[FRIAM] Excellent Beauty

[Jon Zingale]

Dave,

you write:
*The third discussion is of the "Rarity of the Commonplace" — how the
mundane reality about us is so improbable. Part of this discussion deals
with causality (another frequent FRIAM topic, but not so much recently),
part with the fact that only 4% of the Universe is ordinary matter, and
part that is corollary to the discussion of infinities. Part of the latter:
a Real Line is made up of the rational and the irrational numbers, both of
which are infinite in number. The rationals are infinitely dense and so
take up zero percent of the Real Line.  Again, mathematicians, a bear of
little brain asks for an explanation.*

What it means to say that the rationals are infinitely dense on the real
line,
is say that in any neighborhood of a point on the line you will find
infinitely
many rationals. Despite their fecundity, there are still infinitely more
non-rationals on the line (by Cantor's argument). Still, by Diophantine
Approximation we can formally talk about how well we can approximate any
real number with a rational. There is a killer book by Ivan Niven on the
subject. For the record, when you ask for a random number between
zero and one and another gives you back a number you can imagine,
you can reasonably assume that other is biased :)

Jon
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