Advanced High-School Mathematics

435 Pages·2011·3.03 MB·English

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 text1 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
1Peter 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
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-
try standard, LATEX, 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(cid:13)R ACROBAT(cid:13)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
iv
my 40-year career, these students pointed out countless errors in this
document’s original draft. To them I owe an un-repayable debt.
My list of acknowledgements would be woefully incomplete without
special mention of my life-long friend and colleague, Professor Robert
Burckel, who over the decades has exerted tremendous influence on how
I view mathematics.
David Surowski
Emeritus Professor of Mathematics
May 25, 2008
Shanghai, China
[email protected]
http://search.saschina.org/surowski
First draft: April 6, 2007
Second draft: June 24, 2007
Third draft: August 2, 2007
Fourth draft: August 13, 2007
Fifth draft: December 25, 2007
Sixth draft: May 25, 2008
Seventh draft: December 27, 2009
Eighth draft: February 5, 2010
Ninth draft: April 4, 2010
Contents
1 Advanced Euclidean Geometry 1
1.1 Role of Euclidean Geometry in High-School Mathematics 1
1.2 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Basic notations . . . . . . . . . . . . . . . . . . . 2
1.2.2 The Pythagorean theorem . . . . . . . . . . . . . 3
1.2.3 Similarity . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 “Sensed” magnitudes; The Ceva and Menelaus
theorems . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.5 Consequences of the Ceva and Menelaus theorems 13
1.2.6 Brief interlude: laws of sines and cosines . . . . . 23
1.2.7 Algebraic results; Stewart’s theorem and Apollo-
nius’ theorem . . . . . . . . . . . . . . . . . . . . 26
1.3 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.1 Inscribed angles . . . . . . . . . . . . . . . . . . . 28
1.3.2 Steiner’s theorem and the power of a point . . . . 32
1.3.3 Cyclic quadrilaterals and Ptolemy’s theorem . . . 35
1.4 Internal and External Divisions; the Harmonic Ratio . . 40
1.5 The Nine-Point Circle . . . . . . . . . . . . . . . . . . . 43
1.6 Mass point geometry . . . . . . . . . . . . . . . . . . . . 46
2 Discrete Mathematics 55
2.1 Elementary Number Theory . . . . . . . . . . . . . . . . 55
2.1.1 The division algorithm . . . . . . . . . . . . . . . 56
2.1.2 The linear Diophantine equation ax + by = c . . . 65
2.1.3 The Chinese remainder theorem . . . . . . . . . . 68
2.1.4 Primes and the fundamental theorem of arithmetic 75
2.1.5 The Principle of Mathematical Induction . . . . . 79
2.1.6 Fermat’s and Euler’s theorems . . . . . . . . . . . 85
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vi
2.1.7 Linear congruences . . . . . . . . . . . . . . . . . 89
2.1.8 Alternative number bases . . . . . . . . . . . . . 90
2.1.9 Linear recurrence relations . . . . . . . . . . . . . 93
2.2 Elementary Graph Theory . . . . . . . . . . . . . . . . . 109
2.2.1 Eulerian trails and circuits . . . . . . . . . . . . . 110
2.2.2 Hamiltonian cycles and optimization . . . . . . . 117
2.2.3 Networks and spanning trees . . . . . . . . . . . . 124
2.2.4 Planar graphs . . . . . . . . . . . . . . . . . . . . 134
3 Inequalities and Constrained Extrema 145
3.1 A Representative Example . . . . . . . . . . . . . . . . . 145
3.2 Classical Unconditional Inequalities . . . . . . . . . . . . 147
3.3 Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . 155
3.4 The H¨older Inequality . . . . . . . . . . . . . . . . . . . 157
3.5 The Discriminant of a Quadratic . . . . . . . . . . . . . 161
3.6 The Discriminant of a Cubic . . . . . . . . . . . . . . . . 167
3.7 The Discriminant (Optional Discussion) . . . . . . . . . 174
3.7.1 The resultant of f(x) and g(x) . . . . . . . . . . . 176
3.7.2 The discriminant as a resultant . . . . . . . . . . 180
3.7.3 A special class of trinomials . . . . . . . . . . . . 182
4 Abstract Algebra 185
4.1 Basics of Set Theory . . . . . . . . . . . . . . . . . . . . 185
4.1.1 Elementary relationships . . . . . . . . . . . . . . 187
4.1.2 Elementary operations on subsets of a given set . 190
4.1.3 Elementary constructions—new sets from old . . 195
4.1.4 Mappings between sets . . . . . . . . . . . . . . . 197
4.1.5 Relations and equivalence relations . . . . . . . . 200
4.2 Basics of Group Theory . . . . . . . . . . . . . . . . . . 206
4.2.1 Motivation—graph automorphisms . . . . . . . . 206
4.2.2 Abstract algebra—the concept of a binary oper-
ation . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.2.3 Properties of binary operations . . . . . . . . . . 215
4.2.4 The concept of a group . . . . . . . . . . . . . . . 217
4.2.5 Cyclic groups . . . . . . . . . . . . . . . . . . . . 224
4.2.6 Subgroups . . . . . . . . . . . . . . . . . . . . . . 228
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4.2.7 Lagrange’s theorem . . . . . . . . . . . . . . . . . 231
4.2.8 Homomorphisms and isomorphisms . . . . . . . . 235
4.2.9 Return to the motivation . . . . . . . . . . . . . . 240
5 Series and Differential Equations 245
5.1 Quick Survey of Limits . . . . . . . . . . . . . . . . . . . 245
5.1.1 Basic definitions . . . . . . . . . . . . . . . . . . . 245
5.1.2 Improper integrals . . . . . . . . . . . . . . . . . 254
5.1.3 Indeterminate forms and l’Hˆopital’s rule . . . . . 257
5.2 Numerical Series . . . . . . . . . . . . . . . . . . . . . . 264
5.2.1 Convergence/divergence of non-negative term series265
5.2.2 Tests for convergence of non-negative term series 269
5.2.3 Conditional and absolute convergence; alternat-
ing series . . . . . . . . . . . . . . . . . . . . . . . 277
5.2.4 The Dirichlet test for convergence (optional dis-
cussion) . . . . . . . . . . . . . . . . . . . . . . . 280
5.3 The Concept of a Power Series . . . . . . . . . . . . . . . 282
5.3.1 Radius and interval of convergence . . . . . . . . 284
5.4 Polynomial Approximations; Maclaurin and Taylor Ex-
pansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.4.1 Computations and tricks . . . . . . . . . . . . . . 292
5.4.2 Error analysis and Taylor’s theorem . . . . . . . . 298
5.5 Differential Equations . . . . . . . . . . . . . . . . . . . . 304
5.5.1 Slope fields . . . . . . . . . . . . . . . . . . . . . 305
5.5.2 Separable and homogeneous first-order ODE . . . 308
5.5.3 Linear first-order ODE; integrating factors . . . . 312
5.5.4 Euler’s method . . . . . . . . . . . . . . . . . . . 314
6 Inferential Statistics 317
6.1 Discrete Random Variables . . . . . . . . . . . . . . . . . 318
6.1.1 Mean, variance, and their properties . . . . . . . 318
6.1.2 Weak law of large numbers (optional discussion) . 322
6.1.3 The random harmonic series (optional discussion) 326
6.1.4 The geometric distribution . . . . . . . . . . . . . 327
6.1.5 The binomial distribution . . . . . . . . . . . . . 329
6.1.6 Generalizations of the geometric distribution . . . 330
viii
6.1.7 The hypergeometric distribution . . . . . . . . . . 334
6.1.8 The Poisson distribution . . . . . . . . . . . . . . 337
6.2 Continuous Random Variables . . . . . . . . . . . . . . . 348
6.2.1 The normal distribution . . . . . . . . . . . . . . 350
6.2.2 Densities and simulations . . . . . . . . . . . . . 351
6.2.3 The exponential distribution . . . . . . . . . . . . 358
6.3 Parameters and Statistics . . . . . . . . . . . . . . . . . 365
6.3.1 Some theory . . . . . . . . . . . . . . . . . . . . . 366
6.3.2 Statistics: sample mean and variance . . . . . . . 373
6.3.3 The distribution of X and the Central Limit The-
orem . . . . . . . . . . . . . . . . . . . . . . . . . 377
6.4 Confidence Intervals for the Mean of a Population . . . . 380
6.4.1 Confidence intervals for the mean; known popu-
lation variance . . . . . . . . . . . . . . . . . . . 381
6.4.2 Confidence intervals for the mean; unknown vari-
ance . . . . . . . . . . . . . . . . . . . . . . . . . 385
6.4.3 Confidence interval for a population proportion . 389
6.4.4 Sample size and margin of error . . . . . . . . . . 392
6.5 Hypothesis Testing of Means and Proportions . . . . . . 394
6.5.1 Hypothesis testing of the mean; known variance . 399
6.5.2 Hypothesis testing of the mean; unknown variance401
6.5.3 Hypothesis testing of a proportion . . . . . . . . . 401
6.5.4 Matched pairs . . . . . . . . . . . . . . . . . . . . 402
6.6 χ2 and Goodness of Fit . . . . . . . . . . . . . . . . . . . 405
6.6.1 χ2 tests of independence; two-way tables . . . . . 411
Index 418