Sine of (1/55555….)

Take your calculator, and enter a number consisting of several 5’s, such as 5555555. 
Now take the reciprocal of this number. 
Now take the sine of this result (in degrees).

You should get a very interesting number:

3.14159… x 10-9.

Are you surprised?

Presentation Suggestions:
The result is quite striking. (If you did not get a surprising result, be sure your calculator is set to take trig in “degrees”.) Try it for numbers consisting of only 5’s. Challenge yourself to figure out why it is true before looking at the explanation below!

The Math Behind the Fact:
You will find the number in your calculator looks like Pi, scaled by some power of 10. This result depends on the degrees-radians conversion and the “small angle approximation”, which says that sin(A) is approximately A (when A is measured in radians). This approximation is used often in physics and engineering.

Now 1/55555…5 is a number that is approximately 1.8 x 10-n where n is the number of 5’s. To convert from degrees to radians we multiply by and divide by 180. So then the result is a number close to Pi x 10-(n+2), a very small number in radians, and the small angle approximation holds, so taking the sine of this number does not change it very much.

How to Cite this Page: 
Su, Francis E., et al. “Sine of (1/55555….).” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Dan Kalman

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