Recurring decimals

#### Tier: #Higher

Description

A recurring decimal is a decimal where a digit or group of digits repeats indefinitely. It is written using dot notation: $0.\dot{3} = 0.333\ldots$ and $0.\dot{1}\dot{4} = 0.141414\ldots$

Converting a recurring decimal to a fraction:

Let $x = 0.\dot{3}$ $10x = 3.\dot{3}$ Subtract: $9x = 3$, so $x = \frac{3}{9} = \frac{1}{3}$

For a longer repeat, e.g. $x = 0.\dot{1}\dot{4}$: $100x = 14.\dot{1}\dot{4}$ $99x = 14$, so $x = \frac{14}{99}$

The key is to multiply by a power of 10 that aligns the repeating block, then subtract to eliminate it.

All recurring decimals are rational numbers — they can be written as fractions.

Common error: multiplying by the wrong power of 10 — you need to shift exactly one full cycle of the repeating block.

Links

Manipulating algebraic fractions