Why does multiplying or dividing two negative numbers produce a positive?
In: Mathematics
The Khan Academy and others explain it using distributive property of maths, which is therefore a pre-requisite. We did this at age 11 in UK. I’ll show it quickly here:
What is -5 x (3 – 3) ? Ans. 0, since 3 – 3 is 0 and anything multiplied by 0 is 0.
Using distributive property of multiplication: -5 x (3 – 3) = -15 + (-5 x -3) = 0
So it must be true that -5 x -3 = +15.
The other intuition is looking at a pattern such as:
3 x -5 = -15
2 x -5 = -10
1 x -5 = -5
0 x -5 = 0
-1 x -5 = +5
-2 x -5 = +10
And hopefully feel that this makes sense, everything is consistent.
(Ughh, not formatting as it appears on iPad sorry)
One way to understand this is to treat real numbers as ‘phasors’.
A phasor is simply a magnitude and an angle. When you multiply two phasors, you multiply the magnitudes and you add the angles.
So instead of thinking 2 * -2 = -4, instead think of 2 (angle of 0) * 2 (angle of 180 degrees) = 4 (angle of 180 degrees).
A good way to visualize this is to think of a 2d grid rather than a number line. When you multiply numbers, you move your starting point further/closer to the origin while rotating that point around the origin.
With this mental model, it should become clear that multiplying 2 (angle of 180) by 2 (angle of 180) must yield 4 (angle of 0) – you’re just multiplying two positive numbers for the magnitude and adding the angles.
Now, this may seem a bit weird. But it should be obvious that everything you could possibly do with standard real numbers can also be done with phasors – you just represent positive reals with an angle of 0 and negative numbers with an angle of 180. Indeed, phasors are a ‘superset’ of reals – they include every possible real and we can express every possible algebraic combination of reals with them.
As a result, any time you could use reals, you can use phasors instead – and if our multiplication with phasors yields the result you’re pondering, then our multiplication with reals must as well.
My teacher used to say “use the word ‘of’ instead of ‘times’ when you’re multiplying. So 2 of 2 = 4. When it comes to multiplying negatives, think of it like a “opposite negative.”
So, if we have 2 of -2, I’m doubling the amount of deficit. If we have -2 of -2 it’s like reversing amount of deficit since it’s the opposite.
If you are comfortable with multiplying out (a – b)^2 you will find that the -b^2 term i.e. -b x -b MUST be positive.
e.g. suppose a=10 b=3 then result of above expression must be 7^2 = 49. Knowing this, when we expand the terms, a^2 = 100, -2ab = -60 so -b x -b = -b^2 = +9 to give 49 when the terms are summed.
HTH.
I owe you $10. You can interpret that as me having -$10.
I mow your lawn and you cancel my debt. That’s like subtracting that -$10, which is the same as adding $10.
I owe 5 people $10. That like having 5 x -$10 = -$50.
I mow all of their yards. I’m subtracting that debt five times, so -5 x -$10 = $50
Think of it as the number of times the numbers divide, -10/-5 = 2 because negative ten divides by negative five twice not negative twice.
Multiplying is easier to grasp.
Let’s go with a small problem. -2×2
Let’s label the negatives using dots. A ° is a negative.
So -2 = °°
If I have -2 two times that means I have °° °°
So count how many negative dots we have. We have °°°°. Which is 4 negative counters. Or -4.
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