Revisiting Rational Numbers and Their Decimal Expansions

We have studied earlier that rational numbers have either a terminating decimal expansion or a non-terminating repeating decimal expansion. In this section, we are going to consider a rational number, say pq, q ≠ 0, and explore exactly when the decimal expansion of pq is terminating and when it is non-terminating repeating(or recurring).

To come up with some hypothesis we first start with some concrete examples. Let us take some decimals and convert them into pq form and see if we can identify some patterns.

0.375=3751000=375103

0.0875=87510000=875104

23.3456=23345610000=233456104

We can see a clear pattern. All the denominators are powers of .

If we want to rewrite 10 as a product of prime factors we can use the primes and .

Even though, we have worked only with a few examples, you can see that any real number which has a decimal expansion that terminates can be expressed as a rational number whose denominator is a power of 10. Also the only prime factors of 10 are 2 and 5. So, cancelling out the common factors between the numerator and the denominator, we find that this real number is a rational number of the form ,pq where the prime factorisation of q is of the form 2n5m, and n, m are some non-negative integers.

Formally we can define the theorem as:

Now, if we consider any non terminating and recurring rational nu,bers like 1/7, 2/3, 7/11 etc we see that the denominators are not of the form 2n·5m.

So we can define the theorem as

With the help of these theorems we can quickly find out if a rational number of the form p/q is terminating or non terminating. You can do so using the following steps.

1. Find out if q can be represented as a power of the prime factors 2 or 5.
2. If yes, its terminating. If not, it's non terminating.

For example, consider 3/8. Here 8 can be represented as 23. So it is terminating. We can chek it easily as below

3/8=323=3·5323·53=375103=0.375.