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Multiplying Negatives

When You Multiply ...

plus × plus two positives you get a positive: plus 3 × 2 = 6
minus × plus a positive and a negative
you get a negative:
minus (-3) × 2 = -6
plus × minus a negative and a positive
you get a negative:
minus 3 × (-2) = -6
minus × minus two negatives you get a positive: plus (-3) × (-2) = 6

So, a negative times a positive gives a negative ...
but multiplying two negatives gives a positive!

Why does multiplying two negative numbers make a positive?

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

If I say "Eat!" I am encouraging you to eat (positive), but if 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!".

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


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 3 x 2 = 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.

More Examples


Money Owed vs Money Owned

Example: 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 you are $15 better off (-3 × -5 = +15).


Tank Levels Rising/Falling

Example: The tank has 30,000 litres, and 1,000 litres are taken out every day. What was the level 3 days ago?

We know the tank has -1,000 every day, and we need to subtract that 3 times, 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


And That's All !