Adding and subtracting larger numbers
The picture shows a mine, a place where minerals are extracted by digging into the rock. The truck is at the ground level, but the minerals are present both above and below the ground level. There is a fast moving lift which moves up and down in a mineshaft carrying people and ore.
Some of the levels are marked in the picture. The ground level is marked 0. Levels above the ground are marked by positive numbers and levels below the ground are marked by negative numbers.
The number indicates how many metres above or below the ground level it is.
In the mine, just like in the Building of Fun:
Starting Level + Movement = Target Level
For example,
Target Level – Starting Level = Movement needed
For example,
How many negative numbers are there?
Bela's Building of Fun had only six floors above and five floors below. That is numbers –5 to +6. In the mine above, we have numbers from – 200 to + 180. But we can imagine larger buildings or mineshafts.
Just as positive numbers + 1, + 2, + 3, ... keep going up without an end,
Similarly, negative numbers – 1, – 2, – 3, ... keep going down.
Positive and negative numbers, with zero, are called
Complete these expressions.
a.
b.
c.
d.
e.
f.
g.
Check your answers by thinking about the movement in the mineshaft.
To add and subtract even larger integers, we can imagine even larger lifts! In fact, we can imagine a lift that can extend forever upwards and forever downwards, starting from Level 0. There does not even have to be any building or mine around — just an 'infinite lift'!
We can use this imagination to add and subtract any integers we like.
For example, suppose we want to carry out the subtraction + 2000 – (– 200). We can imagine a lift with 2000 levels above the ground and 200 below the ground. Recall that,
Target Level – Starting Level = Movement needed
To go from the Starting Floor – 200 to the Target Floor + 2000, we must press + 2200 (+ 200 to get to zero, and then + 2000 more after that to get to + 2200).
Therefore,
Notice that
Try evaluating the following expressions by similarly drawing or imagining a suitable lift:
a.
b.
c.
d.
e.
f.
g.
h.
In the above step, we saw that + 2000 – (– 200) = + 2000 + (+ 200) =
In other words, subtracting a negative number is the same as adding the corresponding positive number. That is, we can replace subtraction of a negative number by addition of a positive number!
In the other exercises that you did above, did you notice that subtracting a negative number was the same as adding the corresponding positive number?
Take a look at the 'infinite lift' above. Does it remind you of a number line? In what ways?