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\lhead{\color{blue}\large Lecture Notes}
\chead{\Large Fractions and Decimals}
\rhead{\small page \thepage}
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\lfoot{\small \copyright \; Hidegkuti, 2017}
\rfoot{\small Last revised: January 15, 2017}
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\begin{document}
\begin{center}
{\Large Part 1: \ Converting a Fraction to a Decimal}
\end{center}
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This is easy to do if we understand a formal, algebraic definition of a
fraction. \ If $a$ and $b$ are integers, $b$ not zero, then the \textbf{%
fraction} $\dfrac{a}{b}$ is the result of the division $a\div b$. \ In a
sense, fractions are driving instructions. \ They do not tell us the value
of the number, only, how to obtain it. \ To get a decimal, we simply perform
the division.
\textbf{Example 1} \ Convert $\dfrac{3}{8}$ to a decimal.
Solution: \ We perform the division $3\div 8$. \ The result is $0.375.$
\begin{center}
\begin{tabular}{llllll}
& & & $.3$ & $7$ & $5$ \\ \cline{3-6}
& $8$ & $)3$ & $.0$ & $0$ & $0$ \\
& $-$ & $2$ & $4$ & & \\ \cline{3-4}
& & & $6$ & $0$ & \\
& & $-$ & $5$ & $6$ & \\ \cline{4-5}
& & & & $4$ & $0$ \\
& & & $-$ & $4$ & $0$ \\ \cline{5-6}
& & & & & $0$%
\end{tabular}
\end{center}
\textbf{Example 2} \ Convert \ $\dfrac{1927}{11}$ to a decimal.
We perform the division $1927\div 11$. \ The result is $175.\,\allowbreak
\overline{18}.$ \ The bar over the last two digits indicates an infinitely
many times repeating block.
\begin{center}
\begin{tabular}{lllllllllll}
& & & & $1$ & $7$ & $5$ & $.1$ & $8$ & $1$ & $8...$ \\ \cline{3-11}
& $11$ & ) & $1$ & $9$ & $2$ & $7$ & $.0$ & $0$ & $0$ & $0$ \\
& $-$ & & $1$ & $1$ & & & & & & \\ \cline{3-5}
& & & & $8$ & $2$ & & & & & \\
& & & $-$ & $7$ & $7$ & & & & & \\ \cline{5-6}
& & & & & $5$ & $7$ & & & & \\
& & & & $-$ & $5$ & $5$ & & & & \\ \cline{6-7}
& & & & & & $2$ & $0$ & & & \\
& & & & & $-$ & $1$ & $1$ & & & \\ \cline{7-8}
& & & & & & & $9$ & $0$ & & \\
& & & & & & $-$ & $8$ & $8$ & & \\ \cline{8-9}
& & & & & & & & $2$ & $0$ & \\
& & & & & & & $-$ & $1$ & $1$ & \\ \cline{9-10}
& & & & & & & & & $9$ & $0$ \\
& & & & & & & & $-$ & $8$ & $8$ \\ \cline{10-11}
& & & & & & & & & & $2$%
\end{tabular}
\end{center}
\vspace{0.5in}\vspace{1in}\vspace{0.5in}
\begin{center}
{\Large Part 2:\ \ Converting a Terminating Decimal to Fraction}
\end{center}
A decimal is \textbf{terminating} if it has a last digit. \ It is quite easy
to turn a terminating decimal to a fraction of integers.
\textbf{Example 3} \ Convert $0.45$ to a reduced fraction.
Step 1. Write it as a fraction of any kind first.
\begin{equation*}
0.45=\dfrac{0.45}{1}
\end{equation*}%
We can mentally check it as division: any number divided by one results in
the same number.
Step 2. \ We ask ourselves: "How many digits do we need to move the decimal
point to the right in $0.45$\ to obtain an integer"? \ The answer is: two
digits. \ Moving the decimal point to the right by two digits is the same as
multiplication by $100.$ \ Thus, to fix the numerator, we need to multiply
it by $100$. \ Because we also want to preserve the value, we multiply both
upstairs and downstairs by $100$.%
\begin{equation*}
\dfrac{0.45}{1}=\dfrac{0.45\cdot 100}{1\cdot 100}=\dfrac{45}{100}
\end{equation*}
Step 3. We simplify the fraction by dividing upstairs and downstairs by the
greatest common divisor.%
\begin{equation*}
\dfrac{45}{100}=\dfrac{\NEG{5}\cdot 9}{\NEG{5}\cdot 20}=\dfrac{9}{20}
\end{equation*}
\textbf{Example 4} \ Convert $0.0005$ to a reduced fraction.
We will multiply upstairs and downstairs by $10000$.
\begin{equation*}
0.0005=\dfrac{0.0005}{1}=\dfrac{0.0005\cdot 10000}{1\cdot 10000}=\dfrac{5}{%
10000}=\dfrac{\NEG{5}\cdot 1}{\NEG{5}\cdot 2000}=\dfrac{1}{2000}
\end{equation*}
\textbf{Example 5} \ Convert $23.044$ to a reduced fraction.
\begin{equation*}
23.044=23+0.044=23+\dfrac{0.044}{1}=23+\dfrac{0.044\cdot 1000}{1\cdot 1000}%
=23\dfrac{44}{1000}=23\dfrac{4\cdot 11}{4\cdot 250}=23\dfrac{11}{250}
\end{equation*}%
\vspace{0.15in}
\begin{center}
{\Large Part 3: \ (The Fun Stuff)\\[0pt]
Converting a Non-Terminating Decimal to Fraction}\bigskip
\end{center}
A decimal is non-terminating if it has infinitely many digits. \ If there is
a repeating block, we denote it by a bar drawn over the repeating digit. \
For example, the number \ $2.\overline{35}$ \ denotes $2.35\,35\,35\,35.....$%
.
\textbf{Example 6 \ }Re-write each of the given repeating decimals without
the bar notation. \
\qquad \qquad \qquad \qquad \qquad \qquad a) \ $1.201\overline{7}$ \qquad b)
\ $1.20\overline{17}$ \qquad c) \ $1.2\overline{017}$ \qquad\ d) \ $1.%
\overline{2017}$
Solution: \ Only the digit(s) under the bar are repeating. \ The rest is
there as is.
\qquad \qquad a) \ $1.201\overline{7}=1.201\,777777....$ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \qquad\ \ \ \ \ c) \ $1.2\overline{017}=1.2\,017\,017\,017\,017%
\,017....$
\qquad \qquad b) \ $1.20\overline{17}=1.20\,17\,17\,17\,17\,17\,17....$ \ \
\ \ \ \ \qquad\ \ \ d) \ $1.\overline{2017}=1.2017\,2017\,2017\,2017%
\,2017....$
Turning these decimals into fractions of integers is an interesting and fun
application of linear equations.
\vspace{3in}
\textbf{Example 7 \ }Convert the repeating decimal \ $7.\overline{4}$ \ to a
fraction.
Step 1. \ We label our number $x$ and write it without the bar notation. \
The dots are important: they indicate that we have infinitely many $%
4^{\prime }$s there and not just three.%
\begin{equation*}
7.444...=x
\end{equation*}%
Step 2. \ We multiply both sides of this equation by $10.$%
\begin{equation*}
74.444...=10x
\end{equation*}%
Step 3. \ We write these equations together, starting with the second one.%
\begin{eqnarray*}
74.444... &=&10x \\
7.444... &=&x
\end{eqnarray*}%
Step 3. (Chop, chop.) \ We subtract the second equation from the first one.%
\begin{gather*}
~~74.444...=10x \\
-\underline{~~~~~~7.444...=x~~~~~~~~~~} \\
67~~~~~~~~~~~=9x~~
\end{gather*}%
Step 4. We solve the equation for $x$.%
\begin{eqnarray*}
67 &=&9x\text{ \ \ \ \ \ divide by }9 \\
\dfrac{67}{9} &=&x
\end{eqnarray*}%
Thus the answer is $\dfrac{67}{9}$. We can check by long division. \ Indeed,
$67\div 9=\allowbreak 7.\,\allowbreak 444\,444\,44...$
\textbf{Example 8 \ }Convert the repeating decimal \ $0.\overline{405}$ \ to
a fraction.
Step 1. \ We label our number $x$ and write it without the bar notation. \ $%
0.405\,405\,405\,405\,405...=x$
Steps 2 and 3. \ This decimal has a three-digit long repeating block. \ To
obtain proper alignment of the digits, we will move the decimal point by
three digits, i.e. we will multiply by $1000$. \ We multipliy both sides of
this equation by $1000.$ We write these equations together, starting with
the second one.%
\begin{eqnarray*}
405.405\,405\,405\,405... &=&1000x \\
~~~~~~~0.405\,405\,405\,405... &=&x
\end{eqnarray*}%
Step 3. (Chop, chop.) \ We subtract the second equation from the first one.%
\begin{gather*}
~~~~~~~405.405\,405\,405\,405...~~=1000x~~~~~ \\
\underline{~~-~~~~0.405\,405\,405\,405....=x~~~~~~~~~~~~~} \\
405~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=999x
\end{gather*}%
Step 4. We solve the equation for $x$.%
\begin{eqnarray*}
405 &=&999x\text{ \ \ \ \ \ divide by }999 \\
\dfrac{405}{999} &=&x
\end{eqnarray*}%
Please note that the fraction obtained is not reduced. \ However, the
essential point in the problem is to find a fraction, not the reduced form
of it. \ Thus the answer is $\dfrac{405}{999}$. We can check by long
division. \ Indeed, $405\div 999=\allowbreak 0.405\,405\,405....$
\textbf{Example 9 \ }Convert the repeating decimal \ $18.29\overline{04}$ \
to a fraction.
Step 1. \ We label our number $x$ and write it without the bar notation.%
\begin{equation*}
18.29\,04\,04\,04\,04...=x
\end{equation*}%
Steps 2 and 3. \ This decimal has a two-digit long repeating block. \ To
obtain proper alignment of the digits, we will move the decimal point by two
digits, i.e. we will multiply by $100$. \ We multipliy both sides of this
equation by $100.$ We write the two equations together, starting with the
second one.%
\begin{eqnarray*}
1829.04\,04\,04\,04\,04... &=&100x \\
18.29\,04\,04\,04\,04... &=&x
\end{eqnarray*}%
Step 3. (Chop, chop.) \ We subtract the second equation from the first one.%
\begin{gather*}
~~~~~~~~~~~1829.04\,04\,04\,04\,04...=100x~~~~~~~~~~~ \\
\underline{~~~~~-~~~~~~~~18.29\,04\,04\,04\,04...=x~~~~~~~~~~~~~~~~~~~~} \\
1810.75~~~~~~~~~~~~~~~~~=99x~~~~~~~~
\end{gather*}%
It appears that we have a problem: the right-hand side is not an integer
after the subtraction. \ This is quite easy to fix: we just multipliy both
sides by $100$.%
\begin{eqnarray*}
1810.75 &=&99x\text{ \ \ \ \ \ \ multiply by }100 \\
181075 &=&9900x
\end{eqnarray*}%
Step 4. We solve the equation for $x$.%
\begin{eqnarray*}
181075 &=&9900x\text{ \ \ \ \ \ divide by }9900 \\
\dfrac{181075}{9900} &=&x
\end{eqnarray*}%
Thus the answer is $\dfrac{181075}{9900}$. We can check by long division. \
Indeed, $181075\div 9900=\allowbreak 18.\,\allowbreak 290\,40404...$
$%
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$ \ \ {\Large Practice Problems}%
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\begin{enumerate}
\item Perform each of the following conversions.
a) \ Convert the given fraction to decimals.
\qquad i) $\ \dfrac{4}{5}$ \ \qquad \qquad ii) $\ \dfrac{26}{3}$ \qquad
\qquad \qquad iii) $\ \dfrac{26}{25}$ \qquad \qquad \qquad iv) $\ \dfrac{26}{%
7}$ \
How many digits long is the repeating block?
b) \ Convert the given decimal to a fraction of integers. (You do not have
to reduce them!)
\qquad i) $\ 2.\,\allowbreak 18$ \ \ \ \ \ \ \ \qquad ii) $\ 2.\,\allowbreak
\overline{9}$\ \ \ \ \ \ \ \ \qquad iii) $\ 6.\,\allowbreak \overline{47}$ \
\ \ \ \ \ \ \ \ \ \qquad iv) $\ 1.8\overline{705}$
\item Based on your answer for 1b ii), what is a surprising new fact about
decimal presentation of numbers?
\item Consider the fraction $\dfrac{1}{n}$. \ What numbers $n$ will result
in a terminating decimal?
\item The decimal presentation of $\dfrac{2}{13}$ does not appear to be
repeating. Is it?
\item Find a fraction formed of two integers that will result in a
non-terminating, non-repeating decimal.
\end{enumerate}
\pagebreak
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\ {\Large Answers to Practice Problems}
\begin{enumerate}
\item a) \ i) \ $0.8$\ \ \ \ \ ii) \ $8.\overline{6}$ \ \ \ \ iii) $\ 1.04$
\ \ \ \ iv) \ $3.\,\allowbreak \overline{714\,285}$, \ the repeating block
is $6$ digits long
b) \ i) \ $\dfrac{218}{100}$\ \ \ \ \ ii) \ $3$\ \ \ \ \ \ iii) \ $\dfrac{641%
}{99}$ \ \ \ \ \ \ iv) \ $\dfrac{18687}{9990}$
\item The decimal presentation of real numbers is not unique. \ \ \ \ 3. \ $%
n $ \ can only have $2$ and $5$ in its prime-factorization
\item[4.] It is repeating, only the repeating block is $6$ digits long, \ $0.%
\overline{153\,846}$ \ \ \ \ 5. \ It is impossible
\end{enumerate}
\vspace{2in}\vspace{5in}{\Large
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\href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{%
For more documents like this, visit our page at\
https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail
questions or comments to mhidegkuti@ccc.edu.}
\end{document}