Repeating (Recurring) Decimal Expansions

Let’s find the decimal expansion of the number $ \displaystyle\frac{7}{5} $.

$$\frac{7}{5}= \frac{7 \cdot 2}{5 \cdot 2} = \frac{14}{10} = 1.4 $$

Let’s find the decimal expansion of the number $ \displaystyle\frac{14}{10} $.

$$\frac{14}{10}= \frac{140}{100} = \frac{1400}{1000} = \frac{14000}{10000} \cdots $$

$$ 1.4 = 1.40 = 1.400 = 1.4000 = \cdots $$

In the decimal notation of fractions, zeros appended to (or deleted from) the very end of the fractional part do not change the value of the decimal expression. Here, zero is the repeating (recurring) digit. Decimals of this type are called terminating decimals with a repeating zero.

Consequently, every rational number whose denominator is a power of 10 (or can be converted into this form) possesses a decimal expansion with a trailing repeating zero.

Decimal Expansions of Rational Numbers That Cannot Be Converted into Terminating Decimals

Since the denominator of the fraction $\displaystyle \frac{2}{3} $ is not a factor of any power of 10, the fraction $\displaystyle\frac{2}{3}$ cannot be written as a terminating decimal fraction. Accordingly, let’s find the decimal expansion of the number $\displaystyle \frac{2}{3} $ by dividing the numerator of the fraction by its denominator.

\[
\begin{array}{c|c}
\begin{array}{c} 2 \\
\\
\end{array}&
\begin{array}{c}
\quad 3 \\
\hline \\
\end{array}
\end{array}
\quad \quad \quad \begin{array}{c,c}
&\begin{array}{c}
\quad &2.000 \;\;\;\;\\
-\quad &1\phantom{0}8 \;\;\;\; \\
\hline
\;\;\quad &0\phantom{.} 20\\
-\quad \;\;\quad &0\phantom{.} 18\\
\hline
\;\;\quad &0\phantom{.} 2\\\end{array}
\begin{array}{|c}
\quad 3 \\
\hline
\quad 0.666\cdots
\\
\\
\\
\\
\end{array}
\\
\\
\\
\\
\end{array}
\]

In the division operation performed, the digit 6 repeats continuously in the quotient. Therefore:

$$ \frac{2}{3} = 0.666\cdots$$

holds. We can write the number $0.666\dots$ briefly as $ 0.\overline{6} $ by placing a bar over the repeating part. That is, the notation \[ 0.666\cdots = 0.\overline{6} \] is used.

The number $ 0.\overline{6} $ is called the repeating decimal expansion of the rational number $\displaystyle \frac{2}{3} $. Similarly, the fractions $\displaystyle \frac{8}{3}, \displaystyle \frac{8}{15}, \displaystyle\frac{4}{11}, \text{ and } \frac{4}{7} $ cannot be written as terminating decimals either. However, by dividing the numerators of these fractions by their denominators, their repeating decimal expansions are found as:

$$ \frac{8}{3}= 2.666\cdots =2.\overline{6} $$

$$ \frac{8}{15}= 0.5333\cdots =0.5\overline{3} $$

$$ \frac{4}{11}=0.3636\cdots =0.\overline{36} $$

$$ \frac{4}{7}=0.\overline{571428} $$

If the digits in the fractional part of a rational number’s decimal expansion repeat continuously according to a specific rule after a certain place value, this number is called a repeating (periodic) decimal number, and it is written by placing a bar ‘-‘ over the repeating block.

Warning:

1. Every rational number has a decimal expansion.

2. In the decimal expansion of rational numbers that form terminating fractions, the repeating digit is zero, and the repeating zero is written at the end of the expansion as $ \overline{0} $ only when specifically required.

3. In the decimal expansions of rational numbers that do not form terminating fractions, the repeating digit/block is non-zero.

Converting a Repeating Decimal Number into a Rational Fraction:

Let’s find the rational notation of the number $0.2\overline{6}$.

Let \[ x = 0.2\overline{6} \]

If both sides are multiplied by appropriate powers of 10 such that the decimal point falls both right before and right after the repeating part, and side-by-side subtraction is performed:

\[\begin{align*}
100x &= 26.\overline{6} \\
-\quad 10x &= 2.\overline{6} \\
\hline
90x &= 24 \\
\\
x &= \frac{24}{90} = \frac{4}{15} \quad \text{is found.}
\end{align*} \]

Practically, a repeating decimal number can be converted into a rational form using the following formula:

\[
\begin{array}{l l }\quad \quad \quad \quad \quad\quad\quad(\text{The Whole Number}) \quad – \quad (\text{The Non-repeating Part}) \\
\hline
\text{9 for each repeating decimal place, 0 for each non-repeating decimal place}
\end{array}
\]

Here, the rule for the denominator applies strictly to the fractional (post-decimal) part of the number. Expressing this with symbols:

Where $a, b, c, d, e, f$ are digits and \[ a.bcdefdef\dots = a.bc\overline{def} \], then

$$ a.bc\overline{def} = \frac{\text{abcdef} – \text{abc}}{99900} = a + \frac{\text{bcdef} – \text{bc}}{99900} $$

Examples:

\[ \Rightarrow 1.\overline{4} = \frac{14-1}{9} = \frac{13}{9} = 1 \frac{4}{9} \quad \text{or} \quad 1.\overline{4} = 1 + 0.\overline{4} = 1 \frac{4}{9} \]

\[ \Rightarrow 0.8\overline{5} = \frac{85-8}{90} = \frac{77}{90} \]

\[
\Rightarrow 12.48\overline{3} = 12 + \frac{483 – 48}{900} = 12\frac{29}{60}
\]

\[
\Rightarrow 0.\overline{63} = \frac{63}{99} = \frac{7}{11}
\]

Warning:

Where $a$ is a single digit:

$$ 0.\overline{a} = \frac{a}{9} $$

2. If the only repeating digit is 9, the digit immediately to the left of 9 is increased by 1 in its numerical value, and the repeating 9 is dropped.

$$ 2.3\overline{9} = 2.4; \quad 0.1\overline{9} = 0.2; \quad 1.9\overline{9} = 2 $$
*(Note: Corrected textbook typos in the original lines where the expressions incorrectly read $0.1\overline{9}=2$ and repeated the same item with identical outputs due to typographical oversights).*

3. When adding repeating decimals, if the operations inside the repeating blocks do not generate values greater than 9, or when subtracting, if it is not necessary to borrow from the place value to the left, the operations can be performed directly on the decimal numbers without converting them into fractions.

$$ 3.\overline{24} + 1.\overline{65} = 4.\overline{89} $$

$$ 2.\overline{37} + 0.\overline{62} = 2.\overline{99} = 3 $$

$$ 3.\overline{76} – 1.\overline{42} = 2.\overline{34} $$

Rounding Decimal Fractions:

Finding a decimal fraction with fewer digits that can be taken as approximately equal to a given decimal fraction is called rounding a decimal fraction.

To round a decimal fraction to a desired place value, the digit immediately to its right is examined. If the value of this digit is:

5 or greater, 1 is added to the numerical value of the place being rounded, and all digits to its right are dropped.
Less than 5, the digit in the place being rounded is kept exactly as it is, and all digits to its right are dropped.

Examples:

Let’s round the numbers 2.173 and 0.642 to the tenths place.

\[
2.173 \approx 2.2 \quad (7 > 5)
\]
\[
0.642 \approx 0.6 \quad (4 < 5)
\]

Let’s round the numbers 2.72384 and 5.19349 to the third decimal place (thousandths place).

\[
2.72384 \approx 2.724 \quad (8 > 5)
\]
\[
5.19349 \approx 5.193 \quad (4 < 5)
\]