> 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,
Around here somewhere is your Assumption number 1: There exist nonzero integers x and y for which x^2 + y^2 = 0.
> 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.
This is your Assumption number 2.
> 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! > > Why? > > 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.
This is Assumption number 3. You don't know if you can work with this object (call it sqrt(-1), call it i, call it the Harris constant, I don't care) in the same way you can work with the integers.
> ***That leads to a contradiction.***
Maybe so, but which of your three assumptions is untenable? You'd like it to be assumption 1, but you (YOU, James Steven Harris!) have to rule out the other two possibilities. Otherwise your proof is incomplete, as in: unfinished, not done, containing a gap.
> 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.
No. What people are saying is that once you realize that this thing, sqrt(-1) is not an integer, you must stop, and define (or state) what you mean by the operations x + sqrt(-1)*y, x - sqrt(-1)*y and (x + sqrt(-1)*y) * (x - sqrt(-1)*y). And you have to _prove_ for these objects (because they are NOT integers) that you can infer from (x + sqrt(-1)*y) * (x - sqrt(-1)*y) = 0 that either x + sqrt(-1)*y = 0 or x - sqrt(-1)*y = 0. And when you do _that_ proof, you have to make damn sure that it does not somewhere rely on another statement, namely that x^2 + y^2 = 0 has no solutions in the positive integers. Otherwise, your argument is circular.
> ***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! >
But why don't you then simply write x^2 + y^2 = 2*[(x^2)/2 +( y^2)/2] and point out that this pushes you out of the integers already? Remember, no one forces you to introduce sqrt(-1) into the proof, so the only one doing the pushing is yourself.
> 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!!! >
OK. The example has been given several times before. Use the 2x2 matrices and embed the integers in them as follows:
If x is an integer, associate with it the matrix | x 0 | | 0 x |
Addition is done componentwise, like ordinary matrix addition:
| x 0 | + | y 0 | = | x+y 0 | | 0 x | | 0 y | | 0 x+y |,
multiplication is also ordinary matrix multiplication. (I can spell it out for you, but I assume that you can do matrix multiplication on your own.)
Verify for yourself that these objects behave exactly like the integers with respect to addition and multiplication. The two rings (of integers and matrices of this form) are isomorphic, and it does not matter which representation you work with.
Now define the symbol i = | 0 -1 | | 1 0 |.
Verify that i*i = -1. (Again this can be spelled out if you wish.)
Verify further that (x+iy) * (x-iy) = x^2 + y^2.
Accept (or verify for yourself!) that the set of all 2x2 matrices with integer coefficients is a ring with the above operations.