Why is $$\frac {4\sqrt{40}}{\log_{10}{40}} - 4\sqrt{10}\approx 3.1419$$ so close to $\pi$?
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It is a coindent – Sep 30 '16 at 10:34
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4Something out there's gotta be close to $\pi$, right? – barak manos Sep 30 '16 at 10:37
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2For the same reason $,0.333301,$ is so close to $;\frac13;$ : because. – DonAntonio Sep 30 '16 at 10:37
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3This also $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}$ – E.H.E Sep 30 '16 at 10:41
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2You're multiplying the square root of $10$ by a number slightly smaller than $1$, so not a big surprise. – egreg Sep 30 '16 at 10:46
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Why is 3.1415 so close to $\pi$? Note that it closer than your example, despite being shorter. – Ivan Neretin Sep 30 '16 at 11:27
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You may find the page https://en.wikipedia.org/wiki/Mathematical_coincidence#Concerning_powers_of_.CF.80 interesting. – sTertooy Sep 30 '16 at 11:52
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Alright, I just though there may have been an interesting reason. – user85798 Sep 30 '16 at 13:19
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Just the same way as $\sqrt{10}, \frac{22}{7}$ are close to $\pi$.
SchrodingersCat
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@snulty Why? 10 is a convergent of $\pi^2$, which leads to $\pi \approx \sqrt{10}$ – Jaume Oliver Lafont Apr 07 '17 at 03:21
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@JaumeOliverLafont I think you're missing my point, $\frac{22}{7}$ being rational, and a convergent in the sense of continued fractions, means that it's the closest approximation by a rational, given that the denominator is allowed to be no bigger than seven. I mean you could concoct all sorts of irrationals that approximate $\pi$. – snulty Apr 07 '17 at 19:07
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@snulty The first convergents of $\pi$ are $3$ and $\frac{22}{7}$, and the first two convergents of $\pi^2$ are $9$ and $10$.
https://www.wolframalpha.com/input/?i=continued+fraction+pi%5E2
Taking the root at both sides of $\pi^2 \approx 9$ gives $\pi \approx 3$, while taking the square root at both sides of $\pi^2 \approx 10$ gives $\pi \approx \sqrt{10}$.
– Jaume Oliver Lafont Apr 07 '17 at 20:45
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You know $\pi\approx 3.141592$. The interval $[a,b]$ where $a=3.141591$ and $b=3.141593$ contains uncountable many numbers (rational, algebraic and trascendental ones) close to $\pi$. Anyway, to find out closed forms for $\pi$ is a matter of contemporary research and, for instance the number$$\frac{\ln(640320^3+744)}{\sqrt{163}}$$ gives $30$ exact decimal digits of approximation. The number will be much more valuable the greater the approximation be and the number above is not easy to obtain.
Piquito
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By combining up to 15 mathematical symbols from the dozens available, you can make trillions of numbers. Some of them will be close to $\pi$, including $3.1416$, which requires only 6 symbols.
John Bentin
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