Why does multiplying or dividing two negative numbers produce a positive?
In: Mathematics
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.
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