[Computational Complexity] Is Pi defined in the best way?
- &pi, the ratio of the circumference to the diameter of a circle, is one of the most important constants in Math. However, &pi could just as easily have been defined as the ratio of the circumference to the radius of a circle. This would not change math in any serious way, but it would make some formulas simpler. Think about how often `2*&pi' comes up in formulas.
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.
Is the New Volume easier or harder? A little easier in that all of the numerators are 1. But no real pattern. Similar is true for surface area. Are these formulas better? That is a matter of taste.
n Trad Vol/Rn New Vol/n 1 2 2 2 &pi2 (1/4)*PII2 3 (4/3)*&pi3 (1/6)*PII3 4 (1/2)*&pi4 (1/32)*PII4 5 (8/15)*&pi5 (1/60)*PII5 6 (1/6)*&pi6 (1/382)*PII6 7 (16/105)*&pi7 (1/1640)*PII7
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!
Posted By GASARCH to Computational Complexity at 8/07/2007 03:06:00 PM