Decimal Expansions — Terminating and Non-Terminating Repeating
A rational number has either a terminating decimal expansion or a non-terminating repeating decimal expansion.
Practice This ConceptMain explanation
Teacher explanation
When a rational number is written in lowest form, its decimal terminates if the denominator has only factors 2 and/or 5. If the denominator has any other prime factor, the decimal repeats.
Example
7/8 = 0.875 terminates because 8 = 2³. But 1/3 = 0.333... repeats because the denominator has factor 3.
Simple analogy
Only 2 and 5 make the decimal stop.
Common confusion
Students often divide only for a few steps and call a repeating decimal irrational because it does not end.
Exam tip
Before long division, reduce the fraction to lowest form and factorise the denominator.
Answer writing and exam use
1-mark use
Write the exact meaning of decimal expansions — terminating and non-terminating repeating in one clean line.
2-mark use
Define decimal expansions — terminating and non-terminating repeating and add one example or condition.
3-mark use
Explain decimal expansions — terminating and non-terminating repeating, show the method or example, and mention the common mistake.
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