If multiplication is just repeated addition, then why when 2 negatives get multiplied they become a positive?
In: Mathematics
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.
When you add a number more times, the result gets bigger. When you add a number fewer times, it becomes smaller. When you add a number a negative number of times, it crosses past 0 onto the other side of the number line. So, (counting down from 2 by -3 you get -6, -3, 0, +3, +6, …). Minus 2 groups of minus 3 balls is plus 6 balls.
Because you’re adding… negatively.
-2 x 2 = -4. Because that is -2, twice: -2 + -2 = -4.
-2 x -2 = 4.
**Not** because -2 + -2 = -4, which is the opposite of what the second -2 is telling us to do. But because 2 + 2 = 4.
It’s not subtraction (after all, -2 minus -2 is 0), it’s adding negatively.
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