A recurring decimal, also called a repeating decimal, has one or more digits that repeat forever.
The repeated part is usually shown with a bar above it.
1. Recurring decimal notation
The calculator uses the notation \(W.N(R)\), where:
- \(W\) is the whole-number part.
- \(N\) is the non-repeating part after the decimal point.
- \(R\) is the repeating block.
For example:
\[
\begin{aligned}
0.(3)&=0.\overline{3}=0.3333\ldots,\\
1.2(34)&=1.2\overline{34}=1.2343434\ldots.
\end{aligned}
\]
2. Why recurring decimals are fractions
Every recurring decimal is rational. This means it can be written exactly as a fraction:
\[
\begin{aligned}
\text{recurring decimal}
&=
\frac{\text{integer}}{\text{nonzero integer}}.
\end{aligned}
\]
The reason is that the infinite repeating tail can be cancelled by subtracting two shifted copies of the same number.
3. Algebraic method
Suppose:
\[
\begin{aligned}
x&=0.\overline{3}.
\end{aligned}
\]
Multiply by 10 because the repeating block has one digit:
\[
\begin{aligned}
10x&=3.\overline{3}.
\end{aligned}
\]
Now subtract the original equation:
\[
\begin{aligned}
10x-x
&=
3.\overline{3}-0.\overline{3},\\
9x&=3,\\
x&=\frac{3}{9}\\
&=\frac{1}{3}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
0.\overline{3}&=\frac{1}{3}.
\end{aligned}
\]
4. Formula for \(W.N(R)\)
Let \(n\) be the number of non-repeating decimal digits in \(N\), and let \(r\) be the number of repeating digits in \(R\).
Form two integers:
- \(y\): the integer made by joining \(W\), \(N\), and one copy of \(R\).
- \(z\): the integer made by joining \(W\) and \(N\).
Then:
\[
\begin{aligned}
x
&=
\frac{y-z}{10^n(10^r-1)}.
\end{aligned}
\]
After computing this fraction, simplify it using the greatest common divisor.
5. Worked example: \(0.\overline{142857}\)
Here:
\[
\begin{aligned}
W&=0,\quad N=\text{none},\quad R=142857.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
n&=0,\quad r=6.
\end{aligned}
\]
The constructed integers are:
\[
\begin{aligned}
y&=142857,\\
z&=0.
\end{aligned}
\]
So:
\[
\begin{aligned}
x
&=
\frac{142857-0}{10^0(10^6-1)}\\
&=
\frac{142857}{999999}\\
&=
\frac{1}{7}.
\end{aligned}
\]
6. Worked example: \(1.2\overline{34}\)
Here:
\[
\begin{aligned}
W&=1,\quad N=2,\quad R=34.
\end{aligned}
\]
There is one non-repeating decimal digit and two repeating digits:
\[
\begin{aligned}
n&=1,\quad r=2.
\end{aligned}
\]
The constructed integers are:
\[
\begin{aligned}
y&=1234,\\
z&=12.
\end{aligned}
\]
Apply the formula:
\[
\begin{aligned}
x
&=
\frac{1234-12}{10^1(10^2-1)}\\
&=
\frac{1222}{990}\\
&=
\frac{611}{495}.
\end{aligned}
\]
7. Repeating block of one digit
A one-digit repeating block is common:
\[
\begin{aligned}
0.\overline{6}
&=
\frac{6}{9}
=
\frac{2}{3}.
\end{aligned}
\]
If there is a non-repeating digit before the repeating digit, use the full formula.
For example:
\[
\begin{aligned}
0.1\overline{6}
&=
\frac{16-1}{10(10-1)}\\
&=
\frac{15}{90}\\
&=
\frac{1}{6}.
\end{aligned}
\]
8. Negative recurring decimals
Convert the positive value first, then attach the negative sign to the final fraction:
\[
\begin{aligned}
-0.1\overline{6}
&=
-\frac{1}{6}.
\end{aligned}
\]
The repeating pattern and simplification rules are the same.
9. Simplifying the final fraction
After the raw fraction is formed, simplify using the greatest common divisor:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a/\gcd(a,b)}{b/\gcd(a,b)}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{15}{90}
&=
\frac{15\div15}{90\div15}\\
&=
\frac{1}{6}.
\end{aligned}
\]
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Using \(0.3\) when the intended value is \(0.\overline{3}\). These are different numbers.
- Forgetting the non-repeating part in numbers such as \(0.1\overline{6}\).
- Using too few repeated digits from a written decimal like \(0.142857142857\ldots\) without identifying the repeating block.
- Not simplifying the final fraction.
- Forgetting that the repeating block continues forever.
Key idea: multiply by powers of 10 so the infinite repeating tails line up, subtract them away, and simplify the resulting fraction.
