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\title{ Math 3210-4, HW I\\ due Wednesday Sep 04}
\maketitle
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In order to define multiplication of real numbers, it suffices to do so for positive numbers, and here it is convenient to use
cuts of positive rational numbers. Thus a positive real number is defined a non-empty, bounded set $\alpha \subseteq \mathbb Q^+$ such that
\begin{itemize}
\item If $r\in \alpha$ and $s< r$ is a positive rational number, then $s\in \alpha$.
\item $\alpha$ has no maximal element.
\end{itemize}
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An example is ``the cut of square root of 3'': $\sqrt{3}=\{r\in \mathbb Q^+ | r^2<3\}$. The first bullet is clearly satisfied, for the second, note that
the limit of $(r+\frac1n)$ is $r^2$, as the positive integer $n$ tends to infinity, hence for $n$ large enough $(r+ \frac1n)^2 <3$ if $r^2<3$.
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\noindent
1) For $i=0, \ldots, 10$, construct the greatest rational number in the cut of square root of 3 in the form of a (non-reduced) fraction $a_i=x_i/2^i$, and
the least rational number not in the cut in the same form $b_i=y_i/2^i$. For example, $a_0=1$ and $b_0=2$. Their average is $3/2$. Since $(3/2)^2 < 3$,
it follows that $a_1=3/2$ and $b_1=4/2$ etc... Note:
if you had a calculator that ``computes" $\sqrt{3}$ and expresses the answer in binary digits, $a_i$ would be what you get after chopping off all digits
after the $i$-th place right of the point.
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\noindent
2) Let $\alpha$ and $\beta$ be two cuts of $\mathbb Q^+$. Let $\alpha\cdot \beta=\{rs | r\in \alpha, s\in \beta\}$.
Prove that $\alpha\cdot \beta$ is a cut of $\mathbb Q^+$.
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\noindent
3) Prove that $\sqrt{3} \cdot \sqrt{3}= 3^*$, where $\sqrt{3}$ is the cut defined above.
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\noindent
4) Let $1^*=\{r\in \mathbb Q^+ | r<1\}$. Prove that $\alpha \cdot 1^* = \alpha$ for any cut $\alpha$ of $\mathbb Q^+$.
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\noindent
5)
Let $\alpha$ be a cut of $\mathbb Q^+$. Construct a cut $\beta$ such that $\alpha \cdot \beta =1^*$.
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\noindent
6) The set of $2\times 2$ matrices with real coefficients is a non-commutative ring with respect to the usual addition and multiplication of matrices. We can use this information
to quickly construct complex numbers and prove that it is a field. Let $\mathbb C$ be the set of $2\times 2$ matrices
\[
\left( \begin{array}{rr} a & -b \\ b & a \end{array}\right)
\]
where $a$ and $b$ are any real numbers.
If $A,B\in \mathbb C$, prove that $A+B$, $A-B$ and $AB$ are in $\mathbb C$ and that $AB=BA$. This implies that
$\mathbb C$ is a ring (why?). Finally, for every non-zero $A\in \mathbb C$ find $B\in \mathbb C$ such that $AB=1$ i.e. $\mathbb C$ is a field. What fails if we
instead consider $\mathbb C'$, the set of all $2\times 2$ matrices
\[
\left( \begin{array}{rr} a & b \\ b & a \end{array}\right)
\]
where $a$ and $b$ are any real numbers?
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%\noindent {\bf Enrollment Requirement:} Prerequisites: C or better in one of Math 1220, 1250, 1260, 1270, 1311, 1320, 1321, 2210, or AP Calc BC score of~5.
%Requirement Designation: Quantitative Reasoning (Math \& Stat/Logic)
%\section{Teaching Assistant}
%Shiang Tang. LCB loft (4-th floor). Office hours: MW, 9:35 - 10:35. Problem session every Thursday, 10:45 - 11:35, in JWB 333.
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\noindent
{\bf Course content:} This is an Honors section of Math 3210. It is more difficult than other sections, with an emphasis on proofs. It also uses a different text:
Walter Rudin, {\em Principles of Mathematical Analysis}, Third Edition, MacGraw-Hill. We plan to cover the first six chapters:
\begin{itemize}
\item Chapter 1: The Real and Complex Number Systems
\item Chapter 2: Basic Topology
\item Chapter 3: Numerical Sequences and Series
\item Chapter 4: Continuity
\item Chapter 5: Differentiation
\item Chapter 6: The Riemann-Stieltjes Integral
\end{itemize}
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\noindent
{\bf Goal:} This is a course in real analysis in one variable. It includes axioms of real numbers and basic theory of metric spaces and topology necessary for a
proper course in real analysis.
Students are expected to have a good mastery of differentiation and integration techniques.
The course emphasizes proofs and rigorous analytic thinking.
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\noindent
{\bf Homework and Exams:} Homework problems will be regularly assigned and discussed in the class, but not graded. Grade will be based on six take home exams,
one for each chapter. Students will have approximately one week to work on each exam.
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\noindent
{\bf Accommodation:}
The University of Utah seeks to provide equal access to its programs, services and activities for people with disabilities. If you will need accommodations in the class, reasonable prior notice needs to be given to the Center for Disability Services (CDS), 162 Olpin Union Building, 581- 5020 (V/TDD). CDS will work with you and me to make
arrangements for accommodations. All information in this course can be made available in alternative format with prior notification to CDS.
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\noindent
{\bf Student Code:}
All students are expected to maintain professional behavior in the classroom setting, according to the Student Code, spelled out in the Student Handbook. You have specific rights in the classroom as
detailed in Article III of the Code. The Code also specifies proscribed conduct (Article XI) that involves cheating on tests, collusion, fraud, theft, etc. Students should read the Code carefully and know you are responsible for the content. According to Faculty Rules and Regulations,
it is the faculty responsibility to enforce responsible classroom behaviors, beginning with verbal warnings and progressing to dismissal from class and a failing grade. Students have the right to appeal such action to the Student Behavior Committee.
http://regulations.utah.edu/academics/6-400.php
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