Re: ******* TRY THESE SCI.MATH **********

From: Lasse (lasse_rempe_at_yahoo.de)
Date: 01/20/05 

Date: 19 Jan 2005 23:34:24 -0800

Some advice: stop thinking of the real numbers in terms of their
decimal expansions, it only seems to be confusing you. The decimal
expansion only describes a way (one among many) of describe an
approximation to a real number to arbitrary precision.

Also, you seem to be mixing the issues of uncountability of the reals
and computability questions, which are only slightly related.

You already got answers to your questions, as far as they were
well-defined, so I won't bother going through them. You seem to be
asking: is it possible to write a list of numbers which contains
arbitrarily good approximations to every real number? And the answer is
YES: for example, the rationals.

Topologists would say: the real numbers have a countable dense subset
(such spaces are sometimes called separable). This does NOT mean that
the set of real numbers itself is countable - perhaps this is where you
have a problem of understanding?

The issue of computability is worse: you can easily define a single
number whose decimal expansion is not computable: e.g., let the n-th
digit be 1 if the n-th Turing Machine halts, and 0 otherwise. There is
no program which, given input k, outputs the first k digits of this
number.
Hope this helps,
Lasse 

---
(@remove.for.spam.maths.warwick.ac.uk)

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