Proof of -*- = +

Product of two negative numbers always gives a positive number...

How ?

Here how it is.... "-" always means the opposite direction..

Consider the following number line

----|-----|-----|-----|-----|-----|---...
-3 -2 - 1 0 1 2 3 4 5 6

Now we know that what multiplication means...

2X3 means 2 added 3 times 2+2+2=6

Incase of a negative number
-1X2 means -1 added 2 times -1-1=-2

Incase of both the negatives...
-1X-3 is going in the opposite direction on the number line

----|-----|-----|-----|-----|-----|---... (-3 times from 0)
-3 -2 - 1 0 1 2 3 4 5 6
|____________|

So, 0-(-3)=3

graphical explanation

Direction

It is all about direction. Remember the Number Line?

Well here we have Baby krishna taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 3 x 2 = 6 steps forward:

Now, Baby krishna can also step backwards (he is a clever little guy). His Dad puts him back at the start and then krishna steps backwards 2 steps, and does this three times:

Once again krishna's Dad puts him back at the start, but facing the other way.krishna takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:

Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:

So, by walking backwards, while facing in the negative direction, he moves in the positive direction.

And That's All !

and finally

Axiom based proofs

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

2) 0*a = 0*a + 0 = (b + -b)*a + ba + -(ba) = (b+ -b + b)a + -(ba) = ba + -(ba) = 0

3) a*-a = a*-a + 0 = a*-a + (a*a + -(aa)) = a(-a + a) + -(aa) = a*0 + -(aa) = -(aa)

4) -a*-a = -a*-a + 0 = -a*-a + (a*-a + aa) = (-a + a)*-a + aa = 0*-a + aa = aa