Multiplying Negatives

When We Multiply:

      Example
plus × plus two positives make a positive: plus 3 × 2 = 6
       
minus × minus   two negatives make a positive:   plus (−3) × (−2) = 6
       
minus × plus a negative and a positive
make a negative:
minus (−3) × 2 = −6
       
plus × minus a positive and a negative
make a negative:
minus 3 × (−2) = −6

Yes indeed, two negatives make a positive, and we will explain why, with examples!

Signs

Let's talk about signs.

"+" is the positive sign, "−" is the negative sign.

When a number has no sign it usually means that it is positive.

Example: 5 is really +5

And we can put () around the numbers to avoid confusion.

Example: 3 × −2 can be written as 3 × (−2)

Two Signs: The Rules

"Two like signs make a positive sign,
two unlike signs make a negative sign"

 

Example: (−2) × (+5)

The signs are and + (a negative sign and a positive sign), so they are unlike signs (they are different to each other)

So the result must be negative:

(−2) × (+5) = -10

Example: (−4) × (−3)

The signs are and (they are both negative signs), so they are like signs (like each other)

So the result must be positive:

(−4) × (−3) = +12

Why does multiplying two negative numbers make a positive?

Well, first there is the "common sense" explanation:

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.

Direction

It is all about direction. Remember the Number Line?

Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:

Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:

Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:

So, by walking backwards, while facing in the negative direction, he moves in the positive direction.

Try it yourself! Try walking forwards and backwards, then again but facing the other direction.

More Examples

Example: Money Owed vs Money Owned

You owe 3 people $5 each. So you are "Negative 15" (3 × −$5 = −$15).

They then say "we like you so much we forgive the debt" ... you have just had 3 subtractions of −5, so it is like you have added $15 (−3 × −$5 = +$15).

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):

× 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:

× 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 ...

... now we go further to the left, through zero:

× -4 -3 -2 -1 0 1 2 3 4
-4 16 12 8 4 0 -4 -8 -12 -16
-3 12 9 6 3 0 -3 -6 -9 -12
-2 8 6 4 2 0 -2 -4 -6 -8
-1 4 3 2 1 0 -1 -2 -3 -4
0 0 0 0 0 0 0 0 0 0
1 -4 -3 -2 -1 0 1 2 3 4
2 -8 -6 -4 -2 0 2 4 6 8
3 -12 -9 -6 -3 0 3 6 9 12
4 -16 -12 -8 -4 0 4 8 12 16

Same pattern: we can follow along a row (or column) and the values change consistently:

  • Follow the "4" row along: it goes -16, -12, -8, -4, 0, 4, 8, 12, 16. Getting 4 larger each time.
  • Follow the "-4" row along: it goes 16, 12, 8, 4, 0, -4, -8, -12, -16. Getting 4 smaller each time.
  • etc...

So it all follows a neat pattern!

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