How to determine whether a fraction has a terminating or repeating decimal expansion

To identify the type of decimal expansion associated with a given fraction, the first step is to reduce the fraction to lowest terms.

The denominator is then factored into its prime components.

• If the prime factors consist solely of powers of 2 and/or 5, the decimal expansion of the fraction is terminating, meaning that it contains a finite number of decimal digits.

  For example, consider the fraction \( \frac{7}{20} \). Factoring the denominator into prime factors gives $$ \frac{7}{20} = \frac{7}{2^2 \cdot 5} $$ Since the only prime factors are 2 and 5, the quotient is a terminating decimal. Indeed, $$ \frac{7}{20} = \frac{7}{2^2 \cdot 5} = 0.35 $$ The number of digits in the decimal expansion is equal to the highest exponent among the prime factors 2 and/or 5 in the denominator. In this case, the highest exponent is two ( \( 2^2 \) ), so the decimal expansion has two digits.

• If the prime factors of the denominator are all different from 2 and 5, the fraction produces a pure repeating decimal expansion. This type of expansion is characterized by a block of digits, called the period, which repeats indefinitely.

  For example, consider the fraction \( \frac{6}{21} \). Factoring the denominator into prime factors yields $$ \frac{6}{21} = \frac{6}{3 \cdot 7} $$ Since the denominator contains only the prime factors \(3\) and \(7\), the quotient must be a pure repeating decimal expansion: $$ \frac{6}{21} = \frac{6}{3 \cdot 7} = 0.\overline{285714} $$ 

• If the prime factors of the denominator include 2 and/or 5 together with other prime factors, the fraction produces a mixed repeating decimal expansion. In this case, the decimal expansion consists of a non-repeating part, called the antiperiod, followed by a repeating part, known as the period.

  For example, consider the fraction \( \frac{7}{12} \). The denominator has the prime factorization \( 2^2 \cdot 3 \). As a result, the fraction yields a mixed repeating decimal expansion. Indeed, $$ \frac{7}{12} = \frac{7}{2^2 \cdot 3} = 0.58333333333 $$ Here, 58 forms the antiperiod, while 3 is the repeating period of the decimal expansion. It is worth noting that analyzing the prime factorization also allows the number of digits in the antiperiod to be determined before performing the division. Specifically, the highest exponent among the prime factors 2 and/or 5 in the denominator determines the length of the antiperiod. In this case, the highest exponent is two ( \( 2^2 \) ), so the antiperiod consists of two digits (58). 

And so on.