Recurring decimals and geometric series

Writing a recurring decimal as an infinite geometric series and using the limiting sum to convert it to a fraction, including decimals where the repeating block starts later.

The method

A dot over a digit means it repeats forever. Dots over the first and last digits of a block mean the whole block repeats: $0.\dot{1}2\dot{3} = 0.123123123\ldots$

Example 1●

Write $0.\dot{4}$ as a fraction.

$0.4 + 0.04 + 0.004 + \cdots$ has $a = 0.4$ and $r = 0.1$.

$S_\infty = \dfrac{0.4}{0.9} = \dfrac49$

Example 2: the repeat starts later●●●

Write $0.4\dot{1}\dot{2}$ as a fraction.

Split off the part that does not repeat: $0.4 + (0.012 + 0.00012 + \cdots)$.

The bracket is a GP with $a = 0.012$ and $r = 0.01$, so its sum is $\dfrac{0.012}{0.99} = \dfrac{12}{990}$.

$\dfrac{4}{10} + \dfrac{12}{990} = \dfrac{396}{990} + \dfrac{12}{990} = \dfrac{408}{990} = \dfrac{68}{165}$

Flashcards

$0.dot{4}$

$\dfrac49$

Ratio for a 2-digit repeating block

$r = 0.01$

$0.dot{9}$

$1$

Practice

Write $0.\dot{4}$ as a fraction.
Write $0.\dot{2}\dot{7}$ as a fraction in lowest terms.
Write $0.4\dot{1}\dot{2}$ as a fraction in lowest terms.
Write $0.\dot{1}\dot{8}$ as a fraction in lowest terms.
Write $0.1\dot{6}$ as a fraction.
Write $2.\dot{3}$ as a fraction.
Write $0.\dot{1}2\dot{3}$ as a fraction in lowest terms.
In $0.\dot{5}\dot{6} = 0.56 + 0.0056 + \cdots$, the ratio is:
$0.\dot{9}$ equals: