In article <email@example.com>, firstname.lastname@example.org (Randy Poe) wrote: > On Tue, 16 Jan 2001 23:04:15 GMT, email@example.com 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. >
Ok folks, here you are hearing that strange statement again.
I want to emphasize that this guy is arguing that what I have above is not a proof.
> > > >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. >
See folks? I just don't get why I have to repeat over, and over, and over again that x and y are integers.
Given that x and y are integers, how many of you out there have any doubt that x^2 + y^2 = (x+sqrt(-1)y)(x-sqrt(-1)y)?
Well, this guy certainly seems to doubt it, and not only that he's doubts it so much that he decided he'd post that doubt with emphasis.
Remember, he's taking this position as part of saying that my proof of Fermat's Last Theorem is wrong.
That's the big picture, and you should keep it in mind.
> > 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?) > > implies > > 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.
It doesn't matter whether or not a and b are complex or not.
The only possible issue might be made by using something like 3(2)=0 (mod 6), but we're not talking about a finite ring.
As for me proving that if ab = 0, a or b equals 0, I'm not interested in doing that.
If you need that proof to understand what I'm talking about, oh well.
I'll just worry about the people who don't have a problem with that.
> > > >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? >
I did that. You folks just keep asking for more and more detail, like you asking for me to prove that if ab = 0, a or b = 0.
> > > >> > >> > 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.
Isn't it? Like I've said, we don't put down 1+1 to represent 2. The '+' is an operator. The sqr() is an operator. We use the operation to represent the square root of 2, or we write 1.414..., or something similar.
Why you would argue such a simple point is beyond me.
> > >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. >
You don't have to be condescending *and* obnoxious.
I think most folks out there can conceive of a statement as being either true or false. 1=2 is a statement that is false. 2 = 2 is true.
Again, why you would argue over something this simple is beyond me.
And I think it pertinent that you deleted out the statement that was being referred to.
> 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. >
And now we have that again...
> 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... " >
If you want to argue about whether or not ab = 0 means that a = 0, or b = 0, then go ahead, but you'll be doing it without me.