If multiplication is just repeated addition, then why when 2 negatives get multiplied they become a positive?
In: Mathematics
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.
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