COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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countability of the rational numbers
The origin at user(0,0)
© Samuel Dagan
Countability
n =1
n =2
n =3
n =4
n =5
m =
1
2
3
4
5
The blue dots represent
positive
rational numbers: r = n/m .
The numbers extend beyond
n =5
and m =5,
but are not plotted.
All the dots are connected by a
path starting at r = 1/1 .
The non-reduced numbers are
redundant, therefore - eliminated.
By ordering the positive rational
numbers along the path one obtains that
they are countable.
One can also extend the countability
for all the
rational numbers by making the
following changes:
Start from zero, continue
with the positive numbers,
but after each one add the matching
negative one, obtaining:
0, 1, -1, 2, -2, 1/2, -1/2, 1/3, -1/3, 2/3, -2/3, 3/2, -3/2, 3, -3, 4 ...
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