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ex: Solved some other exercises, added code snippets, and more

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Samuel Ortion 2023-02-27 11:07:49 +01:00
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{
"My exercise environment": {
"scope": "latex,tex",
"prefix": "myexercise",
"body": [
"\\begin{myexercise}",
"\t$1",
"\\end{myexercise}"
],
},
"My answer environment": {
"scope": "latex,tex",
"prefix": "myanswer",
"body": [
"\\begin{myanswer}",
"\t$1",
"\\end{myanswer}"
],
},
}

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tex/content/chapters/part01/1.tex
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@ -60,8 +60,6 @@
\end{myanswer} \end{myanswer}
\end{myexercise} \end{myexercise}
\begin{myexercise}{5}{19} \begin{myexercise}{5}{19}
A prime number is an integer $> 1$ that has no exact divisors other than $1$ and itself. A prime number is an integer $> 1$ that has no exact divisors other than $1$ and itself.
@ -268,3 +266,144 @@ T_n = \frac{1}{2} n (n + 1)
% \end{myexercise} % \end{myexercise}
% Dirty false proof... To be continued % Dirty false proof... To be continued
\subsection{Numbers, Powers and Logarithms}
\begin{myexercise}{1}{25}
What is the smallest positive rational number?
\begin{myanswer}
The smallest positive rational number is 0.
\end{myanswer}
\end{myexercise}
\begin{myexercise}{2}{25}
Is $1 + 0.23999999999...$ a decimal expansion?
\begin{myanswer}
0.23999999999... is not a decimal number as it has an infinite decimal expansion.
The decimal expansion of $1 + 0.23999999999...$ is in the form
\[
1 \cdot 10^0 + 2 \cdot 10^{-1} + 3 \cdot 10^{-2} + 9 \cdot 10^{-3} + \sum_{i=1}^{\infty} 9 \cdot 10^{-3-i}
\]
\textit{Nota bene}: This surely does not answer the question...
\end{myanswer}
\end{myexercise}
\begin{myexercise}{3}{25}
What is $(-3)^{-3}$?
\begin{myanswer}
\[
(-3)^{-3} = \frac{1}{(-3)^3} = \frac{1}{(-3)(-3)(-3)} = -\frac{1}{27}
\]
\end{myanswer}
\end{myexercise}
\begin{myexercise}{4}{25}
What is $(0.125)^{-2/3}$?
\begin{equation}
b^{p/q} = \sqrt[q]{b^p}
\end{equation}
\begin{myanswer}
\begin{align*}
0.125^{-2/3} &= \left(\frac{1}{8}\right)^{-2/3} \\
&= 8^{2/3} \\
&= \sqrt[3]{8^2} \\
&= \sqrt[3]{64} \\
&= 4
\end{align*}
\end{myanswer}
\end{myexercise}
\begin{myexercise}{5}{25}
Defining real numbers in terms of binary expansion.
\begin{myanswer}
\[
x = \sum_{i=0}^{\infty} a_i \cdot 2^{-i}
\]
\end{myanswer}
\end{myexercise}
\begin{myexercise}{6}{25}
Let $x = m + 0.d_1d_2\dots$ and $y = n + 0.e_1e_2\dots$.
Give a rule for determining whether $x = y$, $x < y$, $x > y$ based on the decimal representation.
\begin{myanswer}
\begin{algorithm}[H]
\begin{algorithmic}[1]
\Procedure {LowerGreaterEqual} {$X$, $Y$}
\Comment{$X$ and $Y$ are arrays of decimal of numbers $x$ and $y$ respectively.}
\State $i \gets 0$
\While {$i < X.length$ and $i < Y.length$}
\If{$X[i] == Y[i]$}
\State $i \gets i+1$
\ElsIf {$X[i] > Y[i]$}
\State \Return \texttt{GREATER}
\Else
\State \Return \texttt{LOWER}
\EndIf
\EndWhile
\If{$X.length > Y.length$}
\State \Return \texttt{GREATER}
\ElsIf{$X.length < Y.length$}
\State \Return \texttt{LOWER}
\Else
\State \Return \texttt{EQUALS}
\EndIf
\EndProcedure
\end{algorithmic}
\caption{Greater, Lower or Equal Decimal representation}
\end{algorithm}
\end{myanswer}
\end{myexercise}
\begin{myexercise}{7}{25}
Given that $x$ and $y$ are integers. \textit{Prove} the laws of exponents, from definition given by Eq. \ref{eq:rule-of-exponents-2} from Eq. \ref{eq:rule-of-exponents}.
\begin{eqnarray}
b^0 = 1, & b^n = b^{n-1} b \quad \text{ if $n > 0$}, & b^n = b^{n+1} / b \quad \text{ if $n < 0$}.
\label{eq:rule-of-exponents}
\end{eqnarray}
\begin{eqnarray}
b^{x+y} = b^x b^y, & (b^x)^y = b^{xy},
\label{eq:rule-of-exponents-2}
\end{eqnarray}
\begin{myanswer}
We will prove this in $\Reals$
Let us recall the following formula:
\begin{equation}
b^x = e^{x \ln b}
\end{equation}
and
\begin{equation}
\exp(x + y) = \exp(x) \exp(y)
\end{equation}
Thus it comes:
\begin{align*}
b^{x + y} &= e^{(x + y) \ln b} \\
&= e^{x \ln b + y \ln b} \\
&= e^{x \ln b} e^{y \ln b} \\
&= b^x b^y
\end{align*}
Similarly for $(a^x)^y$:
\begin{align*}
(a^x)^y &= e^{y \ln (a^x)} \\
&= e^{xy \ln a} \\
&= a^{xy}
\end{align*}
\end{myanswer}
\end{myexercise}

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tex/content/figures/part01/ch1/sum_odd_numbers_to_square.fig.pdf Executable file
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tex/content/figures/part01/ch1/sum_odd_numbers_to_square.fig.tex Executable file
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\documentclass{article}
\usepackage{graphicx}
\usepackage{caption}
\begin{document}
\begin{figure}
\centering
\includegraphics[scale=1]{sum_odd_numbers_to_square}
\setcounter{figure}{2}
\caption{The sum of odd numbers is a square}
\end{figure}
\end{document}

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tex/content/figures/part01/ch1/sum_odd_numbers_to_square.tex Executable file
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% Draw a 6 by 6 grid of squares, with thicker line on outer borders.
\documentclass[margin=0.5cm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
% Draw grid
\draw (0, 0) grid (6, 6);
% Draw thicker lines on outer borders
\draw[thick] (0, 0) -- (6, 0) -- (6, 6) -- (0, 6) -- (0, 0);
% Draw for each square with bottom left corner at (0, 0) to (6, 6) the thicker border
\foreach \i in {0, 1, 2, 3, 4, 5, 6}
\draw[very thick] (0, 0) -- (\i, 0) -- (\i, \i) -- (0, \i) -- (0, 0);
% For each $i$-th odd number from 1 to 11, add a node at the top right corner of the $i$-th square
\foreach \i [evaluate={\x=int(\i*2+1);}] in {0, 1, 2, 3, 4, 5}
\node at (\i+0.5, \i+0.5) {$\x$};
\end{tikzpicture}
\end{document}

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\chapter*{Introduction} \chapter*{Introduction}
This document gathers some random solution to the problem from the \textit{Art of Computer Programming} by Donald E. KNUTH.
It should not be considered as a reference, but as a personal attempt to solve some of the exercises, there for the answer should not consider the solution as correct unless he find it so.
The answer are presented with page numbers from the third edition of the AOCP.

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\DeclareMathOperator{\Reals}{\mathbb{R}}
\DeclareMathOperator{\Integers}{\mathbb{Z}}
\DeclareMathOperator{\Naturals}{\mathbb{N}}

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% makeindex style file created by the glossaries package % makeindex style file created by the glossaries package
% for document 'main' on 2023-1-3 % for document 'main' on 2023-2-27
actual '?' actual '?'
encap '|' encap '|'
level '!' level '!'

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The LuaLaTeX Chameleon Book Template
% LaTeX Template
% Version 0.0.0 (2022-12-05)
%
% Authors:
% Samuel ORTION <samuel+dev@ortion.fr>
% Compiling this template:
% This template uses biber for its bibliography and makeindex for its index.
% When you first open the template, compile it from the command line with the
% commands below to make sure your LaTeX distribution is configured correctly:
%
% 1) lualatex main
% 2) makeindex main.idx -s indexstyle.ist
% 3) biber main
% 3) makeglossaries main
% 4) lualatex main
% 5) lualatex main
%
% License:
% MIT
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PREAMBLE % PREAMBLE
@ -29,7 +7,8 @@
\documentclass[ \documentclass[
11pt, 11pt,
fleqn, fleqn,
a4paper a4paper,
oneside
]{TheLuaChameleonBook} ]{TheLuaChameleonBook}
% Book PDF metadata % Book PDF metadata
@ -37,7 +16,7 @@
pdftitle={The Art of Computer Programming Vol. 1 - Some Exercises}, pdftitle={The Art of Computer Programming Vol. 1 - Some Exercises},
pdfauthor={Samuel ORTION}, pdfauthor={Samuel ORTION},
pdfsubject={An Attempt to solve some of the exercises from the Book of Donald E. KNUTH}, pdfsubject={An Attempt to solve some of the exercises from the Book of Donald E. KNUTH},
pdfkeywords={computer science, mathematics, algorithmics}, pdfkeywords={computer science, mathematics, algorithmic},
pdfcreator={LaTeX}, pdfcreator={LaTeX},
} }
@ -48,6 +27,7 @@
\makeindex \makeindex
\input{preamble.tex} \input{preamble.tex}
\input{definitions.tex}
\setcounter{tocdepth}{2} \setcounter{tocdepth}{2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -87,7 +67,7 @@
\noindent \texttt{\href{https://samuel.ortion.fr/}{samuel.ortion.fr}}\\ % URL \noindent \texttt{\href{https://samuel.ortion.fr/}{samuel.ortion.fr}}\\ % URL
\noindent Licensed under the Creative Commons Attribution-NonCommercial 4.0 License. \noindent Answers Licensed under the Creative Commons Attribution-NonCommercial 4.0 License.
% You may not use this file except in compliance with the License. You may obtain a copy of the License at \url{https://creativecommons.org/licenses/by-nc-sa/4.0}. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an \textsc{``as is'' basis, without warranties or conditions of any kind}, either express or implied. See the License for the specific language governing permissions and limitations under the License.\\ % License information, replace this with your own license (if any) % You may not use this file except in compliance with the License. You may obtain a copy of the License at \url{https://creativecommons.org/licenses/by-nc-sa/4.0}. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an \textsc{``as is'' basis, without warranties or conditions of any kind}, either express or implied. See the License for the specific language governing permissions and limitations under the License.\\ % License information, replace this with your own license (if any)
% \noindent \textit{First printing, October 2022} % Printing/edition date % \noindent \textit{First printing, October 2022} % Printing/edition date

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tex/preamble.tex
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@ -10,26 +10,22 @@
} }
% Create a new environment {exercise} to refer to exercises from the KNUTH book. % Create a new environment {exercise} to refer to exercises from the KNUTH book.
% Add a TODO switch option to show unresolved exercises
\newenvironment{myexercise}[2] \newenvironment{myexercise}[2]
{ {
\addcontentsline{toc}{subsubsection}{Exercise #1} \addcontentsline{toc}{subsubsection}{Exercise #1}
\begin{minipage}{\textwidth}
\begin{flushleft} \begin{flushleft}
{ {
\Huge \Large
\textbf{Exercise #1.} (p. #2) \textbf{Exercise #1.} (p. #2)
} }
\end{flushleft} \end{flushleft}
} }
{ {
\end{minipage}
} }
% Create a new environment for big quote with `primarycolor' coloured quote sign % Create a new environment for big quote with `primarycolor' coloured quote sign
\newenvironment{myquote} \newenvironment{myquote}
{ {
\begin{minipage}{\textwidth}
\begin{flushleft} \begin{flushleft}
{ {
@ -45,22 +41,22 @@
\end{flushright} \end{flushright}
\end{flushleft} \end{flushleft}
\end{minipage}
} }
\newenvironment{myanswer} \newenvironment{myanswer}
{ {
\begin{minipage}{\textwidth}
\begin{itshape} \begin{itshape}
\begin{flushleft} \begin{flushleft}
{ {
\Large
\textbf{My Answer}: \textbf{My Answer}:
} }
} }
{ {
\end{flushleft} \end{flushleft}
\end{itshape} \end{itshape}
\end{minipage}
} }
\usepackage{algorithmicx}
\usepackage{algorithm}
\usepackage{algpseudocodex}