Steven D'Aprano <steve-remove-t...@cybersource.com.au> writes: > Well, what is the definition of pi? Is it: > > the ratio of the circumference of a circle to twice its radius; > the ratio of the area of a circle to the square of its radius; > 4*arctan(1); > the complex logarithm of -1 divided by the negative of the complex square > root of -1; > any one of many, many other formulae. > > None of these formulae are intuitively correct; the formula C = 2πr isn't > a definition in the same sense that 1+1=2 defines 2. The point that I was > trying to get across is that, until somebody proved the formula, it > wasn't clear that the ratio was constant.
There are several possible definitions of `2'. You've given a common one (presumably in terms of a purely algebraic definition of the integers as being the smallest nontrivial ring with characteristic 0). Another can be given in terms of Peano arithmetic, possibly using an encoding of Peano arithmetic using only the Zermelo-- Fraenkel axioms of set theory: at this point one has only a `successor' operation and must define addition; the obvious definition of 1 and 2 are s(0) and s(s(0)) respectively, and one then has an obligation to prove that s(0) + s(0) = s(s(0)), though this isn't very hard. I think my preferred definition of `pi' goes like this (following Lang's /Analysis I/). Suppose that there exist real functions s and c, such that s' = c and c' = -s, with s(0) = 0 and c(0) = 1. One can prove that a pair of such functions is unique, and periodic. Define pi to be half the (common) period of these functions. (Now we notice that they factor through the quotient ring R/(2 pi) and define `sin' and `cos' to be the induced functions on the quotient ring.) Would the world be a better place if we had a name for 2 pi rather than pi itself? -- [mdw] -- http://mail.python.org/mailman/listinfo/python-list