Which Fractions 1/n have a

terminating decimal expansion?

Answer:

The fraction 1/n will have a terminating decimal expansion if and only if the only prime factors of n are two or five.

Proof:

We must show that The fraction 1/n will have a terminating decimal expansion if and only if the only prime factors of n are two or five. Let k be any possible integer before the decimal point. For example, for .385, k=385. and let M be a real number.
We will now cross multiply.
Now we have
Since K is a whole number, N has to divide 10^M. This is only possible when N has only 2 or 5 as prime factors.

When does 1/n terminate in other number systems?


Before we can tackle this question, we will first talk about our number system. Our number system is a decimal system; this can also be said as a base 10 number system. This means that we count by powers of ten. This seems like a logical base to use because humans have ten fingers. Thus, a decimal number system developed because humans have a habit of counting on their fingers. Before you move on to the next paragraph you should think of the results we acquired in our proof and the prime factors of 10.



Would we get the same result if we were using a different base?

No, we would not. If we were in a base 6 number system, 1/n would only terminate if and only if the prime factors of n are 2 and 3. Notice that 2 and 3 are the prime factors of 6.

Test Question:

In a base 60 number system, which fractions 1/n have a terminating decimal expansion?


cool trivia: the Babylonians had a base 60 number system.

BONUS:

Here is a video that is a bit interesting. It is hypothesized that we use a base ten system because we have ten fingers. What if we had 8 fingers?
Pi and Four Fingers - Numberphile