Homework Assignments for netMA348SU13
30dec11 and 20may13\begin{document} \maketitle
\section{Homework 1}
Chapter 1 of the D'Angelo-West text is meant to be an inventory of your
mathematical readiness for this course. You are not expected to know
all of this material in this chapter. If you did, perhaps you should not take this course. But you are expected to understand the explanations. Chapter 1 ends with 56
problems, all of which you should read
with with a pencil and pad of paper at your side. In preparation for this
homework you should \textit{skim} Chapter 1 so see what's where. Submit a
PDF of a document in which you classify each of the 56 problems according to the
following rubric. As review for the midterm and final, we will spend time in class
[respectively online in the Moodle and the Virtual Office] going over the most popular
problems submitted.
\begin{itemize}
\item [N] Which problems are \textbf{not arranged} in the order in which their
topics appear in the text of Chapter 1?
\item [I] Which of the problems look \textbf{interesting} enough for you to invest time in
understanding a solution?
\item [P] Which of the \textbf{practice problems}, marked (-), can you do right now
\item [R] Which of the \textbf{regular problems}, not those with a (!), would
you be unabe to do even with some studying?
\item (You may list the problems in order and mark each of them with an one or more letter \textbf{N,I,P,R} as appropriate.)
\itme Submit a PDF document in any format. No reduction of credit for the manner in
which the PDF was created for this homework.
\end{itemize}
\section{Homework 2}
\begin{itemize}
\item D'Angelo-West 1.13 (write up the solution done in class [resp. in the Moodle]), 1.20, 1.21, and 1.28.
\item Solve each problem first on scratch paper.
\item Enter the statement and solution in your Journal.
\item Any solution to a problem in this
course which does not state the question, either explicitly or implicitly, receives
at most 3/4 credit.
\item Bring your journal to in class lab [Does not apply online.]
\item Preferred Format (lesser formats are still acceptable for this homework.)
\item Compose a single LaTeX document for all problems, in texWins or otherwise.
\item Print out a copy to bring to class for correction. [Does not apply online.]
\item Upload its .pdf file to the Moodle.
\item Upload its .tex file to the Moodle as well for stylistic correction.
\end{itemize}
\subsection{Lesser acceptible formats.}
\begin{itemize}
\item Note, most important is to get the homework in on time (1/2 of credit).
\item For your submission to receive credit it must be
\begin{itemize}
\item Easily Readable, use complete sentences, justify your answers. See Advice.
\item Immutable (hence PDF or a screen print, but NO
.txt, .doc, or .docx files}
\item Only .pdf files will be returned with corrections and comments.
\end{itemize}
\item A screen print of a legible handwritten homework converted to a PDF.
You can insert the screen print into MS Word and "print it" to the electronic
PDF format.
\item A document prepared with the Equations Editor in MS Word, and then
printed to a PDF.
\item A document prepared in texPad, with pictures roughly aligned, and composed
on the desktop, then made into a screen print. Convert the screen print into a PDF
to submit.
\end{itemize}
\subsection{Advanced Composition Component}
\begin{itemize}
\item Each problem solved must be stated, preferably in your own words.
\item Use complete sentences, but you should use mathematical notation to
shorten the discourse.
\item If you are unable to compose text in LaTeX, you may use a wordprocessor
with mathematical formulas and equations as figures. We recomment MS Word 2010,
whose Equation Editor understands LaTeX equation commands. A good way to learn
LaTeX.
\item If you are unable to use any of the above enumerated lesser, alternate
forms of submission, consult with the instructor. There are extenuating circumstances,
and an oral report in the Virtual Office may be an extraordinary, allowed format.
\end{itemize}
\section{Homework 3}
These are the sections of the chapter your should \textit{study}, and to ask
questions if you do not understand the text. You should \textit{skim} the rest
of the chapter to know where things are discussed.
\subsection{Arithmetic series}
Sections 3.7 and 3.10 treat the sum of consecutive numbers in two different
ways, by induction, and by discovery. The discovery method depends on a
trick. The induction method requires knowledge of the answer before you can
prove its generality. A (finite) sequence of integers for which successive
differences are constant, $a_{n+1}-a_n = d $, is called an \textit{ arithmetic
sequence}. The sum of an arithmetic sequence, called a (finite) \textit{ arithmetic series}, may be discovered by Gauss's trick. Know how to do this, and
given the answer you discovered, prove it by induction. Put this solution
into your Journal.
\subsection{Geometric series}
Similarly, sections 3.13 and 3.14 deal with (finite) \textit{ geometric series}.
Be sure a similar analyis of the simplest case, $\sum_{i=1}^n x^i$ is in
your Journal.
\subsection{Series of powers}
Do not confuse a geometric series with a (finite) sum of the powers of
consecutive numbers, $S_e^n := \sum_{i=1}^n i^e $. You should have inductive
proofs for the following cases in your Journal:
\begin{itemize}
\item $S_0^n = n$ , this is easy!
\item $e=1$, see 3.7
\item $e=2$, see 3.21
\item $e=3$, see Problem 3.16
\end{itemize}
\subsection{Assignemd Problems}
Solutions in LaTeX format of the five problems
3.16, 3.17, 3.19, 3.28, 3.29
are due as stated in the syllabus. Lesser format may still be used, but for
reduced credit.
\section{Homework 4}
There is no separate HW unit number 4. This is the Midterm exam.
\section{Homework 5}
To do the assigned problems you may have to read the textbook, unless you
remember the definitions from elsewhere. The vocabulary
inludes \textit{ relatively prime, greatest common divisor, long division,
Euclid's algorithm, Diophantine equations.} The chief tool you are to use
in solving these problems is the Prime Decomposition Theorem, whenever possible.
\subsection{Practical definitions}
Here are some practical definitions to help in doing the problems.
\begin{itemize}
\item \textbf{ prime decomposition:} Write $pc(a)$ for
$a = a_1 a_2 ... a_n$ as a
product of prime factors, each $1 < a_j \le a$.
\item \textbf{ divides, divisible:} $d|a$ if $(\exists q)(a = dq)$
\item \textbf{ divides, divisible:} $d|a$ if the $pc(d)$ is part or all of $pc(a)$.
\item \textbf{ long division :} We write $divmod(a,d)=(q,r)$ for
$a = dq + r, 0\le r < d$. Thus, the long division of \textit{ dividend}$a$
by \textit{ divisor} $d$ results in the\textit{ quotient} $q$ and \textit{ remainder} $r$. \texttt{ divmod( , )} is the Python function that does long division.
\item \textbf{ greatest common divisor :} $d = gcd(a,b)$ if it is the largest
among all divisors of both $a$ and $b$.
\item \textbf{ relatively prime:} Two integers have no common prime
factor iff $gcd(a,b)=1$ iff none of the $a_i$ is one of the $b_j$ and vice
versa, are said to be relatively prime.
\item \textbf{ greatest common divisor :} $d = gcd(a,b)$ is also the largest
product of primes that cancel out in their fraction,
$\frac{a}{b} = \frac{da'}{db'}=\frac{a'}{b'}$ where $(a',b')=1$.
\item \textbf{ Euclid's Algorithm :} For positive integers $(a,b)$ the
repeated subtraction of the lesser from the greater terminates at zero.
The last positive integer is the $gcd(a,b)$. (This is a theorem).
\item \textbf{Diophantine equation :} has the form $ax + by = c$ where
$a,b,c$ are given integers, and $x,y$ are unknown integers, if they exist.
\end{itemize}
\subsection{Homework assigned for HW 5.}
D'Angelo-West 6.8, 6.9, 6.17, 6.18, 6.28, see repeated here
Bring a printout of your work to class for peer editing.
\subsection{Homework assigned for HW 6}
D'Angelo-West 7.8, 7.9, 7.11, 7.18, 7.24
Bring a printout of your work to class for peer editing. [Does not apply online.]