Example1.5.9

Can we convert the repeating decimal \(9.134343434\ldots=9.1\overline{34}\) to a fraction? The trick is to separate its terminating part from its repeating part, like this:

\begin{equation*} 9.1+0.034343434\ldots\text{.} \end{equation*}

Now note that the terminating part is \(\frac{91}{10}\text{,}\) and the repeating part is almost like our earlier examples, except it has an extra \(0\) right after the decimal. So we have:

\begin{equation*} \frac{91}{10}+\frac{1}{10}\cdot0.34343434\ldots\text{.} \end{equation*}

With what we learned in the earlier examples and basic fraction arithmetic, we can continue:

\begin{align*} 9.134343434\ldots\amp=\frac{91}{10}+\frac{1}{10}\cdot0.34343434\ldots\\ \amp=\frac{91}{10}+\frac{1}{10}\cdot\frac{34}{99}\\ \amp=\frac{91}{10}+\frac{34}{990}\\ \amp=\frac{91\multiplyright{99}}{10\multiplyright{99}}+\frac{34}{990}\\ \amp=\frac{9009}{990}+\frac{34}{990}=\frac{9043}{990} \end{align*}

Check that this is right by entering \(\frac{9043}{990}\) into a calculator and seeing if it returns the decimal we started with, \(9.134343434\ldots\text{.}\)

in-context