GCSE Maths Revision Guide
Surds & Recurring Decimals
Simplify surds, rationalise denominators, and convert recurring decimals. This free GCSE Maths module combines short explanations, worked examples, flashcard-style recall, and timed practice so students can revise the topic without creating an account.
Foundation Skills
Converting Recurring Decimals
Use algebra: let x = 0.777... then 10x = 7.777... Subtract x to get 9x = 7, so x = 7/9.
The Dot Notation
A dot over a number means it repeats forever. Dots over two numbers mean the whole block repeats.
Example: 0.12̇3̇ = 0.1232323...
Higher Skills
Simplifying Surds
Find the largest square factor of the number inside the root.
Formula: √ab = √a × √b
Example: √12 = √4 × √3 = 2√3
Multiplying Surds
Multiply the numbers inside the roots together.
Example: √2 × √8 = √16 = 4
Rationalising (Simple)
Multiply top and bottom by the surd to remove it from the denominator.
Formula: 1/√a = √a / a
Example: 5/√2 = 5√2 / 2
Rationalising (Complex)
Multiply by the conjugate (reverse the sign).
Formula: 1/(a + √b) × (a - √b)/(a - √b)
Example: Sign swap: + becomes −. Middle terms cancel out!