Recurring decimals (Higher)
- A recurring decimal has digits that repeat forever. A dot over a digit (or over the first and last digits of a block) shows the repeating part: 0.333… = 0.3̇ and 0.181818… = 0.1̇8̇.
- Every recurring decimal can be written as a fraction.
Which fractions terminate? (Higher)
- A fraction in its simplest form gives a terminating decimal if the only prime factors of its denominator are 2 and 5.
- 3/8 terminates (8 = 2³), and 7/40 terminates (40 = 2³ × 5). 1/6 recurs because 6 = 2 × 3 has a factor of 3.
Converting to a fraction (Higher)
- One repeating digit: let x = 0.444…, so 10x = 4.444…. Subtract: 9x = 4, so x = 4/9.
- Two repeating digits: let x = 0.181818…, so 100x = 18.1818…. Subtract: 99x = 18, so x = 18/99 = 2/11.
- A non-repeating digit first: let x = 0.1666…. 10x = 1.666… and 100x = 16.666…. Subtract: 90x = 15, so x = 1/6.
Showing your working (Higher)
- For 'prove algebraically' questions, write x = …, multiply by 10 or 100 to line up the repeating parts, subtract, and simplify the fraction.
Key terms
- Recurring decimal
- A decimal in which a digit or block of digits repeats forever.
- Terminating decimal
- A decimal that ends.
- Recurring dot
- A dot placed over the repeating digits of a recurring decimal.
- Prove algebraically
- Show something is true using algebra, not just examples.
- Simplest form
- A fraction that can't be simplified any further.