In article <G7A90y.firstname.lastname@example.org>, "Dik T. Winter" <Dik.Winter@cwi.nl> wrote: > In article <email@example.com> firstname.lastname@example.org writes: > > In article <G78GGv.7ID@cwi.nl>, > > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote: > > > In article <email@example.com> firstname.lastname@example.org writes: > > > > Given x^2 + y^2 = 0, x and y nonzero integers, show that no > > solution > > > > exists. > > > > > > > > Proof by contradiction: > > > > (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. > > I think not. > > > My comment to them is that it's the same thing being done with the > > proof of Fermat's Last Theorem. > > Yup, indeed. You do something you do not prove because it is "obvious", > but it is *not* obvious. You assume that whenever AB = 0 for your funny > things that either A = 0 or B = 0 or both. But you have to give a proof > of that because there are many funny things in mathematics where that > does *not* hold. And it has been *proven* for the complex numbers, but > that proof actually uses that x^2 + y^2 can not be 0 unless both x and > y are 0. Just the thing you want to prove. So because you rely on that > fact for the complex numbers you are reasoning in a circular fashion. It > is like (with either x or y nonzero or both): > x^2 + y^2 != 0 because whenever AB = 0 either A = 0 or B = 0 and the > latter is true because x^2 + y^2 != 0. >
Sigh. Let me remind you of the facts. You are given that x and y are nonzero integers, and that x^2 + y^2 = 0. You notice that
x^2 + y^2 = (x+sqrt(-1)y)(x-sqrt(-1)y), and let me explain that further,
sqrt(-1) here is an object that we can prove is not an integer, but we don't go off and then say that it's complex, because we're in the ring of integers.
What we're doing is noting that if there were this thing that multiplied times itself to give -1, then we'd have that factorization.
We assume that such a thing exists.
At this point that's all I care about.
But you're ready to pull out a textbook on complex numbers. That's not necessary because I know this thing isn't an integer by elementary means, and I don't need to know anything else at this point.
Then I notice that (x+sqrt(-1)y)(x-sqrt(-1)y) = 0 because x^2 + y^2 = 0.
And remember, at this point as far as I'm concerned x and y are still integers!
Because if I know they aren't at this point then I already must have reached the point of contradiction. And then, your argument must be that the proof I've given is too long!!!
So, knowing that I have this odd object that is not an integer, but is *interacting with integers* I follow standard rules for integers.
***That leads to a contradiction.***
What some of you appear to be arguing is that when I realize that this thing, sqrt(-1) is not an integer, I must stop, and pull out a book on complex number theory.
***But I'm doing a proof by contradiction.***
I'm looking for something that pushes me outside of integers because that's what I want to prove must happen!
You say the textbooks do it a different way, and I assume that assaults your sense of order for me to show it this way because if everybody has done it one way, you seem to assume that any other way is wrong.
Then prove it's wrong!!!
So far all any of you have given is convention. Certain human beings have done it one way, and you're saying that by God that's the only way!