Question
What is the rule for determining if a rational number can be expressed as a repeating or terminating decimal?
Answers
GPT-4o mini
A rational number can be expressed as a repeating or terminating decimal based on the prime factorization of its denominator when the rational number is written in its simplest form (i.e., the numerator and the denominator have no common factors other than 1).
The rule is as follows:
- A rational number in the form \( \frac{a}{b} \) (with \( a \) and \( b \) as integers and \( b \neq 0 \)) can be expressed as a terminating decimal if and only if the denominator \( b \) (when simplified) has no prime factors other than 2 and 5. This means that \( b \) can be written in the form \( 2^m \times 5^n \) where \( m \) and \( n \) are non-negative integers.
- Conversely, if the denominator \( b \) has any prime factors other than 2 and 5, then the decimal representation of the rational number will be repeating.
**Examples:**
1. For the fraction \( \frac{3}{8} \):
- The denominator 8 can be factored as \( 2^3 \).
- Since there are no prime factors other than 2, \( \frac{3}{8} \) is a terminating decimal (specifically, 0.375).
2. For the fraction \( \frac{5}{12} \):
- The denominator 12 can be factored as \( 2^2 \times 3^1 \).
- Since there is a factor of 3 (which is not 2 or 5), \( \frac{5}{12} \) is a repeating decimal (specifically, 0.41666...).
3. For the fraction \( \frac{7}{20} \):
- The denominator 20 can be factored as \( 2^2 \times 5^1 \).
- Since the factors are only 2 and 5, \( \frac{7}{20} \) is a terminating decimal (specifically, 0.35).
By using this rule, you can easily determine whether a rational number will have a repeating or terminating decimal representation.
The rule is as follows:
- A rational number in the form \( \frac{a}{b} \) (with \( a \) and \( b \) as integers and \( b \neq 0 \)) can be expressed as a terminating decimal if and only if the denominator \( b \) (when simplified) has no prime factors other than 2 and 5. This means that \( b \) can be written in the form \( 2^m \times 5^n \) where \( m \) and \( n \) are non-negative integers.
- Conversely, if the denominator \( b \) has any prime factors other than 2 and 5, then the decimal representation of the rational number will be repeating.
**Examples:**
1. For the fraction \( \frac{3}{8} \):
- The denominator 8 can be factored as \( 2^3 \).
- Since there are no prime factors other than 2, \( \frac{3}{8} \) is a terminating decimal (specifically, 0.375).
2. For the fraction \( \frac{5}{12} \):
- The denominator 12 can be factored as \( 2^2 \times 3^1 \).
- Since there is a factor of 3 (which is not 2 or 5), \( \frac{5}{12} \) is a repeating decimal (specifically, 0.41666...).
3. For the fraction \( \frac{7}{20} \):
- The denominator 20 can be factored as \( 2^2 \times 5^1 \).
- Since the factors are only 2 and 5, \( \frac{7}{20} \) is a terminating decimal (specifically, 0.35).
By using this rule, you can easily determine whether a rational number will have a repeating or terminating decimal representation.