Number systems: an axiomatic introduction to algebra, number theory,
and analysis
Here is an archival version of the course from 2009. I have a newer, better version.
Text (Old version)
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.
Appendix: Chapter 0. Background.