3

$\pi\approx 3.141592654$

Why is it so close to $3$?

I find this intriguing, this cannot be a coincidence.

Superbus
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    Would you prefer it farther away? – Will Jagy Mar 28 '14 at 05:36
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    Because a regular hexagon is "so close to" a circle? – Blue Mar 28 '14 at 05:36
  • Put the unit circle between two regular $n$-gons, one circumscribed by the circle, and the other inscribing the circle, for $n \ge 12$. The area of the outer polygon tells how close $\pi$ is to 3. – r9m Mar 28 '14 at 05:49
  • I was trying to find who it was that took the fact that both $ \ \pi \ $ and $ \ e \ $ are close to 3 as a sign of the Trinity. I find it more interesting that those numbers are close together. (But there are many interesting irrational -- even transcendental -- numbers, and it is reasonable that humans would locate smaller ones first.) – colormegone Mar 28 '14 at 05:54
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    And why is the golden ratio $\frac{1 + \sqrt{5}}{2} \approx 1.62$? The answer? Because that's what it is. – Jared Mar 28 '14 at 05:58
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    I should also comment that some consider $\pi$ to be the wrong value. Why not define $\pi$ as the ratio of the circumference to the radius? Then, instead of $C = 2\pi r$, we would have $C = \pi r$. People tend to use the radius more often than the diameter. Furthermore, then there would be $\pi$ "radians" in a full circle (why should it be $2\pi$, why multiply by $2$)? Then you'd be asking why is $\pi$ so close to $6$? And the answer would be the same, because it is. – Jared Mar 28 '14 at 06:02
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    Why is $e$ so close to ${ 2718281828 \over 1000000000} $? An accident perhaps? – copper.hat Mar 28 '14 at 06:48
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    @copper.hat This cannot be a coincidence either. :-) – Did Mar 28 '14 at 07:37
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    What do you mean close? It's also close to 0 and 100. – Raskolnikov Mar 28 '14 at 07:39
  • @Did: Its déjà vu all over again. – copper.hat Mar 28 '14 at 14:52
  • Because the following integral is small

    $$\pi-3=2\int_0^1\frac{x(1-x)^2}{1+x^2}dx$$

     – Jaume Oliver Lafont Jan 17 '16 at 07:43
  • and the following series as well $$\pi-3=\sum_{k=1}^{\infty} \frac{3}{(1+k)(1+2k)(1+4k)}$$ – Jaume Oliver Lafont Jan 23 '16 at 07:06

2 Answers2

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As you can clearly see from this figure, $2\pi\simeq6\iff\pi\simeq3$, since the side of the inscribed regular polygon is equal to the radius of the circle:

$\qquad\qquad\qquad\qquad$1

Community
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Lucian
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Let $a<b$ be integers. Pick a number, which we'll call $\pi'$ in $[a,b]$ uniformly at random. The chance that $\pi'$ is within $\pi-3\approx .1415$ of some integer is $2(\pi-3)\approx .283$.

If something has a near $30\%$ chance of occuring at random, I would say that it could definitely just be a coincidence.

Jared
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