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Topic: More Cantor
Replies: 41   Last Post: Aug 11, 1999 1:57 PM

 Messages: [ Previous | Next ]
 Jon Haugsand Posts: 26 Registered: 12/12/04
Re: More Cantor
Posted: Aug 8, 1999 5:13 AM

* vmhjr@frii.com
>
> Obviously, one cannot give any examples of non-constructible numbers.

Here is one. Define pi_hat as follows:

1. If, in the decimal expansion of pi, there are ten thousand zeroes in
a row (i.e. if pi=3.14......00000..000...):
(a) if the decimal following the ten thousandth zero is 5,6,7,8 or
9 pi_hat=3.14....0000..001, that is raise the the thousandth
zero to 1 and take no more decimals (making pi_hat a
terminating decimal).
(b) if the decimal following the ten thousandth zero is 0,1,2,3 or
4 pi_hat=3.14....0000..00, that is make it equal to the
decimal expansion of pi stopping by the ten tousandth 0 (making
pi_hat a terminating decimal).

2. If, in the decimal expansion of pi, there are no ten thousand
zeroes in a row, pi_hat=pi.

Pi_hat is constructible in the sense that you can calculate its
decimal expansion. It is also meaningful.

However, the number rho = pi-pi_hat is not constructible. What is
its first decimal? And is it greater than, equal to, or less than
zero?

I find it, however, meaningful.

(Example found in the excellent book "The Mathematical Experience" by
Davis and Hersh.)

--
Jon Haugsand
Norwegian Computing Center,
<mailto://haugsand@nr.no> Pho: +47 22852608 / +47 22852500,
Fax: +47 22697660, Pb 114 Blindern, N-0314 OSLO, Norway