Non-terminating recurring decimal expansion
If a fraction in lowest form has any prime factor in its denominator other than 2 or 5, its decimal expansion is non-terminating recurring.
Practice This ConceptMain explanation
Teacher explanation
The decimal keeps repeating because the denominator cannot be converted into a power of 10 using only 2s and 5s. Once another prime is present, long division does not end with a clean finite decimal.
Example
1/6 = 0.1666... is recurring because 6 = 2 x 3 and the prime 3 is present.
Simple analogy
Any extra prime makes the decimal loop.
Common confusion
Students think every non-terminating decimal is irrational, but repeating decimals are still rational.
Exam tip
Remember the pair: repeating decimal means rational; endless non-repeating decimal means irrational.
Answer writing and exam use
1-mark use
Write the exact meaning of non-terminating recurring decimal expansion in one clean line.
2-mark use
Define non-terminating recurring decimal expansion and add one example or condition.
3-mark use
Explain non-terminating recurring decimal expansion, show the method or example, and mention the common mistake.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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