Re: Peano's space-filling curve

From: Daniel Grubb (grubb_at_lola.math.niu.edu)
Date: 06/10/04 

Date: 10 Jun 2004 18:05:26 GMT

>> >> learn about limits. How come all the numbers
>> >(1,2,5/2,8/3,65/24,...) are all
>> >> rationals, yet their limit e is transcendental?
>>
>> >Because e is not a member of this series of rationals?
>> > It's its least upper bound isn't it?
>>
>> Close. It is possible for an increasing sequence of
>> rational numbers to have a rational limit. But the idea
>> that limits can give you things that none of the
>> approximations give is an important point. In the
>> same way, for the Peano curve, the point (1/3,2/3)
>> is not on any of the approximating curves, but
>> *is* on the limit curve. In fact, every point of the
>> square is on the limit curve. Once you see why
>> (1/3,2/3) is on, it won't be much of a problem to see
>> why the rest are also.

>I think I had more or less got hold of that point a couple
>of posts ago. This now makes it much clearer, as your posts
>usually do. I also slept on it. a trick that seems to work
>really well for me. But I'm still a little bothered by it.
>If we accept for one moment that 'e' is not a member of that
>series of rationals above in the same sense that 2 is a
>member, then..... well you see where I'm going can't you. I
>have no difficulty persuading myself that 'e' =
>1+1+1/2+1/6+1/24+.... because I can reduce the practical
>impossibility of adding up infinite terms to the solvable
>task of truncating decimal representations to any finite
>number of sig.figs. But none of these truncations
>will_ever_give me 'e'.

I think you are homing in on one of the subtleties in the
concept of 'real number'. The actual, formal, definition
of the collection of real numbers is non-trivial. There are
two standard approaches: Dedekind cuts, and Cauchy sequences.
Both are rather abstract. The essential aspect of both is that
'holes' in the collection of rational numbers have to be 'filled'.
They do so in different, but equivalent ways.

An alternative that is often taken is to *assume* that a set
with the required properties exists (a complete Archimedean ordered
field). This sidesteps an important issue but allows time for
other things. Like calculus. :)

It seems to me that the concept of 'limit' is one that you have
to get a better handle on. You have a certain intuitive feel for
it, but you are getting to the place that a more detailed feel
will become crucial for further understanding. Try finding a book
that covers the epsilon-delta definition of limits. It may be hard
going at first, but it is worth it if you want to grasp some of
the things you have been talking about.

>By_the_by, I have no trouble of this sort with calculus. For
>me dy/dx and f'(x) are not identical in every way. If they
>were, we wouldn't need to separate the infinitesimals as in
>dy = f'{X2}(X1).dX1+f'{X1}(X2).dX2 on one hand, and take a
>limit that effectively involves the no-no of dividing by 0
>in order to get an analytic form f' for the derivative on
>the other. We could just assign one of them to oblivion and
>only ever use the other. Whatever the difference is I can't
>quite put my finger on, though it's something like trying to
>put 'e' in a set of rationals. But my "logic" tells me it's
>more than just notational..

Yes. The problem is your intuition about limits, although there
is also some sloppiness on the part of elementary calculus books
and physicists/chemists. A good, logical treatment (say in an
advanced calculus or elementary analysis book) may well clear
things up.

>Also you wrote "Close." at the
>beginning. How close?

Well, the initial question was a bit vague, so you can be excused.
But an example might help. Consider the sequence 1, 3/2, 7/4, 15/8,
31/16, 63/32,.... which converges to 2. The number 2 is not in the
sequence, yet it is not transcendental. So the reason that e is
transcendental is not *simply* that it isn't in the other sequence.
There is more going on. That 'more' is actually considerable.
You were correct about e being the least upper bound, though.

--Dan Grubb

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