Foundations of the Number System
MATH 600. Foundations of the Number System
Three Credit Hours (3).
Prerequisite: Undergraduate degree in mathematics or by instructor permission.
This course will provide a mature mathematical foundation for the number systems used in secondary and post-secondary mathematics courses, with an emphasis on rigorous logical and set-theoretical foundations of the natural numbers, integers, rational numbers, and real numbers. The course will also cover the common algebraic extensions of the number systems, and familiarize students with the historical development of the number systems.
Detailed Description of Course
This course is a rigorous development of the number systems, from the natural numbers through the real numbers and algebraic extensions. The numbers will be developed through the use of set theory, order types, induction and recursion, equivalence relations, convergent sequences, bounded and unbounded sets of numbers and closed and open sets of numbers. The sequence of topics as follows may include:
• Sets and relations.
o Axiomatic set theory and set operations.
o Cartesian products of sets; ordered pairs.
o Equivalence relations and partitions of sets.
• The Natural Numbers as Von Neumann ordinals.
o Building the Von Neumann hierarchy.
o The Peano axioms.
o The Principle of Mathematical Induction.
o Formal arithmetic.
• The Rational Numbers.
o The rational numbers as equivalence classes of ordered pairs of integers.
o Rational number arithmetic.
• The Real Numbers.
o Convergent sequences.
o Closed and Open sets.
o Dedekind cuts.
o Real number arithmetic.
• Algebraic extensions of the number systems.
o Gaussian integers.
o The complex numbers.
o Modular Arithmetic and finite fields.
o Modules and vector spaces.
• Historical overview of number systems.
Detailed Description of Conduct of Course
The course will be conducted in a traditional lecture format, but may include Socratic dialogue and group discovery learning.
Goals and Objectives of the Course
Students who satisfactorily complete the course will have a mature theoretical understanding of the number systems, and be able to guide their own students’ intuitions and discovery learning from a perspective of authoritative content knowledge.
Students will be assessed on written homework and projects. In addition there will be formal midterms and a final exam.
Other Course Information
Review and Approval
Date Action Reviewed