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Tuesday, August 07, 2007
Is Pi defined in the best way?
Posted by 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 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 &pi (1/4)*PII 3 (4/3)*&pi (1/6)*PII 4 (1/2)*&pi2 (1/32)*PII2 5 (8/15)*&pi2 (1/60)*PII2 6 (1/6)*&pi3 (1/382)*PII3 7 (16/105)*&pi3 (1/1640)*PII3
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!3:06 PM #
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> There is no closed form formula (that I know of)
n even: V(2n) = pi^n / n! for n
n odd: V(2n+1) = pi^n 2^(2n+1) n! / (2n+1)!
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There are some genuine points that he makes. But the issue is at worst a minor pedagogical inconvenience and gets way more attention than it deserves. I've seen it brought up several times already in blogs and magazines.
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Actually V(n) = pi^(n/2)/(n/2)! for all n, even and odd. You should use the Gamma function to interpret non-integral factorials. (And just as pi itself is mistakenly half of what it should be, the Gamma function is mistakenly shifted by one.)
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limited pi, or
Pi, 3.1415676 equals 19 divided by 6.047936
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Currently e to the power pi times i equals minus one. I would hate it if this equation would need mending.
>>>> JL Balcázar
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It may be that 2*pi*i is an even more fundamental constant than 2*pi or pi. It is, after all, the generator of log(1). The fact that so many formulae involving pi^n depend on the parity of n is another clue in this regard.
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Jose' (greetings!), I've thought about your point as a reason to keep the original definition of pi. But the revised version also gives a sense of perfection and completeness. Compare:
e^{\pi*i} + 1 = 0
uses each of {0,1,e,\pi,i,=,+,*,^} exactly once. But
e^{\PII*i/2} + 1 = 0
throws in 2 and division without duplicating anything. Subtraction is still missing, but (IMHO) division really has more of a separate character from multiplication as a fundamental mathematical operation, than subtraction has from addition.
Ah, I see Palais touches on this in his article.
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Another equation for limited pi comes from Nicomachus
2.714258714258714258 = 57/21
and that's 19 digits. According to numbers that leaves three numbers out of 9...369.
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Google says
pi = 3.14159265
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Google doesn't know everything, and neither do I, but I know you can find a perfect circle by following the pattern to limited pi, and it's simple.
Magic squares, although it's not a magick trick.
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Is the upside down Pi symbol used for anything else? If not just use it to denote the ratio of circumference to radius.
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Sometimes the 2*pi combination appears often enough that people do redefine it. In physics h/(2*pi) appears so many times, it is redefined as hbar ---an 'h' with a slash through it ($\hbar$ in LaTeX). h is of course, Plank's constant.
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A lemma in a paper I'm writing deals with the distortion between the circular distance angle(x,y) and the straight-line distance |x - y| between two points x,y on the unit circle. The maximum distortion, for antipodal points, is
\pi/2.
Not only is this fundamental, but one can contend that the distortion angle(x,y)/|x - y| is the first aspect of circular measurement that concerned the ancients.
Hence I am agreed that the "true" value of "pi" is off by a factor of 2---but in the opposite direction! ;->
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With this new constant
e^(i * PII) = 1
However, pi does appear on it's own in a *lot* of equations. You're just picking the ones where it is 2pi and saying "ooo look it would be easier".
It's easier to write n * pi than pi/n in equations so I'd say they made the right decision. If you really can't stand the 2 in '2 pi r' you can use 'pi d' (diameter) instead.
Enough of this nonsense.
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