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Paul
Posts:
676
Registered:
7/12/10


Re: Arbitrariness of pi
Posted:
Mar 3, 2013 9:08 AM


On Sunday, March 3, 2013 10:29:37 AM UTC, Brian Chandler wrote: > 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 60degree '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...?)
I think that 2 * pi comes up quite a bit more in modern maths than pi, but that wasn't necessarily true at the time the pi notation originated. From the standpoint of the contemporary mathematician, it would make more sense to call pi = 6.28...
Is this connected with transcendence? Not really, e is also transcendental. It's very important in mathematics to define notation and concepts to be as simple and transparent as possible. Defining pi as the number that we now call 2 * pi would be an improvement. But the improvement is not significant enough to overcome inertia  there is value in sticking to historical precedents, too.
Having to restate all pibased formulate would be irritating too.
Paul Epstein



