dg wrote:
Sorry, but 5/0 is "undefined" in mathematics. from calculus, the limit of c/x as x approaches 0 is infinity.
True that, by definition, the division by zero is not allowed. It's the multiplication by zero that results in zero.

Permit me to show that (-1) x (+1) = -1:

1) let A = -1

2) then (A/A) = +1

3) thus (-1)/(-1) = +1

4) after multiplying both sides by (-1) yields (-1) = (-1) x (+1) 

5) since also (-2) = (-1) x (+2) and (-n) = (-1) x (+n) we can therefore conclude that any positive number when multiplied by a minus 1 will yield a negative answer.

Also, taking the result (-1) = (-1) x (+1),
we can divide both sides by (-1) to obtain (-1/-1) = (-1/-1) x (+1), thus (+1) = (+1) x (+1) or 1=1, which confirms the proof.
Yes, we can logically determine that any number divided by itself equals 1. But if that is the case, 0 / 0 = 1, and as such, 0=1. This is why the division by zero is not defined.

And your equation here is not exactly proof, but an explanation of the convention through which a negative number is considered to be a positive number multiplied by -1 to make the use of negative numbers possible. It logically makes sense, if you consider, for example, that if you owe 5 people 10 dollars and have zero money, then your worth is -50. The debt, however, is expressed positively as that equal 50$.