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%TCIDATA{<META NAME="Title" CONTENT="Real Numbers">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:08:30 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
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\input{tcilatex}
\begin{document}


\section{Real Numbers\label{Basic operations}}

The real numbers include the integers, fractions (rational numbers), as well
as irrational numbers such as $\sqrt{2}$ and $\pi $ that cannot be expressed
as a quotient of integers.

\strut

\begin{quotation}
\textsl{Reminder\quad }Numbers used in computing must be entered in
mathematics mode and appear red on your screen. If that is not the case,
select each number, or combination of numbers, and click \FRAME{itbpF}{%
0.2992in}{0.2992in}{0.0692in}{}{}{text.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid-file
"F";width 0.2992in;height 0.2992in;depth 0.0692in;original-width
16.0625pt;original-height 16.0625pt;cropleft "0";croptop "1.0179";cropright
"1.0179";cropbottom "0";filename 'Text.wmf';file-properties "XNPEU";}} to
convert to mathematics.
\end{quotation}

\strut

\subsection{\ Basic Operations\label{Evaluate}\label{Floating point form}%
\label{Evaluate numeric}}

The result of applying \textsf{Evaluate} to a combination of integers and
fractions will be a fraction. However, If any of the components of a
combination of numbers is written in floating point form, that is, with a
decimal, the result will be in decimal notation. Symbolic real numbers such
as $\sqrt{2}$ and $\pi $ will retain symbolic form unless evaluated
numerically.

\begin{quote}
$\blacktriangleright $ With the insertion point in a sum, difference,
product, or quotient choose \textsf{Evaluate}
\end{quote}

\begin{quotation}
$235.3+813=\allowbreak 1048.\,3\vspace{6pt}$

$42\left( \dfrac{2}{3}+\dfrac{1}{7}\right) \sqrt{2}\vspace{6pt}=\allowbreak
34\sqrt{2}$

$9.6\pi -2.7\pi =\allowbreak 6.\,\allowbreak 9\pi $

$34+162-\dfrac{25}{4}+\dfrac{13}{7}-29\div 6.97=\allowbreak 187.\,45$
\end{quotation}

\begin{quote}
$\blacktriangleright $ With the insertion point in a sum, difference,
product, or quotient choose \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$235.3+813\vspace{6pt}=\allowbreak 1048.\,3$

$42\left( \dfrac{2}{3}+\dfrac{1}{7}\right) \sqrt{2}\vspace{6pt}=\allowbreak
48.\,083$

$9.6\pi -2.7\pi \vspace{6pt}=\allowbreak 21.\,677$

$34+162-\dfrac{25}{4}+\dfrac{13}{7}-29\div 6.97=\allowbreak 187.\,45$
\end{quotation}

\medskip

You can control the number of decimal places returned by \textsf{Evaluate
Numerically} in the \hyperref{Maple Settings}{}{}{Maple settings}
dialog.\medskip

\subsection{\ Powers and Radicals 
\index{Powers@Powers}\label{Powers}}

To raise numbers to powers, use common notation for powers and apply \textsf{%
Evaluate}. 
\index{Exponents@Exponents}\label{Exponents} 
\index{Radicals@Radicals}\label{Radicals}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$3^{4}=\,81%
\vspace{4pt}$

$3^{-4}=\,\dfrac{1}{81}$

$2^{15}=\,32768$

$\left( \dfrac{2}{5}\right) ^{32}=\,\dfrac{42949\,67296}{232\,83064\,36538%
\,69628\,90625}\vspace{4pt}$

$\left( 0.4\right) ^{32}=\,1.844674407\times 10^{-13}\vspace{4pt}$

$\sqrt{2.34}=\,1.529705854$

$\left( 2.5\right) ^{\frac{4}{5}}=\,2.081383019$

$a^{-1}=\,\dfrac{1}{a}$

$\vspace{4pt}8^{\frac{1}{3}}=\,\sqrt[3]{8}$

$8.0^{\frac{1}{3}}=\,2.0$
\end{quotation}

\strut

Note that \textsf{Evaluate} returns a different answer for $\left( \frac{2}{5%
}\right) ^{32}$ and $\left( 0.4\right) ^{32}$. The fraction displayed for $%
\left( \frac{2}{5}\right) ^{32}$ is the exact answer, and the number
displayed for $\left( 0.4\right) ^{32}$ is the best $10$-digit approximation
to the exact answer.

\strut

Apply \textsf{Evaluate} to get $8^{\frac{1}{3}}=\sqrt[3]{8}$. Apply \textsf{%
Evaluate Numerically} to get $\,\sqrt[3]{8}=\,2.0$.

\strut

\subsection{\ Rationalizing a Denominator \label{Rationalizing a denominator}
}

You can rationalize the denominator 
\index{Rationalizing a denominator@Rationalizing a denominator}of a
fraction. Leave the insertion point in the fraction and from the \textsf{%
Maple} menu, choose \textsf{Factor}.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$%
\dfrac{1}{\sqrt{2}}=\,\dfrac{1}{2}\sqrt{2}$

\strut

$\dfrac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2}$

\strut

$\dfrac{\sqrt{2}+\sqrt{3}}{\sqrt{5}-\sqrt{7}}=\,-\frac{1}{2}\sqrt{2}\sqrt{7}-%
\frac{1}{2}\sqrt{2}\sqrt{5}-\frac{1}{2}\sqrt{3}\sqrt{7}-\frac{1}{2}\sqrt{3}%
\sqrt{5}$
\end{quotation}

\strut

The commands \textsf{Evaluate} and \textsf{Simplify}, which you might expect
to use for this purpose, generally have little or no effect on expressions
of this type.$\,$

\strut

\subsection{\ Numerical Approximations \label{Numerical approximations}\label%
{Evaluate numerically}}

\index{Numerical results@Numerical results}\label{Numerical results}\label%
{Floating point real number}\textsl{Scientific Notebook} returns an exact,
or symbolic, answer whenever appropriate and otherwise returns a numerical
approximation. You can force a numerical result to any evaluation either by
choosing\textsf{\ Evaluate\ Numerically} 
\index{Evaluate numerically@Evaluate numerically}from the \textsf{Maple}\
menu or by starting with numbers in decimal notation. You obtain numerical
approximations in response to any operation when you enter numbers with
decimals because such a number is interpreted as a floating-point real
number and \emph{not} as a rational number.

Numerical analysis may be described as the study of errors introduced by
using\textsl{\ floating point} arithmetic (\textsl{round-off errors}) and by
using a finite number of terms when an infinite number of terms are required
for exactness (\textsl{truncation errors}). Floating point is a data type
that is machine dependent. It is important to understand that a floating
point number is neither rational nor irrational; indeed, each floating-point
number represents an infinite number of possible rational numbers and an
infinite number of possible irrational numbers.

Computer algebra systems use what is called \textsl{infinite precision} or 
\textsl{extended precision} to represent integers and rationals exactly.
Numbers such as $\sin 1$, $%
\sqrt{2}$, and $\pi $ are examples of numbers that are represented exactly.
Numerical evaluation leads to the following approximations, with \textsf{%
Digits Used in Computations} and \textsf{Digits Used in Display} both set to
25 in \hyperref{Maple Settings}{}{}{Maple settings}. 
\begin{eqnarray*}
\sin 1 &=&.\,\allowbreak 841\,470\,984\,\allowbreak
807\,896\,506\,\allowbreak 652\,502\,3 \\
\sqrt{2} &=&1.\,\allowbreak 414\,213\,562\,\allowbreak
373\,095\,048\,\allowbreak 801\,689 \\
\pi &=&1.\,\allowbreak 414\,213\,562\,\allowbreak 373\,095\,048\,\allowbreak
801\,689
\end{eqnarray*}

Notice that the approximations are broken into blocks of length 3 decimal
digits in order to make them more readable. The numbers on the left are
exact, while the numbers on the right are merely approximations. In
particular, 
\[
\left( \sqrt{2}\right) ^{2}=\allowbreak 2 
\]
exactly whereas 
\[
1.\,\allowbreak 414\,213\,562\,\allowbreak 373\,095\,048\,\allowbreak
801\,689^{2}=2.\,\allowbreak 000\,000\,000\,\allowbreak
000\,000\,000\,\allowbreak 000\,001 
\]
with \textsf{Digits Used in Display} set to 25.

Contrast the results of evaluating the following expressions with \textsf{%
Evaluate} and \textsf{Evaluate Numerically}.

\begin{center}
\begin{tabular}{ll}
\textsf{Evaluate} & \textsf{Evaluate Numerically} \\ \hline
$82\div 37=\,\frac{82}{37}$ & $82\div 37=\,2.216216216\vspace{6pt}$ \\ 
$936/14=\,\frac{468}{7}$ & $936/14=\,66.85714286\vspace{6pt}$ \\ 
$936/14.0=\,66.85714286\qquad $ & $936/14.0=\,66.85714286\vspace{6pt}$ \\ 
$\frac{14.2}{83.5}=\,.1700598802$ & $\frac{14.2}{83.5}=\,.1700598802\vspace{%
6pt}$ \\ 
$\frac{2}{3}\div \frac{8}{7}=\,\frac{7}{12}$ & $\frac{2}{3}\div \frac{8}{7}%
=\,.5833333333\vspace{6pt}$ \\ 
$\sqrt{234}=\,3\sqrt{26}$ & $\sqrt{234}=\,15.29705854\vspace{6pt}$ \\ 
$\left( 5^{2}\right) ^{3}=\,15625$ & $\left( 5^{2}\right)
^{3}=\,1.5625\times 10^{4}\vspace{6pt}$%
\end{tabular}
\end{center}

You can change the number of digits displayed in these approximations by
changing the setting in a dialog box. You can also change the number of
digits used in computations and the threshold for scientific notation in
that dialog box. These options are discussed in greater detail in the
following section.

\strut

\subsection{\ Scientific Notation\label{Scientific notation}}

Any nonzero real number $x$ can be written in the form 
\[
x=c\times 10^{n} 
\]
with $1\leq c<10$ and $n$ an integer. A number in this form is in \emph{%
scientific notation}.

\begin{quote}
$\blacktriangleright $ To write a number in scientific notation
\end{quote}

\begin{enumerate}
\item  Enter the number $c$ to as many decimal places as appropriate.

\item  Choose $\times $ from the \textsf{Common Symbols} toolbar or from the 
\textsf{Binary Operations} symbol panel under \FRAME{itbpF}{0.3009in}{%
0.3009in}{0.0701in}{}{}{binops.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1.1563";cropright
"1.1011";cropbottom "0";filename 'Binops.wmf';file-properties "XNPEU";}}.

\item  Enter the number $10$.

\item  Choose \textsf{Insert + Superscript}, or click \FRAME{itbpF}{0.3009in%
}{0.3009in}{0.0701in}{}{}{supscrip.wmf}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1.1563";cropright
"1.1011";cropbottom "0";filename 'Supscrip.wmf';file-properties "XNPEU";}},
and enter the integer $n$ in the input box.
\end{enumerate}

This is illustrated in the following examples.

\begin{eqnarray*}
12 &=&1.2\times 10 \\
1274.9837 &=&1.2749837\times 10^{3} \\
.000001234 &=&1.234\times 10^{-6}
\end{eqnarray*}
\bigskip

\subsection{\ The Display of Numerical Results\label{Maple settings}}

The number of digits used in numerical computations and the display of
numerical results are controlled by settings you can change. From the 
\textsf{Maple }menu, choose \textsf{Settings} to bring up the \textsf{Maple} 
\textsf{Settings} dialog box, and choose the tab \textsf{General.\FRAME{dtbpF%
}{4.1857in}{2.6351in}{0pt}{}{}{maplsetg.wmf}{\special{language "Scientific
Word";type "GRAPHIC";display "PICT";valid_file "F";width 4.1857in;height
2.6351in;depth 0pt;original-width 298.875pt;original-height
187.4375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'maplsetg.wmf';file-properties "XNPEU";}}}The first three effect the display
of numerical results.

\subsubsection{Digits Used in Computations}

The first choice, \textsf{Digits Used in Computations}, is self-explanatory.
The dialog box allows you to increase or decrease this number. If you choose
a large number, you improve accuracy, but computations may be significantly
slower. You can try various settings when doing large computations to see
how speed is affected.

\subsubsection{Digits Used in Display}

Digits used in display 
\index{Settings@Settings!digits used in computations@digits used in computations}%
\label{Settings, digits} \label{Digits used in display} 
\index{Settings@Settings!digits used in display@digits used in display}
refers to the number of digits put on the screen as the result of a
numerical computation. This setting does not affect accuracy in
computations, but if you enter a floating-point number in a document using a
larger number of digits than the setting under \textsf{Digits Used in Display%
}, scientific notation is returned when you choose \textsf{Evaluate} or 
\textsf{Evaluate Numerically}. Recall that when you enter numbers in decimal
(floating-point) notation, \textsf{Evaluate\ }and \textsf{Evaluate
Numerically} have the \emph{same} effect.

Numbers are correctly rounded, depending on the value you have set for 
\textsf{Digits Used in Display}. As a general rule, you could reasonably set 
\textsf{Digits Used in Display} to the number of places determined by your
least accurate measurement.

\subsubsection{Threshold for Scientific Notation}

\label{Settings, scientific notation}The \textsf{Maple Settings} dialog
ensures that the threshold for scientific notation cannot exceed the setting
for the number of displayed digits. For example, when \textsf{Digits Used in
Display} is set to $5$, a setting for \textsf{Threshold For Scientific
Notation} greater than $5$ is not accepted.

If the number of digits to the left of the decimal is greater than the
setting under \textsf{Digits Used in Display}, or if the number of leading
zeroes is greater than or equal to that setting, choosing \textsf{Evaluate
Numerically} always returns the number in scientific notation. \thinspace
Otherwise, the setting 
\index{Settings@Settings!threshold for scientific notation@threshold for scientific notation}%
of \textsf{Threshold for Scientific Notation} determines the number of
digits that leads to a response in scientific notation, based on the number
of digits to the left of the decimal or the number of leading zeroes. The
thresholds for a number and its negative are the same.

Some of the effects of different settings are illustrated in the following
table.

\begin{center}
\begin{tabular}{cccc}
\textbf{Digits Used} & \textbf{Threshold for} & \textbf{Number} & \textbf{%
Evaluate} \\ 
\textbf{in Display} & \textbf{Scientific Notation} & \textbf{Entered} & 
\textbf{Numerically} \\ \hline
6 & 3--6 & $100\pi $ & $314.159$ \\ 
6 & 1--2 & $100\pi $ & $3.14159\times 10^{2}$ \\ 
6 & 3--6 & $-100\pi $ & $-314.159$ \\ 
6 & 1--2 & $-100\pi $ & $-3.14159\times 10^{2}$ \\ 
5 & 5 & $56789$ & $56789.$ \\ 
5 & 1--4 & $56789$ & $5.6789\times 10^{4}$ \\ 
4 & 1--4 & $56789$ & $5.679\times 10^{4}$ \\ 
9 & 6--9 & $.000001111$ & $.000001111$ \\ 
9 & 1--5 & $.000001111$ & $1.111\times 10^{-6}$ \\ 
8 & 1--8 & $.000001111$ & $1.111\times 10^{-6}$%
\end{tabular}

\strut
\end{center}

\subsection{\ Related topics}

\begin{itemize}
\item  
\hyperref{Complex Numbers}{}{}{DM2-5.tex#Complex numbers}

\item  \hyperref{Functions and Relations}{}{}{DM2-4.tex#Absolute value}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Digits used in
computation}{}{}{DMIndex.tex#Digits used in computation}, \hyperref{Digits
used in display}{}{}{DMIndex.tex#Digits used in display}, \hyperref{Exponent%
}{}{}{DMIndex.tex#Exponent}, \hyperref{Numerical approximations}{}{}{%
DMIndex.tex#Numerical approximations}, \hyperref{Evaluate numerically}{}{}{%
DMIndex.tex#Evaluate numerically}, \hyperref{Floating point real number}{}{}{%
DMIndex.tex#Floating point}, \hyperref{Maple settings}{}{}{DMIndex.tex#Maple
settings}, \hyperref{Powers}{}{}{DMIndex.tex#Powers}, \hyperref{Radicals}{}{%
}{DMIndex.tex#Radicals}, \hyperref{Rationalizing a denominator}{}{}{%
DMIndex.tex#Rationalizing a denominator}, \hyperref{Real numbers}{}{}{%
DMIndex.tex#Real numbers}, \hyperref{Scientific notation}{}{}{%
DMIndex.tex#Scientific notation}, \hyperref{Threshold for scientific notation%
}{}{}{DMIndex.tex#Threshold for scientific notation}}

\end{document}
