Re: Rational numbers, irrational numbers: each dense in real numbers

Among the notions of why there are "more" irrationals than rationals
is the (not fundamentally) heuristic notion that if one were to try to
sample at uniform random from the real numbers in [0,1] by flipping
fair coins (independent Bernoulli trials) to form the binary expansion
of a real number, that it is extremely unlikely to have the sequence
terminate in ending with all zeros or ones, or some repeating
sequence.

Consider this then, besides that each particular sequence has the same
probability of selection as any other which is among reasons whyit is
said that there exists a uniform probability distribution of the reals
of the unit interval, that a random sample of the second binary digit
(bit) of a real number's expansion is as well independently a sample
of another real number's first digit. Then, for the i'th binary
sample, there is a bag (multiset) of i random real numbers, to the
precision i, i-1, ..., 1. In sampling one real number, infinitely
many _not-necessarily distinct_ values are sampled.

Then, if for example zero was a sample from the unit interval of
reals, it's sampled infinitely many times. If a rational is sampled,
then its trailing repeating sequence is sampled infinitely many
times. If the number terminates (in zeros) then zero is sampled
infinitely many times. If an irrational is sampled, then infinitely
many distinct irrationals are sampled. So, where it might seem that
the probability of a rational being sampled is infinitely smaller than
that of an irrational being sampled, once sampled that rational number
represents infinitely much more about the population than a particular
irrational number being sampled, as an artifact of the process it
recurred infinitely many times.

Then, particular rationals of smaller numerators, after one and zero,
and denominators, as well in these infinite expansions have various
considerations of why they are sampled more strongly upon occurrence,
with more weight, in partly casual terms.

Consider further, with regards to this notion of an existence of an
injection from an uncountable subset of the irrationals to the
rationals, an alternate proof:

http://groups.google.com/group/sci.math/msg/f218848b9bbe4830

Borel vs. Combinatorics, anyone?

Ross

--
Finlayson Consulting

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Relevant Pages

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  • Re: Rational/Irrational Numbers
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