Wikipedia:Reference desk/Archives/Mathematics/2009 June 4

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June 4

Cirumference of a circle

Pi is unending as I understand it, there is no final digit after the decimal point (there is an internet site giving it to 200 billion decimal places - http://ja0hxv.calico.jp/pai/epivalue.html . Since the circumference of a circle is Pi times the diameter, how can the circumference be finite if the value of pi goes on to infinity? I know I’m missing something obvious but I don’t know what it is. Thanks - Adrian Pingstone (talk) 16:54, 4 June 2009 (UTC)

• Having a decimal expansion in which infinitely many digits are nonzero is not the same as being infinite. Otherwise, even simple numbers like 1/3 = 0.333333... (repeat ad infinitum) would be infinite and arithmetic would be pointless. —David Eppstein (talk) 17:02, 4 June 2009 (UTC)
• (2x edit conflict) Actually, pi does not "go to infinity". It is lesser than 3.2, for instance. So if d is the diameter of a given circle, the circonference is certainly lesser than 3.2 * d. The fact that (a particular way of writing) pi has infinitely many digits has nothing to do with its size. Hope this helps, Goochelaar (talk) 17:04, 4 June 2009 (UTC)
• It might help if you consider this analogy. Let's say that you must walk a mile to your destination, but you can only walk half the distance every time before stopping. That is, the first time, you walk 1/2 mile before stopping. Next, you walk half of that distance before stopping, which is 1/4 mile. Then 1/8 mile, 1/16 mile, and so on ad infinitum. It takes an infinite amount of sequences to reach the 1 mile mark, even though you're only walking a finite distance.

If that is too theoretical for you, simply consider any infinitely-repeating decimal, like 1/3 (which is .3 + .03 + .003 + .0003 + ...). Hope this helps, MuZemike 17:09, 4 June 2009 (UTC)

• (ec, and I even have the same example (but a nicer formula)) The value of pi is not going to infinity, only the length of its decimal representation is infinite. This can be seen as a variant of Xeno's paradox. The important insight is that the sum over an infinite number of summands is not infinite. Consider the simpler case of 1/3, which is 0.333.... It can be written as 0.3+0.03+0.003+0.0003+..., or as $\sum _{i=1}^{\infty }{}3\times 10^{-i}$, but still will always be smaller than 0.4, just like pi is smaller than 3.2 and hence the circles circumference is bounded. --Stephan Schulz (talk) 17:11, 4 June 2009 (UTC)

Many thanks for your answers, now I see my error - Adrian Pingstone (talk) 18:44, 4 June 2009 (UTC)

It's not really your error so much as not having it presented to you as the examples above. It's difficult to understand any mathematical concepts poorly presented. Glad the the folks here were able to help. hydnjo (talk) 20:30, 4 June 2009 (UTC)

That's the basis of mathematical analysis. MuZemike 23:05, 4 June 2009 (UTC)

Let a_n = ${\frac {1}{n^{2}}}$. It may be interesting to note that the sum of all a_n (where n is a natural number), converges; that is, this sum equals to π²/6. This maybe non-trivial but it illustrates how the sum of infinitely many numbers may actually be finite. See Convergence (mathematics) for a simple detailed analysis of this. --PST 04:03, 5 June 2009 (UTC)

You couldn't have used an example that doesn't involve pi, an "infinite" number? —Tamfang (talk) 15:39, 27 June 2009 (UTC)

Adrian is not the first philosopher to argue that infinitely many contributions must produce an infinite total amount. See Zeno's paradoxes. Bo Jacoby (talk) 07:28, 6 June 2009 (UTC).

As we often see with threads on the RD ;) pma (talk) 10:17, 7 June 2009 (UTC)