# Why is $\frac {4\sqrt{40}}{\log_{10}{40}} - 4\sqrt{10}$ so close to $\pi$? [closed]

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

## closed as off-topic by user91500, Did, Watson, TMM, Daniel W. FarlowOct 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
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• It is a coindent – user348832 Sep 30 '16 at 10:34
• Something out there's gotta be close to $\pi$, right? – barak manos Sep 30 '16 at 10:37
• For the same reason $\,0.333301\,$ is so close to $\;\frac13\;$ : because. – DonAntonio Sep 30 '16 at 10:37
• This also $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}$ – E.H.E Sep 30 '16 at 10:41
• You'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

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