Why does multiplying two negative numbers equal a positive number?
In: 1210
There’s many ways to think about it, some are more Eli5, some are less.
One way to think about it is if you think of the first number as a “boss”.
Okay, let’s take a step back. In school we learn that you can change the order of the multiplication, in other words it’s commutative.
Thus, 5*6 equals to 6*5. But is it really? Well, yes, and no.
Multiplication comes from the idea of having something in a package and having several packages. Basically our brain is still thinking like I have two dozens of eggs or five six-packs of beer. The first number tells how many packages you have, the second number tells the package size.
So our intuition tells that five six-packs of beer is represented by 5*6 and not 6*5, although the result of the multiplication is 30 in both cases.
And we can use that intuition to tell that the first number is sort of different from the second. The first number tells how many times I take the second one. The first number is sort of a boss.
But what happens if the boss is negative? Well, then it *swaps* the sign of the next number. If the boss is positive, it *keeps* the sign of the next number.
So if you see -5*6, it’s -30, not because there’s a negative sign somewhere in the equation, it’s because the first negative sign of 5 swapped the next positive sign of 6. If you see 5*-6, it’s -30 because the boss kept the negative sign of the second number.
If you think this way, it’s kind of easy to understand why 5*6 is 30, and why -5*-6 is also 30.
This is not the core mathematical reason, it’s more like an Eli5 level mindset to help you to wrap your head around it.
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