User:Reza.mostafavi

Proof of why a negative real number multiplied by a negative real number is a positive number.

We know that when a positive number is multiplied by a positive number the result is a positive number i.e.

(+ 2) * (+ 3) = + 6 			                              (i)

let’s see what happens when one of the numbers is a negative number, for example the following:

(-2 )*(+ 3) = ? 6.

The result of the above is either + 6 or -6. Let’s assume that the result is + 6 i.e.

(-2 )*(+ 3) = + 6, but from equation (i) we know that (+2) * (+3) = +6. We have the following so far:

(- 2)*(+ 3) = + 6

(+ 2)*(+ 3) = + 6

The right hand sides of the above two equations are the same therefore the left hand sides must be equal to each other i.e.

(- 2)*(+ 3) = (+ 2)*(+ 3).

The above implies that (-2) = (+2) and this can not be the case therefore the following is true:

(- 2)*(+ 3) = - 6				(ii)

The same is true of (+ 2)*(- 3) = - 6.

Now let’s see what is going to happen to the following:

(- 2)*(- 3) = either + 6 or -6. It can not be anything else.

Let’s assume that the result is -6 i.e.

(- 2) *(- 3) = - 6

Comparing the above with equation (ii) we have:

(- 2) *(- 3) = - 6

(- 2) *(+ 3) = - 6

In the above equations the right hand sides are the same and therefore the left hand sides must be equal i.e.

(- 2) *(- 3) = (- 2) *(+ 3)					(iii)

In equation (iii) we have (- 2) on both sides and therefore for these two equations to be equal we should have (- 3) = (+3) and this is certainly not true and therefore

(- 2) *(- 3) = + 6

The above implies that the product of two negative numbers is always positive (QED).