# 10-millionth rational in McDonald scheme?: A Loophole In Cantor's Argument?

Thu May 1 14:37:21 CEST 2003

```-Newsgroups: sci.math
-Date: 1999/05/02
-Help.1-1 Map from Q onto Z+. (fwd)
- Date:   1998/03/03
- Forum:   sci.math

The set of Rationals, Q, comprises integers -1, 0, +1
plus all continued fraction convergents pn/qn > 0, ..  > 1.
pn/qn = [a0,a1,a2,...,an]
=  a0 +1/(a1+1/(a2 +... 1/an))
with all ai > 0
and final an >= 2, n >=0. #######

Together with the additive and multiplicative inverses of
pn/qn.

A 1-1 map F from Q onto the set of all positive integers  Z+ is

[-infinity -> 00    ?]

-1   -> 01
0    -> 10
+1   -> 11

(3)x 1s ...  (7-1)x0s, binary

Append "sign / inverse" for 4 subintervals of
real number line,

[-infinity, -1)  append "00"
[-1,0)             "01"
[0,+1)             "10"
[1,+inf)           "11".

Where 1 means 'to the right of'
and 0 means   'to the left of.'

Has this map been proposed before?
I have known of it for 30+ years. Help.

Surreal nos.  Farey Sequence.

E.g. [1,2,3,4]  -> "1 00 111 000"  + "sign/inverse".

subj:  M(n)=2^n-1,Mn-th rational=(n-1) in McDonald scheme.
date:  02.05.99  13:59  pm.  nzst.
ngs:   sci.math
ref.   actrix.Mn->n-1.

M(n) = 2^n-1,  Mersenne numbers nos.

Ex.  M(6) = 2^6-1 = 63.
Mn maps to ->  (n-1).

63-rd rational.
Binary Repunit  111 111      ->  continued fraction  [6-2+1]
"sign-inverse"

1-1 Map Q onto Z+  (Mar. 1998.)

10-millionth rational  ->  - 8220 /  5831.
cf  =  [1,2,2,3,1,2,1,1,2, 1,1,7-2+1].

Continued fractions; a code for hashing your PIN no.
BEEBLET 1987?88?    BBC microcomputer User Group of NZ (Inc.) newsletter.

63/*3  hutchison rd
Wellington 2, New Zealand.

/ don.  (loto)
--