing nothing, which comes before 
1.  The number 0 has a very important history in India and now in the 
world. For example, around the world we learn to write numbers in 
the Indian number system using the digits 0 to 9, allowing us to write 
numbers however large or however small using just these 10 digits. 
We then learned about more numbers that exist between the 
numbers 0, 1, 2, 3, 4, … , such as 
1
2
, 
3
2
, and 
13
6
. These are called 
fractions.
But are there still more numbers? Well, 0 is an additional number 
that we didn’t know about earlier, and it comes before 1 and is less 
than 1. Are there perhaps more numbers that come before 0 and are 
less than 0?
Phrased another way, we have seen the number line:
0 1 2 3 4 5 6 7 8 9 10
However, this is actually only a number ‘ray’, in the language we 
learned earlier in geometry; this ray starts at 0 and goes forever to 
the right. Do there exist numbers to the left of 0, so that this number 
ray can be completed to a true number line?
That is what we will investigate in this chapter!
Chapter 10_The Other Side of Zero.indd   242 13-08-2024   17:32:05
Reprint 2025-26
The Other Side of Zero
243
 Can there be a number less than 0? Can you think of any ways to 
have less than 0 of something?
10.1 Bela’s Building of Fun
Children ??ock to Bela’s ice cream factory to see 
and taste her tasty ice cream. To make it even 
more fun for them, Bela purchased a multi-
storied building and ??lled it with attractions. 
She named it Bela’s Building of Fun. 
But this was no ordinary building! 
Observe that some of the ??oors in the 
‘Building of Fun’ are below the ground. What are the shops that you 
??nd on these ??oors? What is there on the ground ??oor?
A lift is used to go up and down between the ??oors. It has two 
buttons: ‘+’ to go up and ‘–’ to go down. Can you spot the lift ?
To go to the Art Centre from the ‘Welcome Hall’, you 
must press the ‘+’ button twice. 
We say that the button press is + + or + 2.
To go down two ??oors, you must press the ‘–’ 
button twice, which we write as – – or – 2.
So if you press + 1 (i.e., if you press the ‘+’ 
button once), then you will go up one ??oor and if 
you press – 1 (i.e., if you press the ‘–’ button 
once), then you will go down 1 ??oor.
Lift button presses and numbers:
+++ is written as + 3
 – – – – is written as –  4
 What do you press to go four ??oors up? 
What do you press to go three ??oors down?
Chapter 10_The Other Side of Zero.indd   243 13-08-2024   17:32:06
Reprint 2025-26
Ganita Prakash | Grade 6
244
Numbering the ??oors in the building of fun
Entry to the ‘Building of Fun’ is at the ground ??oor level and is called 
the ‘Welcome Hall’. Starting from the ground ??oor, you can reach the 
Food Court by pressing + 1 and can reach the Art Centre by pressing 
+ 2. So, we can say that the Food Court is on Floor + 1 and that the Art 
Centre is on Floor + 2.
Starting from the ground ??oor, you must press – 1 to reach the Toy 
Store. So, the Toy Store is on Floor – 1 similarly starting from the ground 
??oor, you must press – 2 to reach the Video Games shop. So, the Video 
Games shop is on Floor – 2.
The ground ??oor is called Floor 0. Can you see why?
 Number all the ??oors in the Building of Fun.
Did you notice that + 3 is the ??oor number of the Book 
Store, but it is also the number of ??oors you move 
when you press + 3? Similarly,  – 3 is the ??oor number 
but it is also the number of ??oors you go down when 
you press – 3, i.e., when you press – – – .
A number with a ‘+’ sign in front is called a positive 
number. A number with a ‘–’ sign in front is called a 
negative number. 
In the ‘Building of Fun’, the ??oors are numbered 
using the ground ??oor, Floor 0, as a reference or starting 
point. The ??oors above the ground ??oor are numbered 
with positive numbers. To get to them from the ground 
??oor, one must press the ‘+’ button some number of 
times. The ??oors below the ground are numbered with 
negative numbers. To get to them from the ground ??oor, 
one must press the ‘–’ button some number of times.
Zero is neither a positive nor a negative number. 
We do not put a ‘+’ or ‘–’ sign in front of it.
Chapter 10_The Other Side of Zero.indd   244 13-08-2024   17:32:07
Reprint 2025-26
The Other Side of Zero
245
Addition to keep track of movement
Start from the Food Court and press + 2 in the lift. Where will you 
reach? ____________
We can describe this using an expression:
Starting Floor + Movement = Target Floor.
The starting ??oor is + 1 (Food Court) and the number of button 
presses is + 2. Therefore,  you reach the target ??oor (+?1)?+?(+?2)?=?+?3 
(Book Store).
 Figure it Out
1.  You start from Floor + 2 and press – 3 in the lift. Where will you 
reach? Write an expression for this movement.
2. Evaluate these expressions (you may think of them as Starting 
Floor + Movement by referring to the Building of Fun).
a. (+ 1) + (+ 4) = _______ b. (+ 4) + (+ 1) = _______
c. (+ 4) + (–  3) = _______ d. (– 1) + (+ 2) = _______
e. (– 1) + (+ 1) = _______ f.      0 + (+ 2) = _________
g.      0 + (– 2) = _________
3.  Starting from different ??oors, ??nd the movements required to 
reach Floor – 5. For example, if I start at Floor + 2, I must press – 7 
to reach Floor – 5. The expression is (+ 2) + (– 7) = – 5.
  Find more such starting positions and the movements needed to 
reach Floor – 5 and write the expressions.
Combining button presses is also addition 
Gurmit was in the Toy Store and wanted to go down two ??oors. 
But by mistake he pressed the ‘+’ button two times. He realised his 
mistake and quickly pressed the ‘–’ button three times. How many 
??oors below or above the Toy Store will Gurmit reach?
Gurmit will go one ??oor down. We can show the movement 
resulting from combining button presses as an expression:  
(+ 2) + (– 3) = – 1.
Chapter 10_The Other Side of Zero.indd   245 13-08-2024   17:32:07
Reprint 2025-26
Ganita Prakash | Grade 6
246
  Figure it out
Evaluate these expressions by thinking of them as the resulting movement 
of combining button presses:
a.  (+ 1) + (+ 4) = _____________ b. (+ 4) + (+ 1) = _____________
c. (+ 4) + (–  3) + (– 2) = _______ d. (– 1) + (+ 2) + (– 3)  = _______
Back to zero!
On the ground ??oor, Basant is in a great hurry and by 
mistake he presses +3. What can he do to cancel it and 
stay on the ground ??oor? He can cancel it by pressing  
– 3. That is, (+3) + (– 3) = 0.
We call – 3 the inverse of +3. Similarly, the inverse of 
– 3 is +3.
If Basant now presses +4 and then presses – 4 in the 
lift, where will he reach?
Here is another way to think of the concept of 
inverse. If you are at Floor +4 and you press its inverse 
– 4, then you are back to zero, the ground ??oor! If you 
are at Floor – 2 and press its inverse +2, then you go to 
(– 2) + (+2) = 0, again the ground ??oor!
  Write the inverses of these numbers:     
+4, –4, –3, 0, +2, –1.
  Connect the inverses by drawing lines.
+9
+7
–8
–5
–7
+8
+5
–9
Comparing numbers using ??oors
 Who is on the lowest ??oor? 
1. Jay is in the Art Centre. So, he is on Floor +2. 
2. Asin is in the Sports Centre. So, she is on Floor ___.
3. Binnu is in the Cinema Centre. So, she is on Floor ____. 
4. Aman is in the Toys Store. So, he is on Floor ____.
Chapter 10_The Other Side of Zero.indd   246 13-08-2024   17:32:08
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