Virtual Math Circle Report: Terminating and Repeating Decimals

Have you ever noticed that some decimals terminate and that others repeat?

a) Find six fractions, three that turn into terminating decimals and three that turn into repeating decimals.

b) Play around some more to see if you can determine the general pattern for which fractions terminate and which fractions repeat.

I assigned this puzzle in a Zoom math circle to Claire, Rowan, and Sage.

This is what they came up with for how to define a fraction that converts into a terminating decimal:

I most often have seen "terminating decimals" as defined by only having powers of 2 and powers of 5 in the denominator.

I think their definition as "the denominator goes evenly into 10, 100, 1000, etc." is an interesting one. It's correct and it's a novel way of looking at it.

For the ones drawn above, 1/2 terminates because 2 goes evenly into 10.

1/20 terminates because 20 goes evenly into 100.

1/200 terminates because 200 goes evenly into 1000.

The denominator of the fraction must go evenly into a power of 10 (or a multiple of that power).

Let's look at the fraction 3/40:

Thus, 3/40 terminates because 40 goes evenly into 3000. I hadn't thought about it that way before!