Why does multiplying two negative numbers equal a positive number?
In: 1210
The way that makes sense to me is negative means “opposite sign”. So a negative times a positive is negative because you’re taking the opposite sign for that number. If you do negative times negative, it is positive because positive is the opposite of negative. I had to think of it this way for some physics equations to make sense.
2 x 3 = I will give you 2 boxes, three times. You now have 6 boxes.
-2 x 3 = I will give you -2 boxes, three times. You now have -6 boxes, because I took 2 boxes, three times.
-2 x -3 = I will give you -2 boxes, -three times. You now have 6 boxes. I took 2 boxes, and then gave them back, three times.
The opposite of plus is minus. The – indicates the opposite. Instead of giving you boxes, I took them. Instead of keeping them, I gave them back.
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.
If you turn around twice, what direction are you facing?
We want to show -a * – b = a * b, where a and b are real numbers.
-a * – b
= -1 * a * -1 * b
= -1 * -1 * a * b
So what we need is to show that -1 * -1 = 1.
Well 1 – 1 = 0
And -1 * (1 – 1) = -1 * 0 = 0
Which means -1 * 1 + -1 * -1 = 0
Thus -1 + – 1 * -1 = 0
Which means -1 * -1 must be equal to 1.
Think of it instead as “opposites”.
-5 x -6 is “the opposite of 5 times the opposite of 6”
So it’s the opposite of the opposite of 5×6.
This is a bit beyond ELI5, but I guess I will go for it.
So let’s think about what a negative number is. A negative number is a number so that when you add it to its positive counterpart, it adds up to zero. Due to the structure we are dealing with (technically called a ring), we also force that there is **only one such number.** So the number that you have to add to number *a* to get zero is *-a.* This also implies, since we have to add *-a* to *a* to get zero *-(-a)=a.*
So let’s take it a bit more general. We are going to focus on multiplying two numbers. So let’s now consider the following addition
*(-ab + (-a)(-b))*
But using the distributive property, we can rewrite this.
*(-ab + (-a)(-b)) = ((-a)b + (-a)(-b)) = (-a)(b + (-b))*
But *-b* **means** the number that when we add it to *b* we get zero. So we end up getting
(-*ab + (-a)(-b)) = (-a)0 = 0*
So we have shown that (*-a)(-b)* is precisely that number that when we add it to *-ab* we get zero. But by definition, that number is *ab*, so
*(-a)(-b) = ab*
Q.E.D.
It doesn’t have to be. There’s a book by Alberto A. Martínez called “Negative Math: How Mathematical Rules Can Be Positively Bent” that shows you can build a consistent mathematical system where a minus times a minus equals a minus:
https://press.princeton.edu/books/paperback/9780691133911/negative-math
the best way I’ve heard it explained is to pretend it’s instructions for walking on a number line. The first number determines whether you’re facing left or right and the second number determines whether you’re walking forward or walking backwards. so
3 * 3 = 9 you’re facing right and walking forward so you’ve walked right on the number line
-3 * 3 = -9 you’re facing left and walking forward so you’ve walked left on the number line
3 * -3 = -9 you’re facing right but walking backwards so you’ve walked left of on the number line
-3 * -3 = 9 you’re facing left but walking backwards so you’ve walked right on the number line
If you think of the two numbers in a multiplication as (A) the size of step you take on a line and (B) the number of steps you take on a line then…
2×5 means each step is two units long, and you’re taking 5 steps, so you move forward ten units.
-2×5 means each step is two units long, but the minus means you’re pointed the opposite direction on your line. You still take 5 steps. So you move ten units in the opposite direction.
2x-5 means you’re facing the original direction and your steps are two units long. But this time each of the five steps you take is a backwards step. So you move ten units backwards.
Finally…
-2x-5 means you turn to point yourself backwards for your two unit steps but then you take 5 backwards steps. This moves you ten units just the same as if you had just walked forward to begin with.
TLDR: “walk forward” moves you to the same place as “turn around and walk backward” as long as the number and size of the steps are the same.
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