Eli5 Multiplying Negative Numbers

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I have always heard that multiplication is fast adding of the same number. This made sense until we got to negative numbers. Why would two negative numbers become a positive number when they are multiplied if it is indeed just adding the same number? I have asked several people this and never got a straight answer. Please help me!

In: Mathematics
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Multiplication being repeated addition isn’t the definition of multiplication, it’s just a metaphor to explain its function. That metaphor doesn’t work when referring to the product of two negative numbers.

4 x 4 = take four steps forward, do this four times.

-4 x 4 = take four steps backwards, do this four times.

-4 x -4 = take four steps backwards, but *turn around* and do this four times.

I don’t know how to put it in really simple terms but you can’t multiply a negative number by a negative number as the negative signs cancel each other out.

> multiplied if it is indeed just adding the same number

Because if the number is negative you don’t add, you substract.

so for example -1 * -4.

We need to add `-1` *minus four times*: aka substract -1 four times.
So `-1 * -4 = – (-1) – (-1) – (-1) – (-1) = 1 + 1 + 1 + 1 = 4`

Let’s say you and I are doing business. I sell you some items at some price. The value of the transaction would be amount * price. But what if instead of me selling you items, you sold me items? We can just make the amount negative. Suddenly the result of the equation is a negative value, which means money is going in the opposite direction.

But then, what if instead of buying items, it was a refund. The price goes from positive to negative, and money is flowing back in the original direction.

I think the other answers here cover to ELI5 part, so I’ll take it a step further and prove that (-a)*(-b)=a*b for any two numbers a and b. To do this I first show that (-a)*b=-a*b:

(-a)*b + a*b = (-a+a)*b = 0*b = 0,

so subtracting a*b on both sides gives (-a)*b=-a*b. Now since -(-x)=x for any number x, we get

(-a)*(-b)=-a*(-b)=-(-a*b)=a*b, and we’re done.