# 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).