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Decimal Expansion — Definition, Formula & Examples

Decimal expansion is the way a number is written out in decimal form, showing all the digits to the right of the decimal point. For example, the decimal expansion of 14\frac{1}{4} is 0.250.25, and the decimal expansion of 13\frac{1}{3} is 0.333…0.333\ldots

The decimal expansion of a real number is its representation in base 10, expressed as an integer part followed by a decimal point and a sequence of digits (possibly infinite) representing the fractional part. A decimal expansion is called terminating if the digit sequence ends, and repeating (or recurring) if a fixed block of digits repeats indefinitely.

How It Works

To find the decimal expansion of a fraction, divide the numerator by the denominator using long division. If the remainder eventually becomes zero, the expansion terminates (like 38=0.375\frac{3}{8} = 0.375). If the remainder starts repeating before reaching zero, the expansion repeats (like 211=0.18‾\frac{2}{11} = 0.\overline{18}). A bar over the digits indicates the repeating block. Every fraction produces either a terminating or repeating decimal, while irrational numbers like π\pi have decimal expansions that never terminate and never repeat.

Worked Example

Problem: Find the decimal expansion of 5/6.
Divide: Perform long division: 5 divided by 6.
5÷6=0.8333…5 \div 6 = 0.8333\ldots
Identify the pattern: After the first digit (8), the digit 3 repeats forever. Write this with a repeating bar.
56=0.83‾\frac{5}{6} = 0.8\overline{3}
Answer: The decimal expansion of 56\frac{5}{6} is 0.83‾0.8\overline{3}, a repeating decimal.

Why It Matters

Understanding decimal expansions helps you compare and order numbers quickly, which matters in science measurements and financial calculations. Recognizing whether a fraction produces a terminating or repeating decimal also deepens your number sense when working with ratios and proportions in pre-algebra and algebra courses.

Common Mistakes

Mistake: Assuming all decimal expansions eventually end.
Correction: Only fractions whose denominators (in lowest terms) have no prime factors other than 2 and 5 produce terminating decimals. Fractions like 13\frac{1}{3} or 17\frac{1}{7} produce repeating decimals that go on forever.