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.
(** What is the word for a rational number with small numerator and denominator...?)