Question

Without actually performing the long division, state whether the following rational number will have the terminating decimal expansion or a non-terminating repeating decimal expansion $\frac{1}{8}$

Answer 1

Definition:

$\frac{p}{q}$is terminating if

1. p and q are co-prime.
2. q is of the form $2^{n}and{5}^m$
   Where n and m are non-negative integers.

Co-primes:

A pair of numbers is called co-primes if those numbers have only one common factor, namely 1.

Example:

3,5 are co-primes as they have one common factor which is 1.

623e0a0588db7c968919e44b623e0a0588db7c968919e44b

Explanation:

Checking co-prime:

1. 1 is a prime number.
2. Factors of 8 are $8=2\times 2\times 2$
3. 1 and 8 has no common factors.
4. So, 1 and 8 are co-primes.

Denominator:

1. Here $8=2^3$$\times 5^0$ $(As,5^0=1)$
2. Which is in the form $2^n{5}^m$ where, n=3 and m=0

Conclusion:

Hence, $\frac{1}{8}$is a terminating decimal number.