Multiplying Negatives
When You Multiply ...
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Example |
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two positives you get a positive: |
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3 × 2 = 6 |
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a positive and a negative
you get a negative: |
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(-3) × 2 = -6 |
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a negative and a positive
you get a negative: |
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3 × (-2) = -6 |
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two negatives you get a positive: |
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(-3) × (-2) = 6 |
So, a negative times a positive gives a negative ...
but multiplying two negatives gives a positive!
An easy way to remember:
"Two like signs become a positive sign, two unlike signs become a negative sign"
Example: (-4) × (-3)
The signs are - and -, 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:
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.
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 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 it is like you have added $15 (-3 × -$5 = +$15).
You now have no debt: -$15 + $15 = $0
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 (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 was 33,000 litres.
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