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Arbitrariness of pi
Posted:
Mar 3, 2013 5:29 AM
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There's an exceedingly tedious argument going on at
http://en.wikipedia.org/wiki/Talk:Pi#It_is_time_to_move_Tau_out_of_the_In_popular_culture_section
about the "tauist" movement, which argues that if only we used a
symbol for 2pi instead of pi, the world would become numerate,
textbooks would be thinner and everything would be free. Well, this is
all a bit silly.
But it seems to me that there is an important point somewhere: it is
rather arbitrary to use a symbol for periphery/diameter, rather than
periphery/radius, or the ratio of the curved arc of a 60-degree 'fan'
to its side. In other words, using a symbol for any reasonable "small
fraction"** of pi would give us exactly the same mathematical
expressiveness, with only notational changes, and for many of these
variants there would be cases (formula for the volume of a sphere,
anyone?) in which the different symbol gave a simpler form.
This does not seem to be true of e, for example, and obviously isn't
true of i. (I suppose we could have a name for the fourth root of -1,
and square it all the time?) I have an intuition that the
transcendence of pi is related to this, because in algebraic terms,
extending the rationals (or the algrebraic numbers for that matter) by
any rational multiple of pi gives exactly the same structure. (I can't
give a simple justification why this shouldn't also imply that e could
be similarly known as f/2, except that in practice e doesn't keep
appearing with small fractional multiples.)
Can anyone help me write this more precisely, and (optimally) as
understandable as possible to people who have no idea what a field is?
I want to add a dispassionate bit about how the choice of 'pi' is
rather arbitrary, even though historically it seems to have arisen at
least twice (?? I think) independently, in Babylonia or whatever, and
in China. Any suggestions appreciated.
Brian Chandler
(** What is the word for a rational number with small numerator and
denominator...?)
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