A real proof: Two negatives equal a positive

[Noumenon84]

let x = a*b + -a*b + -a*-b

x = b( a + -a) + -a*-b
x = b(0) + -a*-b
x = -a*-b

x = a*b + -a(b + -b)
x = a*b + -a(0)
x = a*b

-a*-b = a*b



I figured for every paradoxical "proof", the world needs to be balanced out by an actual proof.

[The Drunken Actuary]

Thanks!

[Bamafan]

how do you know that a*b is positive?

[Noumenon84]

To prove anything in life, nay to have any system of logic you must start with axioms that need no proof. However, this one can be empirically demonstrated from the definition of multiplication. Therefore it needs no thorough logical argument.

[The Drunken Actuary]

Originally Posted by Noumenon84

To prove anything in life, nay to have any system of logic you must start with axioms that need no proof. However, this one can be empirically demonstrated from the definition of multiplication. Therefore it needs no thorough logical argument.

What does it mean to multiply a negative number by a negative number? I mean, you know, by the definition of multiplication.

Originally Posted by Noumenon84

To prove anything in life, nay to have any system of logic you must start with axioms that need no proof. However, this one can be empirically demonstrated from the definition of multiplication. Therefore it needs no thorough logical argument.

Are you just stringing together words because you like the way they sound? Because otherwise what you said makes no sense at all, and just makes you sound like an ignorant ass trying to trick people into thinking that you know what you're talking about. Seriously, who do you fool with this kind of crap?

[Vorian Atreides]

So what is the definition of multiplication in the case of -a*-b? For simplicity, assume that both a and b are positive.

[Vorian Atreides]

I would also like a proof that a value that is right multiplied can be left-factored.

And if you think that what I wrote is garbage, just consider matrix multiplication.

.

[Vorian Atreides]

Originally Posted by Noumenon84

To prove anything in life, nay to have any system of logic you must start with axioms that need no proof. However, this one can be empirically demonstrated from the definition of multiplication. Therefore it needs no thorough logical argument.

1. It's not that the axiom needs no proof but rather they're assumed to be true (or given). If it so happens that one axiom is the logical result of the other axioms in the system, then at least one of the axioms is redundant and should be removed as unnecessary luggage.

2. So what are the axioms from which you're working?

3. And (as someone else queried), how does one demonstrate empirically the multiplication of two negative numbers?

A counterexample:

Quaternions:

i^2 = j^2 = k^2 = ijk = -1

=>

(-j)*(-i) = -k