10-millionth rational in McDonald scheme?: A Loophole In Cantor's Argument?
Don McDonald
parabola at paradise.net.nz
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
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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