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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.

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Student-friendly 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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    1/6 = 0.1666... is recurring because 6 = 2 x 3 and the prime 3 is present.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

Example

1/6 = 0.1666... is recurring because 6 = 2 x 3 and the prime 3 is present.

Rule to remember

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

Memory hook

Any extra prime makes the decimal loop.

Examples and method

Worked example

5/12 = 0.416666... so the digit 6 repeats forever. Since 12 = 2^2 x 3, the extra prime 3 causes repetition.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

Students think every non-terminating decimal is irrational, but repeating decimals are still rational.

Common wrong answer

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

Exam tip

Remember the pair: repeating decimal means rational; endless non-repeating decimal means irrational.

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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 lowest form, if the denominator has a prime other than 2 or 5, the decimal is recurring. 1/6 = 0.1666... is recurring because 6 = 2 x 3 and the prime 3 is present.

3-mark answer

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. In lowest form, if the denominator has a prime other than 2 or 5, the decimal is recurring. 5/12 = 0.416666... so the digit 6 repeats forever. Since 12 = 2^2 x 3, the extra prime 3 causes repetition. Classify a fraction's decimal expansion, or decide whether it terminates or repeats after simplification. Calling 1/6 terminating because it has only one extra prime is wrong; the prime 3 forces repetition.
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