How do fractions convert into repeating decimals?
How do fractions convert into repeating decimals?
Fractions represent division. When you convert a fraction like $\frac{3}{4}$ into a decimal, you divide the numerator (3) by the denominator (4). Sometimes this division terminates cleanly (e.g., $3 \div 4 = 0.75$), but other times, the digits go on forever in a repeating pattern (e.g., $\frac{1}{3} = 0.333...$).
The Role of Prime Factors
To understand why some fractions repeat and others terminate, we look at the denominator in its simplest form.
Our number system is base-10, meaning 10 is the base. The prime factors of 10 are 2 and 5 ($2 \times 5 = 10$).
- Terminating Decimals: A fraction in simplest form will result in a terminating decimal if and only if the prime factors of its denominator are exclusively 2, 5, or both. For example, $\frac{3}{8}$ has a denominator whose only prime factor is 2 ($2^3$), so it terminates ($0.375$).
- Repeating Decimals: If the denominator has any prime factor other than 2 or 5 (such as 3, 7, 11, 13, etc.), the division process will never leave a remainder of zero. Instead, the remainders must eventually repeat, creating a repeating decimal.
Example: Converting $\frac{1}{3}$
- Set up long division: $1.000... \div 3$.
- 3 goes into 10 three times ($3 \times 3 = 9$), leaving a remainder of 1.
- Bring down a zero to make 10. 3 goes into 10 three times, leaving a remainder of 1.
- Because the remainder is always 1, the quotient repeats indefinitely as $0.333...$, written as $0.\bar{3}$.
Example: Converting $\frac{2}{7}$
The denominator 7 is not 2 or 5, so we expect a repeating decimal:
- $2 \div 7 = 0.285714285714...$
The block of digits 285714 repeats endlessly because the remainders in the long division cycle through a predictable sequence of numbers before repeating.