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As we all know, $\pi$ is the ratio of a circle's circumference to its diameter. When you divide the circumference by the diameter, the result is $\pi$. But, here's my question:

When you enter the numbers into a PC, main frame, or whatever, what numbers do you use? It seems to me if you do not get them PRECISELY correct, the calculated value of $\pi$ will be incorrect?

Please explain.

Widawensen
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Robert Benson
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  • What do you mean by "using" numbers here? – Wojowu Mar 19 '15 at 17:05
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    Since $\pi$ is a transcendental number, it can never be stored 'precisely' in any machine. If there were to exist integers $m,n$ such that $m=n\pi$ then this would be a different story – jameselmore Mar 19 '15 at 17:05
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    @jameselmore: It really depends on what you mean by "precisely". – tomasz Mar 19 '15 at 17:07
  • @tomasz, I mean without error – jameselmore Mar 19 '15 at 17:07
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    @jameselmore: What do you mean by without error? – tomasz Mar 19 '15 at 17:09
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    @jameselmore An algorithm for computing the digits is precisely stored, for example. There are real numbers for which there is no precise storage, but $\pi$ isn't one of them. – Thomas Andrews Mar 19 '15 at 17:10
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    With negligible error (say to $10$ decimal places) may be good enough for all practical purposes. – André Nicolas Mar 19 '15 at 17:10
  • @ThomasAndrews: that's my point. I can't think of a reasonable definiton of "precisely stored" number that would exclude nonrational computable numbers (except those which explicitly exclude irrational numbers, but that's not very reasonable). Even less likely is a reasonable definition that would include $\sqrt 2$ but not $\pi$. – tomasz Mar 19 '15 at 17:12
  • In base $\pi$, the number can be represented in 2 digits without any loss of precision. – Jonny Mar 19 '15 at 17:19
  • @Jonny: "Accuracy" of your $10$ is the same as that of $\pi=\sum_{k=1}^{\infty}\frac{3^k-1}{4^k}\zeta (k+1)$ where $\zeta$ is the zeta function. Regards. – Piquito Jul 01 '16 at 16:57

2 Answers2

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I think the thought behind the question might be the notion that if $\pi$ is the ratio of circumference to diameter, then why can't I measure the circumference of a circular object and its diameter, throw these numbers into a computer, and get $\pi$? If we could somehow precisely measure these two attributes, then perhaps we could get precisely $\pi$.

The first issue with this line of thinking is the idea that a perfect circle actually exists somewhere in nature. I don't believe this is the case -- the best one could ever do is construct/find some polygon with a very very large number of sides. Perhaps even an arbitrary number of sides.

The second issue with this line of thinking is the idea that it is possible to precisely measure something. Even with a perfect circle at the ready, the best one could ever do is measure its circumference and diameter to some arbitrary level of precision. One could only ever say that its circumference or diameter is at least this, and at most that.

Jonny
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Speaking as a computer scientist, it is more standard to use an infinite series to approximate pi. That way, you can control the precision to an arbitrary number of digits (or bits). For applications that do not require high mathematical precision (the majority of computer science), the underlying programming language (e.g. Java) already comes with packages that contain these pre-defined constants. Note that the same holds for any irrational number. To guarantee arbitrary precision in a computational application, you have to use some kind of 'generator', the most common of which is an infinite series, but also computing square roots etc.

M.Dax
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