Repeating decimals

When the denominator can be expanded to 10, 100, 1000 …, the fraction is easy to write as a decimal: $\displaystyle {\frac{3}{4}}={\frac{75}{100}}={0.75}$.

$\displaystyle \frac{1}{3}$ cannot be expanded to 10, 100 or 1000 – no multiple of 3 is one of them. So divide: $\displaystyle {\frac{1}{3}}={1\div3}$. Every step of the division leaves the same remainder, 1, so it never ends: $\displaystyle {\frac{1}{3}}={0.333\ldots}$

Such a decimal is repeating: the same digits come back again and again. A line over them shows which: $\displaystyle {0.333\ldots}={0.\overline{3}}$, and $\displaystyle {\frac{1}{7}}={0.142857142857\ldots}={0.\overline{142857}}$. Sometimes the repeating starts later: $\displaystyle {\frac{5}{6}}={0.8333\ldots}={0.8\overline{3}}$.

For calculations, 3 or 4 digits after the point are enough: $\displaystyle \frac{1}{3}\approx0.333$ ($\displaystyle \approx$ means “about”). The last digit is rounded: if the digit after it is 5 or more, add 1 to it – $\displaystyle {\frac{2}{3}}={0.6666\ldots\approx0.667}$.

Expand to 10, 100 or 1000, and write as a decimal.

Divide (a calculator is fine), write 4 digits after the point, and mark which digits repeat.

$\displaystyle \frac{2}{3} \qquad \frac{1}{6} \qquad \frac{5}{9} \qquad \frac{4}{11}$

Investigate: which of $\displaystyle \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \ldots \frac{1}{12}$ give a decimal that ends, and which a repeating one?
Can you find a rule?