This theme was explored by Bob Palais in this article. He makes a good case. I look at two examples not in the article, one of which supports his case, and the other is a matter of taste. During this blog I will denote the ratio of Circumference to Radius by PII.
EXAMPLE ONE: Consider the volume and surface area of an n-dim sphere. There is no closed form formula (that I know of) but there is a recursive formula. See this. The following table shows, for each n, the volume of an n-dim sphere divided by Rn.
|n||Trad Vol/Rn||New Vol/n|
EXAMPLE TWO: The Zeta Function is
&zeta(n) = &sum r-n (The sum is from r=1 to infinity.)
It is known that
&zeta(2n) = (-1)n-1 ((2*&pi)2n/2(2n)!)B2n
where Bn is the nth Bernoulli Number. If we use PII instead we get the simpler
&zeta(2n) = (-1)n-1 ((PII)2n/2(2n)!)B2n
This is BETTER!