Converting Fractions to Decimals
To convert a fraction to a decimal, divide the numerator by the denominator. The result will either be a terminating decimal (ends after a finite number of digits) or a repeating decimal (a pattern of digits repeats infinitely).
Types of Decimal Results
Terminating Decimals
The decimal ends after a finite number of digits. Occurs when the denominator's only prime factors are 2 and/or 5.
Repeating Decimals
A block of digits repeats infinitely. Occurs when the denominator has prime factors other than 2 and 5.
Bar Notation
A line (vinculum) placed over the repeating digits to indicate the repeating block.
Mixed Repeating
Some decimals have non-repeating digits before the repeating block begins.
The Long Division Method
Long division is the standard method for converting a fraction to a decimal by hand. Here is the process:
- Divide the numerator by the denominator.
- Write the quotient (whole number part) followed by a decimal point.
- Multiply the remainder by 10 and divide again.
- Repeat the process, tracking remainders.
- If a remainder repeats, you have found the repeating block.
- If the remainder becomes 0, the decimal terminates.
How to Detect Repeating Decimals
During long division, keep track of each remainder. If you encounter a remainder that you have seen before, the decimal digits from that point onward will repeat. The repeating block is the sequence of digits between the two occurrences of the same remainder.
A fraction in lowest terms has a terminating decimal if and only if the denominator has no prime factors other than 2 and 5. Otherwise, the decimal will repeat.
Common Fraction-to-Decimal Conversions
- 1/2 = 0.5
- 1/3 = 0.3
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.16
- 1/7 = 0.142857
- 1/8 = 0.125
- 1/9 = 0.1
- 1/11 = 0.09