Math 378: Number Systems: An Axiomatic Approach
This is the website for Math 378 (for Spring 2010).
Last updated: 19 January 2010.
Lecture Notes
Chapter 0. Background. This is still in the draft stage.
Chapter 1.
Peano's Axioms. Natural Numbers N.
Chapter 2.
Further exploration of the Natural Numbers.
Chapter 3.
The construction of the integers Z.
Chapter 4.
Exploring Z.
Chapter 5.
Modular Arithmetic, including the
ring of integers modulo n, the field of integers mod p, units and exponentiation.
Chapter 6.
The field Q and ordered fields in general.
Chapter 7.
The construction of R.
Chapter 8.
Exploring R. This discusses monotonic sequences,
decimal expansions, nth roots, and uncountability.
Chapter 9.
The Complex numbers C.
Chapter 10.
Polynomials and the Fundamental Theorem of Algebra.
Course Description
This course carefully and rigorously develops several number systems
including the set of natural numbers, the ring of integers, the ring
of integers modulo m, the field of rational numbers, the field of real number,
and the field of complex numbers. Other topics include:
Cauchy sequences, limits, De Moivre's theorem, and the fundamental theorem of algebra.
A major purpose of this course is to illustrate the axiomatic method. As such
it will emphasize careful proof in a mostly familiar setting.
This course also develops foundational ideas needed for
more advanced courses
including Algebra (470), Analysis (430), Number Theory (422).
Prerequisites
Basic proficiency with logic, proofs, set theory,
functions (injective, surjective, bijective, inverses), and relations (order relations and equivalence relations).
At CSUSM, this material is covered in Math 350 or Math 370.
Web CT or Moodle
I will use webCT or Moodle in addition to the current site. This other site will
contain announcements, study guides, grades, and such, and is limited to current students.
Textbook Policy
The course will be mainly taught from handouts.
Our goal is to rigorously prove all the basic properties of the number systems.
The handouts supply some of the proofs, and the others
proofs are given as exercises in the handouts.
In addition, there is one required textbook. It will be used as a reference to
the logical tools used in this course, and as an additional source of homework problems
for these tools.
How to Prove It: A Structured Approach
Daniel J. Velleman
Cambridge University Press (2nd Edition)
My course notes follow a long tradition of developing the number systems axiomatically.
I have chosen some such older classics as optional texts:
The Number Systems: Foundations of Algebra and Analysis,
by Solomon Feferman
Foundations of Analysis, by Edmund Landau
These textbooks have the same objectives
as our course, and might help you gain perspective.
However, they will not help much with
your day-to-day work in Math 378 since our approach is different in
many small
ways, so there much "translation" required to relate their approaches with ours.
Also, I want you to work out the proofs on your own: I want your proofs to be exercises in thinking, not in looking things up other books.
Instructor
Professor Wayne Aitken
e-mail:
waitken@csusm.edu