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Question
Hi there sir. How are you? I would like to know - why is a negative times a positive a negative and why is a negative times a negative a positive? Is there a nice concise proof for this or is it just a mathematical law and just is so. Thank you for reading.

Answer
Hello, Aubreyepain.

I'm sorry for the delay in responding.

A negative number times a negative equaling a positive number can (and a negative times a positive equaling a negative) is determined to be a convenient mathematical convention.

Let's begin with a negative times a negative equaling a positive number.

What is -3 x -3? The answer is 9, but why?

First, we can write (again, based on an agreed-upon convention), that -3 = -1(3). (This is sort of like arguing in a circle, because I'm taking it for granted that -1 x 3 = -3, but the reason for that is to be shown in a second.)

3(-1) x 3(-1) = ?

(-1)(-1)(3)(3) = ?

(-1)(-1)(9) = ?

-1 x -1 has been decided to be equal to 1, for the reason that the distributive property for multiplication wouldn't work otherwise.

Example: -1(-1 + 1) = (-1)(-1) + (-1)(1)

-1(0) = (-1)(-1) - 1

If (-1)(-1) = -1, then (-1)(0) = -1 - 1, or 0 = -2. This is clearly false, so (-1)(-1) must be equal to +1.

If it is, then -1(0) = 1-1, or 0=0, which, of course, is true.


The same reasoning can be applied to the question of why a negative times a positive equals a negative.

I hope that this helps.

John

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