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

[Don McDonald]

-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 
 
22/7 = [3,7]  -> "111 000 000"  + "sign inverse" 
 (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" 
 ->  +5.   Answer Whole number. 
 
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) 
--
don.mcdonald at paradise.net.nz
formerly   don.mcdonald at welcom.gen.nz, mcdonald_d at kosmos.wcc.govt.nz
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