10 THE OTHER SIDE OF
ZERO
Integers
More and more numbers!
Recall that the very first 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 , 3 , and 13 . These are called
2 2 6
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!
Reprint 2025-26
, The Other Side of Zero
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 flock 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 filled it with attractions.
She named it Bela’s Building of Fun.
But this was no ordinary building!
Observe that some of the floors in the
‘Building of Fun’ are below the ground. What are the shops that you
find on these floors? What is there on the ground floor?
A lift is used to go up and down between the floors. 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 floors, 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 floor and if
you press – 1 (i.e., if you press the ‘–’ button
once), then you will go down 1 floor.
Lift button presses and numbers:
+++ is written as + 3
– – – – is written as – 4
What do you press to go four floors up?
What do you press to go three floors down?
Reprint 2025-26
243
, Ganita Prakash | Grade 6
Numbering the floors in the building of fun
Entry to the ‘Building of Fun’ is at the ground floor level and is called
the ‘Welcome Hall’. Starting from the ground floor, 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 floor, you must press – 1 to reach the Toy
Store. So, the Toy Store is on Floor – 1 similarly starting from the ground
floor, you must press – 2 to reach the Video Games shop. So, the Video
Games shop is on Floor – 2.
The ground floor is called Floor 0. Can you see why?
Number all the floors in the Building of Fun.
Did you notice that + 3 is the floor number of the Book
Store, but it is also the number of floors you move
when you press + 3? Similarly, – 3 is the floor number
but it is also the number of floors 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 floors are numbered
using the ground floor, Floor 0, as a reference or starting
point. The floors above the ground floor are numbered
with positive numbers. To get to them from the ground
floor, one must press the ‘+’ button some number of
times. The floors below the ground are numbered with
negative numbers. To get to them from the ground floor,
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.
244 Reprint 2025-26
, The Other Side of Zero
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 floor is + 1 (Food Court) and the number of button
presses is + 2. Therefore, you reach the target floor (+ 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 floors, find 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 floors.
But by mistake he pressed the ‘+’ button two times. He realised his
mistake and quickly pressed the ‘–’ button three times. How many
floors below or above the Toy Store will Gurmit reach?
Gurmit will go one floor down. We can show the movement
resulting from combining button presses as an expression:
(+ 2) + (– 3) = – 1.
Reprint 2025-26
245
ZERO
Integers
More and more numbers!
Recall that the very first 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 , 3 , and 13 . These are called
2 2 6
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!
Reprint 2025-26
, The Other Side of Zero
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 flock 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 filled it with attractions.
She named it Bela’s Building of Fun.
But this was no ordinary building!
Observe that some of the floors in the
‘Building of Fun’ are below the ground. What are the shops that you
find on these floors? What is there on the ground floor?
A lift is used to go up and down between the floors. 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 floors, 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 floor and if
you press – 1 (i.e., if you press the ‘–’ button
once), then you will go down 1 floor.
Lift button presses and numbers:
+++ is written as + 3
– – – – is written as – 4
What do you press to go four floors up?
What do you press to go three floors down?
Reprint 2025-26
243
, Ganita Prakash | Grade 6
Numbering the floors in the building of fun
Entry to the ‘Building of Fun’ is at the ground floor level and is called
the ‘Welcome Hall’. Starting from the ground floor, 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 floor, you must press – 1 to reach the Toy
Store. So, the Toy Store is on Floor – 1 similarly starting from the ground
floor, you must press – 2 to reach the Video Games shop. So, the Video
Games shop is on Floor – 2.
The ground floor is called Floor 0. Can you see why?
Number all the floors in the Building of Fun.
Did you notice that + 3 is the floor number of the Book
Store, but it is also the number of floors you move
when you press + 3? Similarly, – 3 is the floor number
but it is also the number of floors 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 floors are numbered
using the ground floor, Floor 0, as a reference or starting
point. The floors above the ground floor are numbered
with positive numbers. To get to them from the ground
floor, one must press the ‘+’ button some number of
times. The floors below the ground are numbered with
negative numbers. To get to them from the ground floor,
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.
244 Reprint 2025-26
, The Other Side of Zero
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 floor is + 1 (Food Court) and the number of button
presses is + 2. Therefore, you reach the target floor (+ 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 floors, find 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 floors.
But by mistake he pressed the ‘+’ button two times. He realised his
mistake and quickly pressed the ‘–’ button three times. How many
floors below or above the Toy Store will Gurmit reach?
Gurmit will go one floor down. We can show the movement
resulting from combining button presses as an expression:
(+ 2) + (– 3) = – 1.
Reprint 2025-26
245
