Integers, Fractions and Decimals
Recurring Decimals
Pearson Edexcel International GCSE Mathematics A
Recurring decimal notation
- A dot over one digit: that digit repeats for ever.
- Dots over the first and last digits of a block: the whole block repeats, including the digits between the dots.
- Digits before the first dot do not repeat.
Without dots, write a recurring decimal to at least 5 significant figures, followed by …
Recurring decimal to a fraction
Method: use algebra to show that a recurring decimal equals a fraction
- Let x equal the recurring decimal.
- Multiply x by two powers of 10 so that both results have exactly the same digits after the decimal point: one power of 10 moves the point to the end of the first repeating block, the other moves it to the start of that block.
- Subtract the smaller equation from the larger: the recurring part cancels, leaving a whole number.
- Divide to write x as a fraction.
- Simplify to the fraction given in the question.
- Write two equations in x and subtract them. Never write a decimal check with no equations in x.
- With a whole number part, the same method gives an improper fraction: convert it to a mixed number if the given answer is one.
- A letter for a recurring digit uses the same subtraction, giving the fraction in terms of the letter.
Worked example: recurring decimal with a non-recurring digit
Use algebra to show that 0.572 = 63110
x = 0.572727…
1000x = 572.7272…
10x = 5.7272…
1000x − 10x = 567
990x = 567
x = 567990 = 63110
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