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Tim Peters wrote:
> [jstevh@xxxxxxx]
> [...]
> For the rest, I must say this is one of the clearest posts I've seen from
> you in years. I don't know what you think the /point/ of it is, but at
> least most of it was intelligible.
Yes, it's clear he _doesn't_ know what a ring is. My attempt at
educating him (posting the definition) seems to have gone unknown.
> > have floundered on the issue of convergence and what it means with
> > rings like the ring of algebraic integers, where I introduce the
> > concept of an exclusionary ring.
> >
> > The concepts are simple, luckily, as I can mostly use something basic:
> >
> > S = 1 + x + x^2 + x^3 +...
> >
> > where the issue is convergence, so that if S converges in whatever ring
> > you're in--notice none given yet--you can go to
> [...]
> > S = 1 + x*S
And now it's obvious that JSH doesn't know anything about infinite
series, either. I mean, come on, this is Calc II we're talking about
here, Jim! Surely you had to take Calc II to get your physics degree!
JSH is making mistakes along the line of the following "derivation" of
0 = 1:
1 = 1 + 0 + 0 + 0 + ...
= 1 + ((-1) + 1) + ((-1) + 1) + ...
= (1 + (-1)) + (1 + (-1)) + ...
= 0 + 0 + 0 + ...
= 0
as well as other similar mistakes which were made 400 years ago.
> > and solve to get
> >
> > S = 1/(1-x)
> >
> > where to get convergence in the ring of complex numbers, which of
> > course is also the field of complex numbers,
JSH has of course forgetten the key phrase "|x| < 1", which is a
requirement for S to converge. Otherwise, we get silly results like
1 + 2 + 4 + 8 + ... = -1,
where a sum of positive terms "adds up" to a negative term.
> > let's choose x = sqrt(5) - 2, considering only the positive solution,
>
> ? sqrt(5)-2 ~= 0.236 /is/ positive. That's like saying "chose x = 3,
> considering only the positive solution". The longer half of it sounds like
> gibberish.
He's still on his "sqrt(x) is actually two numbers" kick again.
> > then in complex numbers, I have
> >
> > S = 1/(1 - sqrt(5))
>
> No, you have S = 1/(1-x) = 1/(1-(sqrt(5)-2)) = 1/(3-sqrt(5)). You should be
> able to see instantly that the expression you came up with for S is
> negative -- which doesn't work all that well since you also want to believe
> it's an infinite sum of strictly positive terms ;-)
He's typing all of this in on the fly, remember? "Thoroughly tested"
for JSH means he pressed the POST MESSAGE button.
> [...]
> > Now I feel confident there are posters who will want to dispute me on
> > the ability to use convergent infinite series in the ring of algebraic
> > integers, but I want more than namecalling, or other childish rants,
> > and I want more than someone saying they read it different in some
> > number theory text.
>
> You can use infinite sums all you like, provided you're careful to use a
> sensible meaning for convergence, and spell out what that is.
Now JSH is really showing his ignorance. Ring theory is introduced in
Abstract Algebra class, not Number Theory.
But what are we to do if JSH can't use infinite series correctly? And
what if JSH refuses to acknowledge this? Or what if (as frequently
happens) he does not reply to a clear refusal of whatever he's on
about?
I gave a link to MathWorld's page on rings, and I even posted the
definition, pointing out that nowhere in it does convergence play a
part. He did not reply to that. (He probably won't reply to this,
because it's one of my posts.)
> [...]
> Which specific result of any established mathematical theory do you believe
> is in error?
Ideal theory.
> Note that I want "more than namecalling, or other childish
> rants", I'm asking a specific question about mathematics. You claim
> established theory is wrong. You say that endlessly. But you never say
> /which result/ in established theory is wrong.
Ideal theory.
> If you can, please name a
> specific theorem that's wrong in your view of the world.
Ideal theory. ["Jimmy wants a cracker!"]
> Or, if you can't
> name a single specific result that's wrong, please explain the basis for
> your claim that math professors are teaching wrong results.
Probably because he flunked out of Calculus and has harbored a revenge
fantasy for years or decades.
--- Christopher Heckman
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