Is there a proof od negative numbers?

[Geekchick]

Hi everyone!

This may be a stupid question but is there a proof of negative numbers? Specifically why does this work -a(-b)=c I was trying to explain it to someone and drew a blank.

Thanks!

 

[JG89]

Are you asking why two negatives multiplied produces a positive? If so, here's a proof I once read:


Let x = ab + (-a)(b) + (-a)(-b)


Then, x = ab + -a(b-b)
= ab -a(0)
= ab - 0
= ab

However, we also have x = b(a -a) + (-a)(-b)
= (-a)(-b)

Therefore, ab = (-a)(-b). QED.

Hope that's what you wanted.

 

[Geekchick]

Thanks i just couldn't logically reason it!

 

[robert Ihnot]

1 is the identity and -1 can only change the sign. Since (-1) ab = -ab. Thus the absolute value of (-1)(-1) = 1. So we have only the choices (-1)(-1) = 1 or
(-1)(-1) = -1.