# Surprise pi! Explain this phenomenon

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.

• Attribution: Arthur Benjamin's book (I forgot the title) Mar 28 at 16:38

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 $$\frac{\pi}{10^{(n+2)}}$$.