Freshman Seminar 21u --Calculating Pi
Syllabus -- Fall 2004
Revised September 15, 2004
Instructor:
Dr. Paul Bamberg
Office: Science Center 423 (phone 617-495-1748)
Harvard apartment: Quincy House 102 (phone 3-3100)
Office hours: In Science Center 423, Tuesdays 2:30-3:30
In Quincy 102 (phone 3-3100 first to make sure that I am home).
Tu 10AM-noon
Wed 9AM-12:30PM
Th 8AM-noon
Tu evenings in Quincy 102
Phone: 493-3100 (Quincy House - late Monday through mid-afternoon Thursday)
Home Phone 508-460-6569 (rest of the week)
Email: bamberg@tiac.net
Schedule:
Meeting place and time: Bullitt Room, Quincy House basement, Wednesdays 3-6 PM. Enter through the Quincy House gate at 58 Plympton St. Walk across the courtyard and slightly to the right. Enter the house, and go through the door to the right of the elevators and down one flight of steps. Turn left into the basement corridor. The Bullitt room is the first on the left.
If enrollment is small, we will meet in the conference room near the Quincy House Senior Tutor's office. This is down the short corridor to the right of the same elevators. It's the first room on the right.
There will be a 20-minute refreshment break in Quincy 102.
First class: Wednesday, September 29
We will not meet on the day before Thanksgiving. Instead, there will be a short class 1-2:30 on the Friday afternoon of parents' weekend, Friday, October 29, so that your parents can see you actually participating in a class.
Books:
Required (at the Harvard Coop)
Beckmann, A History of Pi (entertaining, but stops in the 1970s)
Bold, Famous Problems of Geometry (a great, cheap book)
You will also need a textbook on single-variable calculus. Whatever book you used when you studied that subject will be fine.
Recommended: (also at the Coop)
Berggren, Borwein, and Borwein, Pi: A Source Book (everything you wanted to know plus lots more)
This is strongly recommended, although we will have time only for portions of it.
Knopp, Theory and Application of Infinite Series (heavy but Teutonically thorough, may arrive late)
Doerrie, 100 Great Problems of Elementary Mathematics
Both these books are quite inexpensive and useful to own as reference works.
Other useful books (not at the Coop)
Weisstein, CRC Encyclopedia of Mathematics (in the Cabot reference section)
Although just a single 2000-page volume, this is a mine of information, with lots of up-to-date references.
Jane Muir, Of Men and Numbers
E.T. Bell, Men of Mathematics
The titles may seem politically incorrect, but these are good sources for biographical reports. Find them on the shelves at Cabot, and dozens of other useful books will be next to them.
Software:
Mathematica Version 5 (free download available for any Harvard student). Versions are available for Windows, Mac, or UNIX. Allow a day or two to get your password.
An alternative for those with a little programming experience:
EasyPHP and HAPEdit (for creating mathematical Web pages). An installation CD will be provided.
Prerequisites: You should have taken AP calculus(either exam) or received a placement recommendation of either Math 1b or Math 21a. This course will quickly review much of calculus and infinite series, but it will assume that you have seen the material in the past. There will be much more emphasis on proofs of key results and less on problem-solving techniques than in most calculus courses. After you complete this seminar, you should be well prepared for courses that require Math 1b, except perhaps in the area of differential equations.
If you do well on the online placement test called "math1b-mastery" and enjoy the online test called "more fun math," you will be very well prepared for this seminar and almost certain happy in it. Please take these tests before the end of the second week of classes. They are not a requirement for admission to the seminar, but knowing how this year's students have done on them will make it easier to establish advising guidelines for the future.
Multivariable calculus and linear algebra are irrelevant for this subject, but a good background in geometry is a real plus.
Much of the reading will be from primary sources and from textbooks that are more advanced (and difficult) than the typical calculus textbook.
It is helpful, but by no means required, to have programming experience in C++ or Java so that you can learn PHP. A Computer Science AP course would be perfect. Experience with HTML and/or PHP would also be useful. However, all computer work can be done in Mathematica, which you can learn during the seminar.
Technology Tree:
The first part of the course is based on a set of 17 outlines, called "From Euclid to Spigotry," that cover, in historical order, the topics that every previous group of students has chosen to do in this seminar. These are the entire content of QR44, but we will aim to do them all by early November.
On the Web site, in the Basic_topics folder, are .rtf files for these. Printed versions will be available before the first seminar meeting. You should arrange to pick one up in Quincy 102 any time on or after Thursday, September 29 and to get your assignment for the first meeting.
The second part of the course consists of a set of outlines on advanced topics that depend on the first part. There are more than we can possibly cover, so the class will choose ones that look interesting. These are also on the Web site, in the Advanced_topics folder, in .rtf format, but they are subject to revision.
Homework Problems:
Following the advice of students who have taken this seminar in the past, I have written a few problems. These are fairly easy, but they check that you have understood the material presented in class. For the first set of outlines, they are numbered 1 through 21. For the advanced topics, there are problems for only some of the outlines.
Since the pace and content of the seminar is established by the students, due dates for these assignments cannot be known in advance. The general policy is that an assignment is due the week after we finish the topic to which it pertains. A weekly email to the class will specify what is due each week. My hope is to find a grader who likes junk food, who will come to our weekly refreshment break to collect new assignments and return graded ones.
These assignments are in the Assignments folder on the Web site. Those on advanced topics are subject to revision.
Course Organization:
Each outline will be presented by a team of students, which will on occasion include the instructor. Everyone is expected to do the reading in advance. People are encouraged to concentrate on whatever aspect of the subject - pure mathematics, computation, or history and biography -- they find most interesting.
We will cover one, two, or three outlines per week.
The team of people doing a topic will be responsible for presenting some or all of the following:
1. Short presentations that summarize the most crucial mathematics.
2. Computational exercise -- generally writing a demonstration program to compute pi.
3. Biographical report on a famous mathematician.
Some of the presentations topics are quite difficult, which is not surprising since many of them are about topics that broke new mathematical ground. It is a good idea to get together with the instructor for a "dry run" if you are presenting a tricky topic.
Grading policy:
The seminar is graded credit - noncredit. You are expected to attend all meetings, to prepare your presentations carefully, and to hand in the assignments. If you do well, the seminar report that is submitted at the end will explicitly recommend that the seminar should count for mathematics, computer science, physics, and/or history and science concentration credit.
The Pi Technology Tree (text version)
Note: the topics will not be covered in the order that they are listed here, but within each general area the order is a reasonable one.
Topics marked with an asterisk will be include in the first part of the course. The others are available as advanced topics in the second part.
Computation:
Mathematica
The programming language Mathematica has infinite-precision arithmetic built into it and is frequently the fastest way to get many digits of pi out of a new computational approach. You can use it like C or C++, or you can use a list-based or rule-based programming style. No prior experience required.
AGM Methods
Around 1800, Gauss discovered the "arithmetic-geometric mean" identity for elliptic
integrals. When rediscovered two centuries later, this led to some of the most efficient
techniques for computing pi to millions of decimal places.
Ramanujan Series
During his short life in the early twentieth century, the brilliant Indian mathematician
Ramanujan invented several novel infinite series for computing pi. It took decades for
lesser mathematicians to understand how he devised these series, and we won't even try.
It is exciting, though, just to use them and see how fast they converge to pi.
Calculus:
*Derivatives and Antiderivatives
Requires: nothing Leads to: Antiderivatives, Newton's Method
About 1500 years after the Greeks' initial attempts to "square the circle" and compute pi,
the invention of the calculus led to a host of new techniques. Almost all of them rely,
directly or indirectly, on the differentiation rules for the sine and cosine functions. You
probably remember how to use these rules, but can you derive them from first principles?
*Series and Products
There are many ways to express pi as the sum of an infinite series. Review what this
means in terms of the concept of convergence. Some of the best-known series for pi are
only conditionally convergent and cannot be rearranged without changing their sum: you
will meet the most famous example. Since two of the most famous formulas for pi involve
infinite products rather than infinite series, you must also learn to test whether an infinite
product is convergent.
*Power Series
The binomial theorem, invented by Newton and first proved by Euler, leads to numerous
infinite-series expressions for pi. The Taylor-series expansions of the exponential, sine,
and cosine functions, also power series, provide a way of defining pi that does not rely on
geometry.
Riemann Sums
Since pi can be defined geometrically in terms of arc length or area, which in calculus
terms are expressed as definite integrals, it is not surprising that many expressions for pi
involve definite integrals. Approximating these integrals numerically, for example by using
Simpson's rule, is not a bad way to compute pi to ten or twenty digits of precision. But
how do you assess the accuracy of such an estimate?
Improper Integrals
An integral is "improper" if one of the limits of integration is infinite. In some cases, an
improper integral of a function can be shown to involve pi, even though this value cannot
be obtained by the usual antidifferentiation techniques. These examples do not lead to
efficient ways to calculate pi, but they raise fascinating issues of convergence.
Geometry:
*Functional Identities
Since pi can be defined in terms of properties of the circle, relationships among the
"circular functions" (sine, cosine, tangent, etc.) -- such things as addition formulas and
double-angle formulas -- are important. Euler's discovery that the exponential of an
imaginary number can be expressed in terms in sines and cosines placed the exponential
function in the same category.
*Ruler and Compass
Leads to: Archimedes' Recursion, Beyond Ruler and Compass, Pentagon and Heptagon
The ancient Greeks were fascinated by the problem of what can be constructed with only
a straightedge and a compass. Later developments in algebra showed that this question is
closely related to the concept of a "field." You will learn to solve a quadratic equation
geometrically, using only straightedge and compass.
*Beyond Ruler and Compass
By relaxing the strict Greek rules for ruler-and-compass constructions, it becomes possible
to achieve constructions that trisect an arbitrary angle and "square the circle." These methods generally "cheat" either by using a marked ruler (Euclid allows only an unmarked straightedge) or by requiring the limit of an infinite sequence of operations (Euclid allows only a finite number of operations).
*Quadratrix of Hippias
In ancient times, the Greek mathematician Hippias used bisection of segments and angles to construct the quadratrix, a curve with which he was able to trisect any angle and to construct two straight line segments whose lengths are in the ratio of p to 1. Although Hippias used only compass and straightedge, he violated one of Euclid's rules by requiring an infinite number of operations.
*Pentagon Construction
The Greeks knew how to construct a regular pentagon with ruler and compass, and so
soon will you. But they failed to construct a regular heptagon. There is a reason:
constructing a pentagon is equivalent to solving a quadratic equation, while constructing a
heptagon is equivalent to solving an irreducible cubic equation.
*Heptagon Construction
The reason that a heptagon cannot be constructed with compass and straightedge is that
its construction is equivalent to solving an irreducible cubic equation. This puts the problem in the same category as angle trisection. Since angle construction can be accomplished with the aid of a marked ruler, why not heptagon construction? Archimedes had a complicated marked-ruler construction, but a simple construction was finally discovered and published in 1975.
*Gauss's Heptadecagon
At the age of nineteen, Gauss made the most remarkable advance in ruler-and-compass
constructions since the ancient Greeks by showing how to construct a regular
seventeen-sided polygon. The algebra behind the construction is fascinating.
Number Theory:
Computing Real Numbers
What does it mean to "know" an irrational number like or p? We must be able to "trap" the number within an infinite sequence of nested intervals whose endpoints are rational numbers. A computer program can in principle calculate as many of the nested intervals as we wish; so it implements this sort of definition.
In order to see what is going on in a program that does a lot of computation, it is
useful to be able to display the values of variables as you step through a program and to generate tables of intermediate results through which you can scroll.
When you use C++ in an environment with good graphical user interface ("GUI") tools, this is easy to do. You end up with a fairly professional-looking application, but almost all the GUI code gets written for you by the development tools The only C++ code that you need to write entirely from scratch is what implements the actual mathematics.
Uncountability and Transcendence
One of the most interesting properties of p is that it is a transcendental number, which means that it does not satisfy any polynomial equation with integer coefficients. Although is was a challenge for mathematicians to exhibit even one transcendental number, and a greater challenge to prove that p is transcendental, there is an ingenious counting argument due to Cantor that shows that among the real numbers, transcendental numbers are the rule rather than the exception.
cleverest ones involves ingenious changes of variable or integration by parts. Review these
fundamental techniques.
Diophantine Approximation
Almost everyone knows that p is nearly 22/7. Over the years other even better rational approximations like 335/113 have been discovered. It turns out that for any irrational number, there are infinitely many good rational (or "Diophantine") approximations. What
is more, if you insist that a number be more that just irrational, for example, that is satisfies no polynomial equation of degree less than 5, then there are infinitely many superbly good rational approximations.
This fact led to Liouville's construction of the first transcendental number, one that can be approximated so well by rational numbers that it cannot satisfy any polynomial equation.
*Niven Irrationality
Everyone knows that 22/7 is only an approximation to pi, but might there be a rational
number whose value is precisely pi? A short proof by the twentieth-century mathematician
Ivan Niven shows that the answer is "no." Niven's approach to this problem is to show
that if pi were rational, there would have to exist a positive integer smaller than one.
Transcendence of e
The numbers e and pi are not merely irrational, they are "transcendental," which means
that they do not satisfy any polynomial equation with integer coefficients. Again the proof
involves the strategy of reaching the contradiction of a positive integer that is less than one.
Transcendence of Pi
In the nineteenth century, mathematicians finally succeeded in showing that "squaring the
circle" is impossible by proving that pi, like e, is transcendental. The proof has been
reworked and reworked, but it is still intricate. Of course, what is shown is that if pi
satisfied a polynomial equation, there would exist a positive integer less than one!
History of Pi
*Archimedes' Recursion
The Greek mathematician Archimedes, working without the benefit of any good algebraic
notation, devised a method for computing pi as a limit involving polygons with an
ever-increasing number of sides which remained the state of the art for about 1500 years.
His approach of recursively updating two sequences of numbers lends itself naturally to
computation.
Wallis's Product
While attempting to do "interpolation" in an era when techniques of integral calculus were
not yet well established, the English mathematician and theologian Wallis discovered a
famous infinite product for pi. Put into modern notation, Wallis's discovery is an ingenious
application of integration by parts.
*Newton's Arc Sine
With characteristic brilliance, Newton started with the definition of pi in terms of the area
of a circle and converted it into an infinite series for pi that converged fast enough to be
useful for calculation. His approach is closely related to the inverse trigonometric functions,
which are reviewed here.
Brouncker's Fraction
Inspired by the infinite product of his friend Wallis, Lord Brouncker devised a continued
fraction that converges to pi. A century later, Euler showed that this fraction is equivalent
to a well-known infinite series for pi. Continued fractions, once a hot topic in mathematics
but now almost forgotten, provide a way of getting rational approximations to pi that are
better than 22/7.
*Vieta's Product
The French Renaissance mathematician Vieta, famous for his algebraic solution of the
general cubic equation, ushered in the modern era of pi in Europe with an infinite product
based on the old Greek idea of approximating a circle by means of polygons. In modern
notation this becomes an exercise in trigonometric identities.
*Arc Tangentry
John Machin, Professor of Astronomy in London, was the first to find an identity involving the arc tangent series that makes it possible to compute pi by means of rapidly convergent series. Similar identities were later discovered and rediscovered by Euler and others, and debate rages over which is the best. With a little twentieth-century insight involving complex numbers, you can quickly rediscover them all.
Fourier Series
Euler's discovery that any periodic function can be expanded in a series whose terms are sine and cosine functions leads to many different series whose terms are rational numbers and whose sum involves pi.
Euler Sums and Products
With inspired intuition, the eighteenth-century Swiss mathematician Euler devised
infinite-product formulas for the trigonometric functions which lead to Wallis' product as
one of many special cases, then used one of them to find the sum of an infinite series that
had defied the best efforts of the entire Bernoulli family. Proving these formulas correct is a
challenging exercise in dealing with the infinite.
Zeta Series
Having invented much of the theory of infinite series, Euler went on to explore series
whose terms involve only the prime numbers and developed some series for pi which,
although fairly useless for computation, are fascinating to number theorists.
*Spigotry
Contemporary mathematician Stan Wagon, starting from an infinite series discovered by
Euler, devised a "spigot" algorithm that generates digits of pi one at a time. Wagon's novel
approach to the problem of computing decimal digits of pi makes it easy to compute
thousands of digits using ordinary integer arithmetic on a personal computer.
Hex Expansions
In the 1990s the Canadian Borwein brothers stumbled across a power-series expansion
for pi that makes it possible to calculate an arbitrary digit in the expansion of pi without
calculating all the preceding digits -- provided the calculation is done in base-sixteen
arithmetic.