Repeating (Recurring) Decimals

Related calculator: Long Division Calculator

Periodic (recurring) decimal is a decimal that has infinite number of digits, i.e. its digits repeat forever.

Until now, we converted fractions into decimals and divided decimals without a problem.

But there are situations, when we can't finish division.

For, example let's try to convert fraction $$$\frac{{55}}{{3}}$$$ into a decimal (which is equivalent to long division):

$$$\begin{array}{r}18.333\\3\hspace{1pt})\overline{\hspace{1pt}55.000}\\-\hspace{1pt}\underline{3}\phantom{5.000}\\25\phantom{.000}\\-\underline{24}\phantom{.000}\\1\phantom{.}0\phantom{00}\\-\hspace{1pt}\underline{\phantom{1.}9}\phantom{00}\\10\phantom{0}\\-\underline{9}\phantom{0}\\10\\-\underline{\hspace{1pt}9}\\\color{red}{1}\end{array}$$$

Notice, that last three steps were same. We are making circles. And this process will last forever. We keep getting remainder 1 and write 3 into decimal.

To write such decimals, following notation is used: $$${\color{brown}{{\frac{{55}}{{3}}={18.333}\ldots}}}$$$. Ellipsis means "goes on forever".

But, there is also another notation for such decimals. Since 3 is repeating, we write line over it: $$${\color{purple}{{\frac{{55}}{{3}}={18}.{\overline{{{3}}}}}}}$$$.

It is also worth noting, that repeating pattern can consist of more than one digit.

For example, $$$\frac{{19}}{{7}}={2}.{\color{brown}{{{714285}}}}{\color{green}{{{714285}}}}\ldots={2}.{\color{purple}{{{\overline{{{714285}}}}}}}$$$.

Exercise 1. Convert $$$\frac{{1}}{{3}}$$$ into repeating decimal.

Answer: $$${0.333333}\ldots.={0}.{\overline{{{3}}}}$$$.

Exercise 2. Divide 25 by 7.

Answer: $$${3.5714285714285}\ldots={3}.{\overline{{{571428}}}}$$$.

Exercise 3. Find $$$-{1.2}\div{55}$$$.

Answer: $$$-{0.02}{\overline{{{18}}}}$$$.

Exercise 4. Find $$$-{0.77}\div{\left(-{0.6}\right)}$$$.

Answer: $$${1.28}{\overline{{{3}}}}$$$.

Exercise 5. Find $$${57}\div{\left(-{1.1}\right)}$$$.

Answer: $$$-{51}.{\overline{{{81}}}}$$$.