1. ## Proof

how do you prove

(-a) * b = -ab

2. Originally Posted by metlx
how do you prove

(-a) * b = -ab
This seems like an application of the associative law:

$(-a)\cdot b = (-1\cdot a)\cdot b = -1\cdot(a\cdot b)=-1(ab)=-ab$

3. Originally Posted by metlx
how do you prove

(-a) * b = -ab
Another proof slightly different from the one given by Resoxfan: une uniqueness of additive inverse: -(ab) is the inverse of ab, and we're gonna prove (-a)*b is also an inverse of ab and thus they're equal:

ab + (-a)b = (a +(-a))b = 0*b = 0.

Tonio

4. Originally Posted by tonio
Another proof slightly different from the one given by Resoxfan: une uniqueness of additive inverse: -(ab) is the inverse of ab, and we're gonna prove (-a)*b is also an inverse of ab and thus they're equal:

ab + (-a)b = (a +(-a))b = 0*b = 0.

Tonio
Infact in the proof by Resoxfa, I think one step needs to be incorporated is that: (-a) = -1.a

Which again will require the use of distributive property (used by Tonio)

The reason I wrote this is that Resoxfa proofs gives an impression that result depends on Associative Property. I think that is erroneous it depends ONLY on distributive property. [Off course you have to convince yourself 0.a = 0as well]. Infact even if associative property was not one of the axioms of Rings the result will still hold true (this is my argument and can very well be wrong)

Will appreciate if any error in my argument can be pointed out. In-these kind of exercises I think it is important to realize which is/are the base axioms that result in the property we have to prove. And I feel this is by no means a trivial task (more so because of our old habits we pick in school years)
Thanks

5. I'm sorry for confusing everybody. I was infact wrong - we will need assosciative property as well. Existence of unique inverse (which is used by Tonio and will be needed in this prove) require associative axiom. Sorry again
Thanks