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[Computational Complexity] Is Pi defined in the best way?

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  • GASARCH
    &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
    Message 1 of 1 , Aug 7, 2007
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      &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.
      nTrad Vol/RnNew Vol/n
      122
      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
      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.

      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
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