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
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.
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