In this course we will look at the axiomatic method --- the cornerstone of modern mathematics --- through the historical development of several important mathematical topics. In particular, we will discuss the following topics: The development of geometry concentrating mostly on Euclidian axioms; Numbers, including the Peano axioms for natural numbers, integers and the fundamental theorem of arithmetic, rational numbers as an example of a field, and the long history to the rigorous notion of real numbers; Polynomials as the ``nicest'' functions, including the discovery of complex numbers, fundamental theorem of algebra, and ``infinite'' polynomials, aka power series, that were invented by Isaac Newton; and finally Probability theory, including the axioms of Kolmogoroff, classical probability, and the (basic version of) central limit theorem.

• Classes: MWF 8:00am-8:50am, NDSU Minard Hall, Rm 208
• Office hours: MWF 11:00am-11:50am (Minard 408E32)
• Syllabus
• Textbook: David M. Burton, The History of Mathematics, McGraw-Hill Education; 7 edition (2010) Amazon.

• Week 1. A bird-eye view on the history of mathematics.
• Week 2. The beginning of Greek geometry.
• Week 3. Euclid and the Elements. Euclidian geometry.
• Week 4. Euclidian geometry. Non-Euclidian geometry.
• Week 5. On the way to rigor. Cantor and set theory. Peano's axioms.
• Week 6. Axiomatic development of number sets.
• Week 7. Euclid's number theory. Prime numbers.
• Week 8. Diophantine Equations. Fermat, Euler, and Gauss.
• Week 9. Rational and real numbers.
• Week 10. Spring break.
• Week 11. Solving polynomial equations. Cubic equations and complex numbers.
• Week 12. The fundamental theorem of algebra.
• Week 13. Newton and ``infinite'' polynomials. Birth of calculus.
• Week 14. Axioms of probability theory. Basic combinatorics.
• Week 15. The law of large numbers and the central limit theorem.
• Week 16. Review and additional topics.
• Week 17. Dead week. Presentations.
• Week 18. Final exam (May 10th, Friday, 10:30am). Presentations.

• Basic Euclidean geometry and review of proof techniques
• Project 1: Pythagorean theorem
• Euclid, axioms, and mathematical rigor
• Project 2: Straightedge and compass constructions
• Mathematical induction and Peano's axioms
• Project 3: Proofs my mathematical induction
• From Euclid to RSA algorithm
• Project 4: All Pythagorean triples
• Project 5: Fermat's last theorem (n=4)
• Polynomials
• Project 6: Geometry and complex numberss
• Project 7: Pi is irrational
• The main invention of Newton: Infinite series