Introduction
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
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
We then learned about more numbers that exist between the numbers 0, 1, 2, 3, 4, … , such as
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:
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
That is what we will investigate in this chapter!
Can there be a number less than 0? Can you think of any ways to have less than 0 of something?
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 multistoried 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? Arrange them in correct order.
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
Lift button presses and numbers:
So if you press +1 (i.e., if you press the ‘+’ button once), then you will go
If you press –1 (i.e., if you press the ‘–’ button once), then you will go
+++ 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?
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.
A number with a '+' sign in front is called a positive number. A number with a '–' sign in front is called a negative number.w
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.
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) =
1. You start from Floor +2 and press –3 in the lift. Where will you reach? Write an expression for this movement.
Answer We will reach Floor
Expression: Starting Floor + Movement = Target Floor (+2) + (–3) =
2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun).
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.
Answer:
a. Starting from Floor 0: Movement needed =
Expression :(0) + (-5) = –5
b. Starting from Floor +3: Movement needed =
Expression :(+3) + (-8) = –5
c. Starting from Floor –2: Movement needed =
Expression :(–2) + (-3) = –5
d. Starting from Floor +5: Movement needed =
Expression :(+5) + (-10) = –5
e. Starting from Floor –1: Movement needed =
Expression :(–1) + (-4) = –5
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
We can show the movement resulting from combining button presses as an expression:
Expression:
Evaluate these expressions by thinking of them as the resulting movement of combining button presses:
__ a.__
b.
c.
d.
Back to zero!
On the ground floor, Basant is in a great hurry and by mistake he presses +3. What can he do to cancel it and stay on the ground floor? He can cancel it by pressing – 3. That is,
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? floor
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 floor! If you are at Floor – 2 and press its inverse +2, then you go to
Write the inverses of these numbers:
| Numbers | Inverses |
|---|---|
| +4 | |
| –4 | |
| –3 | |
| 0 | |
| +2 | |
| –1 |
Connect the inverses by drawing lines
Comparing numbers using floors
Who is on the lowest floor?
- Jay is in the Art Centre. So, he is on Floor +2.
- Asin is in the Sports Centre. So, she is on Floor
. - Binnu is in the Cinema Centre. So, she is on Floor
. - Aman is in the Toys Store. So, he is on Floor
.
Floor +3 is lower than Floor +4. So, we write +3
Should we write –3
Floor –4 is lower than Floor –3. So, –4
1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >.
a. (+ 40) +
b. (+ 40) +
c. (– 50) +
d. (– 50) +
e. (– 200) – (– 40) =
f. (+ 200) – (+ 40) =
g. (– 200) – (+ 40) =
Notice that all negative number floors are below Floor 0. So, all negative numbers are less than 0. All the positive number floors are above Floor 0. So, all positive numbers are greater than 0.
2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >:
a. – 10
b. + 17
c. 0
d. + 9
e. – 25
f. + 15
3. If Floor A = – 12, Floor D = – 1 and Floor E = + 1 in the building shown on the right as a line, find the numbers of Floors B, C, F, G, and H.
Answer:
B -
C -
F -
G -
H -
4. Mark the following floors of the building shown on the right.
a. – 7
b. – 4
c. + 3
d. – 10
Number Line.