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Advanced High-School Mathematics
David B. Surowski
Shanghai American School
Singapore American School
January 29, 2011
i
Preface/Acknowledgment
The present expanded set of notes initially grew out of an attempt to
flesh out the International Baccalaureate (IB) mathematics “Further
Mathematics” curriculum, all in preparation for my teaching this dur-
ing during the AY 2007–2008 school year. Such a course is offered only
under special circumstances and is typically reserved for those rare stu-
dents who have finished their second year of IB mathematics HL in
their junior year and need a “capstone” mathematics course in their
senior year. During the above school year I had two such IB math-
ematics students. However, feeling that a few more students would
make for a more robust learning environment, I recruited several of my
2006–2007 AP Calculus (BC) students to partake of this rare offering
resulting. The result was one of the most singular experiences I’ve had
in my nearly 40-year teaching career: the brain power represented in
this class of 11 blue-chip students surely rivaled that of any assemblage
of high-school students anywhere and at any time!
After having already finished the first draft of these notes I became
aware that there was already a book in print which gave adequate
coverage of the IB syllabus, namely the Haese and Harris text
1
which
covered the four IB Mathematics HL “option topics,” together with a
chapter on the retired option topic on Euclidean geometry. This is a
very worthy text and had I initially known of its existence, I probably
wouldn’t have undertaken the writing of the present notes. However, as
time passed, and I became more aware of the many differences between
mine and the HH text’s views on high-school mathematics, I decided
that there might be some value in trying to codify my own personal
experiences into an advanced mathematics textbook accessible by and
interesting to a relatively advanced high-school student, without being
constrained by the idiosyncracies of the formal IB Further Mathematics
curriculum. This allowed me to freely draw from my experiences first as
a research mathematician and then as an AP/IB teacher to weave some
of my all-time favorite mathematical threads into the general narrative,
thereby giving me (and, I hope, the students) better emotional and
Peter Blythe, Peter Joseph, Paul Urban, David Martin, Robert Haese, and Michael Haese,
Mathematics for the international student; Mathematics HL (Options),
Haese and
Harris Publications, 2005, Adelaide, ISBN 1 876543 33 7
1
ii
Preface/Acknowledgment
intellectual rapport with the contents. I can only hope that the readers
(if any) can find some something of value by the reading of my stream-
of-consciousness narrative.
The basic layout of my notes originally was constrained to the five
option themes of IB: geometry, discrete mathematics, abstract alge-
bra, series and ordinary differential equations, and inferential statistics.
However, I have since added a short chapter on inequalities and con-
strained extrema as they amplify and extend themes typically visited
in a standard course in Algebra II. As for the IB option themes, my
organization differs substantially from that of the HH text. Theirs is
one in which the chapters are independent of each other, having very
little articulation among the chapters. This makes their text especially
suitable for the teaching of any given option topic within the context
of IB mathematics HL. Mine, on the other hand, tries to bring out
the strong interdependencies among the chapters. For example, the
HH text places the chapter on abstract algebra (Sets, Relations, and
Groups) before discrete mathematics (Number Theory and Graph The-
ory), whereas I feel that the correct sequence is the other way around.
Much of the motivation for abstract algebra can be found in a variety
of topics from both number theory and graph theory. As a result, the
reader will find that my Abstract Algebra chapter draws heavily from
both of these topics for important examples and motivation.
As another important example, HH places Statistics well before Se-
ries and Differential Equations. This can be done, of course (they did
it!), but there’s something missing in inferential statistics (even at the
elementary level) if there isn’t a healthy reliance on analysis. In my or-
ganization, this chapter (the longest one!) is the very last chapter and
immediately follows the chapter on Series and Differential Equations.
This made more natural, for example, an insertion of a theoretical
subsection wherein the density of two independent continuous random
variables is derived as the convolution of the individual densities. A
second, and perhaps more relevant example involves a short treatment
on the “random harmonic series,” which dovetails very well with the
already-understood discussions on convergence of infinite series. The
cute fact, of course, is that the random harmonic series converges with
probability 1.
iii
I would like to acknowledge the software used in the preparation of
these notes. First of all, the typesetting itself made use of the indus-
A
try standard, LTEX, written by Donald Knuth. Next, I made use of
three different graphics resources:
Geometer’s Sketchpad, Autograph,
and the statistical workhorse
Minitab.
Not surprisingly, in the chapter
on Advanced Euclidean Geometry, the vast majority of the graphics
was generated through Geometer’s Sketchpad. I like Autograph as a
general-purpose graphics software and have made rather liberal use of
this throughout these notes, especially in the chapters on series and
differential equations and inferential statistics. Minitab was used pri-
marily in the chapter on Inferential Statistics, and the graphical outputs
greatly enhanced the exposition. Finally, all of the graphics were con-
verted to PDF format via ADOBE
R
ACROBAT
R
8 PROFESSIONAL
(version 8.0.0). I owe a great debt to those involved in the production
of the above-mentioned products.
Assuming that I have already posted these notes to the internet, I
would appreciate comments, corrections, and suggestions for improve-
ments from interested colleagues and students alike. The present ver-
sion still contains many rough edges, and I’m soliciting help from the
wider community to help identify improvements.
Naturally, my greatest debt of
gratitude is to the eleven students
(shown to the right) I conscripted
for the class. They are (back row):
Eric Zhang (Harvey Mudd), Jong-
Bin Lim (University of Illinois),
Tiimothy Sun (Columbia Univer-
sity), David Xu (Brown Univer-
sity), Kevin Yeh (UC Berkeley),
Jeremy Liu (University of Vir-
ginia); (front row): Jong-Min Choi (Stanford University), T.J. Young
(Duke University), Nicole Wong (UC Berkeley), Emily Yeh (University
of Chicago), and Jong Fang (Washington University). Besides provid-
ing one of the most stimulating teaching environments I’ve enjoyed over

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