This topic summary covers Knowledge Organiser: Recurring Decimals within Recurring Decimals for GCSE Mathematics. Revise Recurring Decimals in Number for GCSE Mathematics with 14 exam-style questions and 11 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 6 of 6 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Recurring Decimals
Key Terms
- Recurring decimal: A decimal that repeats a digit or group of digits forever
- Dot notation: A dot over a digit means it repeats (e.g. 0.3̇ = 0.333…)
- Two-dot notation: Dots over first and last digit of a repeating block (e.g. 0.1̇2̇ = 0.121212…)
- Terminating decimal: A decimal that ends (not recurring)
Must-Know Facts
- 1/3 = 0.3̇; 1/9 = 0.1̇; 2/9 = 0.2̇; 1/6 = 0.16̇
- 0.3̇ ≠ 0.3 — the dot means the 3 repeats forever
- One repeating digit → multiply by 10, subtract
- Two repeating digits → multiply by 100, subtract
- Pattern: for 0.ababab… use 99 in denominator; for 0.abcabc… use 999
- Always simplify the resulting fraction
Key Method: Algebraic Conversion
- Step 1: Let x = the recurring decimal
- Step 2: Multiply by 10 (single digit) or 100 (two digits) to shift repeating block
- Step 3: Subtract x from the multiplied version
- Step 4: Solve for x and simplify
- Example: 0.7̇ → 10x = 7.7̇, subtract: 9x = 7, so x = 7/9
Key Formulas
- Single recurring digit: x = d/9 (e.g. 0.3̇ = 3/9 = 1/3)
- Two recurring digits: multiply by 100, subtract, divide by 99
- Non-recurring then recurring: multiply to shift all recurring part, subtract
Common Mistakes
- Wrong multiplier: Single repeating digit → multiply by 10; two repeating digits → multiply by 100
- Subtraction error: Make sure both expressions have the recurring part perfectly aligned before subtracting
- Not simplifying: Always simplify the resulting fraction to its lowest terms
- Mixed recurring: For 0.16̇, only the 6 recurs — multiply by 100 and by 10, then subtract
Practice questions for Recurring Decimals
Which of these fractions gives a recurring decimal when you divide?
Explain why 1/6 gives a recurring decimal. You must refer to prime factors in your answer.
Quick recall flashcards
What does dot notation mean in recurring decimals?
A dot above a single digit: that digit repeats. 0.3̇ = 0.333...
Dots above two digits: the entire block between them repeats. 0.1ї8ї = 0.181818...
What is a recurring decimal?
A decimal where one or more digits repeat infinitely in a regular pattern.
Examples: 0.333... = 1/3, 0.090909... = 1/11