Math 378: Number Systems
This is the website for Math 378.
Last updated: 10 May 2008.
Handouts
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: Polynomial Rings. (Under construction)
Chapter 11: Fundamental Theorem of Algebra. (Under construction)
Web CT
I use webCT in addition to the current site.
The webCT site is only accessible to current students.
WebCT site
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: polynomial rings,
unique factorization of polynomials and integers,
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 need for
other courses including Algebra (470), Analysis (430), Number Theory (422) at CSUSM.
Prerequisites
Basic proficiency with logic, proofs, set theory,
functions (injective, surjective, bijective, inverses), and relations.
At CSUSM, this material is covered in Math 350 or Math 370.
Textbook
There is no required textbook for this course.
Optional textbooks include some older classics:
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
way, 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.
The course will be 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.
Instructor
Professor Wayne Aitken
e-mail:
waitken@csusm.edu