Adding and subtracting directed numbers
Numbers can be shown on a number line which extends above and below zero. This gives positive and negative numbers.
The signs + and – show the direction from 0. Positive and negative numbers are called directed numbers.
To add a positive number, move to the right on the number line.
Example
(+1) + (+3) = +4
(-3) + (+5) = +2
To subtract a positive number, move to the left on the number line.
Example
(+5) – (+3) = +2
(+3) – (+7) = -4
(-1) – (+2) = -3
To add a negative nm=umber, move to the left on the number line. This is equivalent to subtracting a positive number of the same value.
Multiplication of directed numbers
Positive multipliers
Multiplication is a short way of writing repeated additions. For example
3 x 4 = 3 lots of 4
= 4 + 4 + 4
= 12
With directed numbers.
(+4) + (+4) + (+4) = 3 lots of (+4)
= 3 x (+4)
The multiplier is. It is positive. Thus,
(+3) x (+4) = (+4) + (+4) + (+4) = +12
(+3) x (+4) = +12
Negative multipliers
Find the next four terms in each of the following patterns.
- +15, + 12, + 9, +6, _, _, _, _
Example: Tank Levels Rising/Falling
The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?
We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change will be:
−3 × −1,000 = +3,000
The full calculation is:
30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000
So 3 days ago there were 33,000 liters of water in the tank.
Multiplication Table
Here is another way of looking at it.
Start with the multiplication table (just up to 4×4 will do):
| x | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
Now see what happens when we head into negative territory!
Let’s go backwards through zero:
| x | 1 | 2 | 3 | 4 |
| -4 | -4 | -8 | -12 | 16 |
| -3 | -3 | -6 | -9 | -12 |
| -2 | -2 | -4 | -6 | -8 |
| -1 | -1 | -2 | -3 | -4 |
| 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 2 | 3 | 4 |
| 2 | 2 | 4 | 6 | 8 |
| 3 | 3 | 6 | 9 | 12 |
| 4 | 4 | 8 | 12 | 16 |
Look at the “4” column: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
Look over that table again, make sure you are comfortable with how it works, because …
What About Multiplying 3 or More Numbers Together?
Multiply two at a time and follow the rules.
Example: What is (−2) × (−3) × (−4) ?
First multiply (−2) × (−3). Two like signs make a positive sign, so:
(−2) × (−3) = +6
Next multiply +6 × (−4). Two unlike signs make a negative sign, so:
+6 × (−4) = -24
Result: (−2) × (−3) × (−4) = −24
Division with directed numbers
When directed numbers are multiplied together two like signs give a b positive result;
Two unlike signs give a negative result.
Example
(+3) x (+8) = +24
(-3) x (-8) = +24
(+3) x (-8) = -24
(-3) x (+8) = -24
The same rule is true for division. For example
(+24) ÷ (-3) = (+8)
(-24) ÷ (-3) = (+8)
(+24) ÷ (-3) = (-8)
(-24) ÷ (+3) = (-8)
Example
Divide -36 by 9
-36 ÷ 9 = – 36/+9 = – (36/9) = -4
