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 `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)(55/3=18.333...)`. 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)(55/3=18.bar(3))`.

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

For example, `19/7=2.\color(brown)(714285)color(green)(714285)...=2.color(purple)(bar(714285))`.

Exercise 1. Convert `1/3` into repeating decimal.

Answer: `0.333333....=0.bar(3)`.

Exercise 2. Divide 25 by 7.

Answer: `3.5714285714285...=3.bar(571428)`.

Exercise 3. Find `-1.2-:55`.

Answer: `-0.02bar(18)`.

Exercise 4. Find `-0.77-:(-0.6)`.

Answer: `1.28bar(3)`.

Exercise 5. Find `57-:(-1.1)`.

Answer: `-51.bar(81)`.