Name: Mary

Who is asking: Student

Level: Elementary

Question:

I have a question about adding and multiplying positive and negative numbers. When we add two negative numbers the answer is negative BUT when we multiply two negative numbers the answer is positive. I don't understand. Why?

Thank you very much!

Hi Mary,
I am not surprised that this is difficult to understand. Negative numbers have not always been well accepted. In fact they were not always called negative numbers. They have been called fictitious numbers, defective numbers and even absurd numbers.

I think that the difficulty with understanding a negative times a negative is that this is not something we do in our everyday lives. We multiply positives times positives and even positives times negatives. (If you are keeping track of your expenses then you might treat money you earn as positive and money you spend as negative. If so and you buy 3 items, each for $2, then in your records you would enter 3 times -$2 or -$6).

**The reason that, for example, -5 times -3 is +15 rather than -15 is that the choice of +15 makes arithmetic work the way we think it should.** To show you what I mean let me do +5 times -3 first.

Start with the true fact that

-3 + 3 = 0.
Multiply both sides by 5

5(-3 + 3) = 5(0) = 0.
Using the distrubutive law I get

5(-3) + 5(3) = 0
But 5(3) = 15 so

5(-3) + 15 = 0
Thus, whatever the number 5(-3) is, if you add 15 to it you get zero. Thus 5(-3) must be -15. That is 5(-3) = -15.
Now, start again with

-3 + 3 = 0.
This time multiply each side by -5.

-5(-3 + 3) = -5(0) = 0.
Again the distributive law gives

-5(-3) + (-5)(3) = 0.
But we know that (-5)(3) = -15 so

-5(-3) - 15 = 0
Thus, whatever the number -5(-3) is, if you subtract 15 from it you get zero. Thus -5(-3) must be 15. That is -5(-3) = 15.
I hope this helps,

Penny