medium difficulty4 min

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.

In one line: Any extra prime makes the decimal loop.

Practice This Concept

Understand it

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 × 3 and the prime 3 is present.

Formula and rule

In lowest form, if the denominator has a prime other than 2 or 5, the decimal is recurring.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

15/12 = 0.416666... so the digit 6 repeats forever.
2Since 12 = 2² × 3, the extra prime 3 causes repetition.

Method you can reuse

1Reduce the fraction to lowest form.
2Factorise the denominator.
3Look for primes other than 2 and 5.
4If any appear, the decimal is recurring.

Visual tip

A long division layout helps show the repeating remainder pattern clearly.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

After simplification, what type of decimal does 5/12 produce?

Think first, then show answer

It produces a non-terminating recurring decimal because 12 contains the prime factor 3.

Common wrong answer

Calling 1/6 terminating because it has only one extra prime is wrong; the prime 3 forces repetition.

Don't mix this up

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Classify a fraction's decimal expansion, or decide whether it terminates or repeats after simplification.

1
Enough for 1 mark
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.
2
Add for 2 marks
Example: 1/6 = 0.1666... is recurring because 6 = 2 × 3 and the prime 3 is present.
3
Add for 3 marks
The decimal keeps repeating because the denominator cannot be converted into a power of 10 using only 2s and 5s.

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