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Take any scientific calculator and follow these steps:

  1. Ensure the calculator is in degrees mode.
  2. Enter some number of 5's (more than 4 is ideal).
  3. Take the reciprocal of step 2.
  4. Take the sine of step 3.

You should find that the result is an approximation of $\pi$ (that becomes more accurate as more 5's are entered), multiplied by a negative exponent of 10.

Why does this occur?

Attribution: I found this puzzle on Numberphile.

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  • $\begingroup$ Attribution: Arthur Benjamin's book (I forgot the title) $\endgroup$
    – mathlander
    Commented Mar 28, 2023 at 16:38

1 Answer 1

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A string of $n\ 9$s is $10^n-1$, so a string of $n\ 1$s is $\frac{10^n-1}{9}$, so a string of $n\ 5$s is $\frac{5\times(10^n-1)}{9}$; the reciprocal of that is $\frac{9}{5\times(10^n-1)}$, or approximately $\frac{9}{5\times10^n}$. That many degrees is $\frac{\pi}{180}\times\frac{9}{5\times10^n}=\frac{9 \pi}{900\times10^n}=\frac{\pi}{100\times10^n}=\frac{\pi}{10^{(n+2)}}$ radians. For very small $x$, $\sin{x}$ is approximately $x$ (when one works in radians, as all right-thinking people do), so the sine computed in step 4 is also approximately $\frac{\pi}{10^{(n+2)}}$.

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  • 1
    $\begingroup$ Using 55.55555555... instead of 5555555555... removes the exponent and yields pi directly. $\endgroup$ Commented Mar 29, 2023 at 17:06

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