If multiplication is just repeated addition, then why when 2 negatives get multiplied they become a positive?
In: Mathematics
I actually tried to explained this idea to an actual 5 year old (okay, 6 year old) with: temperature and time travel!
Multiplication is repeated addition. A x B is B plus B plus B … A times. You can see it as having a machine that do “plus B” to the temperature of the house, every second, and you wait A seconds to get the right result.
Let’s do 3 x 4. It’s “plus 4”, “plus 4”, “plus 4”: so plus 12. Okay, it’s getting warm.
Let’s do 3 x -4. It’s “minus 4”, “minus 4”, “minus 4”: so minus 12. Okay, cool.
Now, what if we rewind time. Instead of waiting A seconds, we go back A seconds early. This is what happen with a negative A.
Let’s do -3 x 4. It’s undo “plus 4”, undo “plus 4”, undo “plus 4”: so undo plus 12, aka. minus 12. Okay, the revert if getting warm is getting cool!
Let’s do -3 x -4. It’s undo “minus 4”, undo “minus 4”, undo “minus 4”: so undo minus 12, aka. plus 12. It’s getting warm!
The only issue with that explanation is that it feel like multiplication is not symmetric, even though it works perfectly fine either way.
A bunch of mathematicians, such as [this guy](https://www.maa.org/external_archive/devlin/devlin_06_08.html), are adamant that multiplication is in fact not repeated addition, but that multiplication makes for a shortcut for doing repeated addition.
Adding 2 is like taking two steps forward.
Adding -2 is like taking two steps backward.
Doing 2*2 is saying add 2, twice. This is saying to take 2 steps forward, twice. So effectively, you took 4 steps forward, i.e. 2*2 = 4
Doing -2*2 is like saying turn around (because of the negative), then take 2 steps forward twice. So effectively, you took 4 steps backward, i.e. -2*2 = -4
Doing 2*-2 is like saying take 2 steps backward twice. So effectively, you took 4 steps backward, i.e. 2*-2 = -4
Doing -2*-2 is like saying turn around, then take 2 steps backwards twice. So effectively, you took 4 steps forward, i.e. -2*-2 = 4
If multiplication is just repeated addition, how can you multiply by a negative number at all?
Saying that multiplication is repeated addition works fine when you’re just talking about positive integers, but how can you repeat something a negative number of times? That doesn’t make any sense.
So, what mathematicians do about that is they say “What properties does repeated addition have, and how can we preserve those properties when using other numbers?”
One property that multiplication has is that x*(y+1) = x*y + x. This holds for all positive integers. For example: 12 = 4*3 = 4*(2+1) = 4*2 + 4 = 8+4 = 12.
And you can use that property to extend multiplication into the negative integers. For example: 0 = 4*0 = 4*(-1+1) = 4*-1 + 4 = *?* + 4 = 0. So, from this we know that 4*-1 is equal to some number that when you add 4 to it, you get 0. What number is that? -4. (I assume you’re comfortable with adding and subtracting negative numbers.)
And you can keep going. 4*-1 = 4*(-2+1) + 4 = 4*-2 + 4 = *?* +4 = -4. So, 4*-2 is equal to some number that when you add 4 to it, you get -4. What number is that? -8.
That’s with one negative number. What if both numbers are negative? Well, let’s try the same thing.
0 = -4*0 = -4*(-1 + 1) = -4*-1 + -4 = *?* + -4 = 0. So, -4*-1 is equal to some number that when you add -4 to it, you get 0. What number is that? 4. Not -4, because -4 + -4 = -8, not 0.
It’s a double negative.
Think of it like words:
He IS smart = positive
He is NOT smart = negative (has 1 negative)
He isn’t not smart = positive (has 2 negatives)
2 x 2 = 2 + 2 = 4 = 4
2 x -2 = -(2 + 2) = -(4) = -4
-2 x 2 = -2 + -2 = -2 – 2 = -4
-2 x -2 = -(-2 + -2) = -(-2 – 2) = -(-4) = 4
Try it with 3s
3 x 3 = 3 + 3 + 3 = 9
3 x -3 = -(3 + 3 + 3) = -(9) = -9
-3 x 3 = -3 + -3 + -3 = -3 – 3 – 3 = -9
-3 x -3 = -(-3 + -3 + -3) = -(-3 – 3 – 3) = -(-9) = 9
At least that is how I always envisioned it
Say you pay $2 for something. You do this twice, so you lose $4. This is -2*2=4.
But there was a mistake, it was supposed to be free! You’ve already lost your $2 (-2), but you’re going to get to undo those losses (-2). That’s -2*-2=4, so you get your $4 back.
If you have a debt of $2, and I take that debt away from you twice, how much money did I give you?
Let’s look at a few examples.
2 * 3 = 2 + 2 + 2
2 * -3 = -2 + -2 + -2
-2 * -3 = (-)(-)2 + (-)(-)2 + (-)(-)2
The two minus signs cancel each other out, so you’re left with
-2 * -3 = 2 + 2 + 2
If something **is** positive, you could say it is *not* **not** positive.
If you say `x * y` means that you add the number `x` to the result `y` times, then the interpretation is that if `y` were actually negative then you’re subtracting `x` from the result `|y|` times (the “absolute”, or non-negative version, of y).
So logically if both `x` and `y` are negative, then you’re subtracting a negative number multiple times, and subtracting a negative number is the same as adding a positive number. So it’s like normal multiplication without the negative signs.
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