Fractions and decimals are two different ways to write the same number.
A fraction shows division explicitly, while a decimal shows the value in base ten.
1. Fraction to decimal
A fraction has the form:
\[
\begin{aligned}
\frac{p}{q},
\qquad q\ne0.
\end{aligned}
\]
To convert it to a decimal, divide the numerator by the denominator:
\[
\begin{aligned}
\frac{p}{q}=p\div q.
\end{aligned}
\]
2. Terminating decimals
A decimal terminates if the long-division remainder eventually becomes zero.
For example:
\[
\begin{aligned}
\frac{5}{8}
&=5\div8\\
&=0.625.
\end{aligned}
\]
The decimal \(0.625\) stops after three decimal places.
3. Recurring decimals
A decimal recurs if a long-division remainder repeats.
When a remainder repeats, the same digits repeat forever.
Example:
\[
\begin{aligned}
\frac{1}{3}
&=0.3333\ldots\\
&=0.\overline{3}.
\end{aligned}
\]
The bar over \(3\) means that the digit \(3\) repeats forever.
In the calculator input/output, the same value may be written as 0.(3).
4. Longer repeating blocks
Some fractions repeat with more than one digit:
\[
\begin{aligned}
\frac{22}{7}
&=3.142857142857\ldots\\
&=3.\overline{142857}.
\end{aligned}
\]
The repeating block is \(142857\), and its period is 6.
5. Why remainders detect repetition
During long division, each next digit is determined by the current remainder.
If the same remainder occurs again, all later steps repeat exactly.
\[
\begin{aligned}
\text{same remainder}
\quad\Longrightarrow\quad
\text{same next digit pattern}.
\end{aligned}
\]
6. Decimal to fraction: terminating decimals
A terminating decimal can be written over a power of 10.
Count the decimal places, then use \(10^n\) as the denominator.
\[
\begin{aligned}
0.625
&=
\frac{625}{1000}.
\end{aligned}
\]
Then simplify:
\[
\begin{aligned}
\frac{625}{1000}
&=
\frac{5}{8}.
\end{aligned}
\]
7. Decimal to fraction: recurring decimals
For a recurring decimal, use multiplication by powers of 10 so that the repeating tails line up.
Example:
\[
\begin{aligned}
x&=0.\overline{3}.
\end{aligned}
\]
Multiply by 10:
\[
\begin{aligned}
10x&=3.\overline{3}.
\end{aligned}
\]
Subtract the original equation:
\[
\begin{aligned}
10x-x
&=3.\overline{3}-0.\overline{3},\\
9x&=3,\\
x&=\frac{3}{9}=\frac{1}{3}.
\end{aligned}
\]
8. Worked example: \(1.2\overline{34}\)
Let
\[
\begin{aligned}
x&=1.2\overline{34}.
\end{aligned}
\]
There is one non-repeating digit after the decimal point and two repeating digits.
Multiply by \(10^3=1000\) and by \(10^1=10\):
\[
\begin{aligned}
1000x&=1234.\overline{34},\\
10x&=12.\overline{34}.
\end{aligned}
\]
Subtract:
\[
\begin{aligned}
990x&=1222,\\
x&=\frac{1222}{990}\\
&=\frac{611}{495}.
\end{aligned}
\]
9. Simplification
Fractions should be simplified by dividing numerator and denominator by their greatest common divisor:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a/\gcd(a,b)}{b/\gcd(a,b)}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{625}{1000}
&=
\frac{625\div125}{1000\div125}\\
&=
\frac{5}{8}.
\end{aligned}
\]
10. When does a fraction terminate?
In lowest terms, a fraction terminates exactly when the denominator has no prime factors other than 2 and 5.
This is because base ten is built from:
\[
\begin{aligned}
10=2\cdot5.
\end{aligned}
\]
For example, \(5/8\) terminates because \(8=2^3\).
But \(1/3\) recurs because the denominator has a factor of 3.
11. Negative values
The sign does not change the conversion method. Convert the positive value first, then attach the negative sign.
\[
\begin{aligned}
-\frac{1}{3}
&=
-0.\overline{3}.
\end{aligned}
\]
12. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
13. Common mistakes
- Rounding a recurring decimal and treating the rounded value as exact.
- Forgetting to simplify the fraction after converting a decimal.
- Confusing \(0.3\) with \(0.\overline{3}\). The first is \(3/10\), while the second is \(1/3\).
- Using a denominator of zero in a fraction.
- Forgetting that repeating notation 0.(6) means the digit 6 repeats forever.
Key idea: fractions become decimals by division; decimals become fractions by using powers of 10 and then simplifying.
