On Tue, 16 Jan 2001 23:04:15 GMT, firstname.lastname@example.org wrote:
>> > (x+sqrt(-1)y)(x-sqrt(-1)y) = x^2 + y^2 = 0, so >> > >> > x = sqrt(-1)y *or* x = -sqrt(-1)y. >> >> James, you have to *prove* that last step. You cannot rely on the >standard >> proof because it uses: x^2 + y^2 <> 0 whenever x != 0 or y != 0. > >I think some people may be surprised that you are essentially calling >this a gap.
Quick poll: Anyone who is surprised that this is being called a "gap" (remove the word "essentially"), chime in.
> >My comment to them is that it's the same thing being done with the >proof of Fermat's Last Theorem.
Yes. This proof was introduced into this forum because the logical fault is similar.
> >As for your other statement, it doesn't make any sense to me. > >What's going on is simple though. I have that x^2 + y^2 = 0, and I >know that x^2 + y^2 = (x+sqrt(-1)y)(x-sqrt(-1)y),
You don't know this unless you define the context in which you are operating.
> so > > (x+sqrt(-1)y)(x-sqrt(-1)y) = 0,
> so > > x+sqrt(-1)y = 0 or x-sqrt(-1)y=0,
And you don't know this automatically. > >and that goes to the question of how long a proof has to be.
No, it goes to the question of whether a statement preceded by the word "so" actually follows logically from the statement that precedes it.
> >As you can see, some people will force you to outline simple steps that >others would find unnecessary.
And the person providing the proof should be able to outline those steps, even if he/she finds it unnecessary.
So humor us. Even though it's unnecessary, prove that
ab = 0, for complex a, b (they're COMPLEX, not integers, OK?)
a = 0 or b = 0.
It's not an axiom, it's a theorem. So it's provable. Just for grins, offer up a proof. Even though it's unnecessary. Waste a few electrons on us. It can't hurt.
> >That bit of arbitrariness is used by some people to claim that a proof >isn't.
So isn't the obvious answer to add a few lines to prove the assertion? If you did, everyone would shut up. Saying "I don't have to" doesn't do anything to answer the objections. What's the harm in explaining what mathematical principle justifies the conclusion?
> >> >> > First off, it's worth noting that I'm treating the sqrt(-1) as an >> > *operation*. I think some of you live under the false notion that >> > something like sqrt(2) is the actual number. >> >> I must be dense, but what the heck do you mean? If it is not a number >> you have to define how to do multiplication of operations with >numbers. >> > >The sqrt(2) is a representation of the number that multiplies times >itself to give 2 (notice how circular that is).
That makes it a number, not an operation. It stands for the NUMBER. That is not at all circular.
>I think I should mention that there's also a question of trueness.
Don't know what you mean by this, except that perhaps you are looking for a vote on the truth value of an assertion.
In mathematics, normally a sequence of logical deductions serves to demonstrate truth. Got that?
SEQUENCE of LOGICAL deductions.
For instance, please offer a sequence of logical steps showing that for two complex numbers a and b, ab=0 implies a=0 or b=0.
Even a one-liner would be better than saying "it's obvious" over and over. Here, I'll start you out: "We can conclude that either a=0 or b=0 because... "