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Why is $$\frac {4\sqrt{40}}{\log_{10}{40}} - 4\sqrt{10}\approx 3.1419$$ so close to $\pi$?

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closed as off-topic by user91500, Did, Watson, TMM, Daniel W. Farlow Oct 1 '16 at 0:07

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – user91500, Did, Watson, TMM, Daniel W. Farlow
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ It is a coindent $\endgroup$ – user348832 Sep 30 '16 at 10:34
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    $\begingroup$ Something out there's gotta be close to $\pi$, right? $\endgroup$ – barak manos Sep 30 '16 at 10:37
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    $\begingroup$ For the same reason $\,0.333301\,$ is so close to $\;\frac13\;$ : because. $\endgroup$ – DonAntonio Sep 30 '16 at 10:37
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    $\begingroup$ This also $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}$ $\endgroup$ – E.H.E Sep 30 '16 at 10:41
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    $\begingroup$ You're multiplying the square root of $10$ by a number slightly smaller than $1$, so not a big surprise. $\endgroup$ – egreg Sep 30 '16 at 10:46
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Just the same way as $\sqrt{10}, \frac{22}{7}$ are close to $\pi$.

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    $\begingroup$ $\frac{22}{7}$ is a bit more special being a convergent to $\pi$. $\endgroup$ – snulty Sep 30 '16 at 10:40
  • $\begingroup$ @snulty Why? 10 is a convergent of $\pi^2$, which leads to $\pi \approx \sqrt{10}$ $\endgroup$ – Jaume Oliver Lafont Apr 7 '17 at 3:21
  • $\begingroup$ @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 $\pi$. $\endgroup$ – snulty Apr 7 '17 at 19:07
  • $\begingroup$ @snulty The first convergents of $\pi$ are $3$ and $\frac{22}{7}$, and the first two convergents of $\pi^2$ are $9$ and $10$. 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 $\pi \approx \sqrt{10}$. $\endgroup$ – Jaume Oliver Lafont Apr 7 '17 at 20:45
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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.

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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.

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