The Modern Math Academy with Dr. W

This is part two of a four part series called Introduction to Modern Mathematics. The intention of this series is to bring the un-initiated up to speed with what the modern mathematical methodologies, paradigms, and objects of study are as quickly as possible by using the two most important mathematical texts of all time Euclid's Elements & Gauss' Disquitiones Arithmeticae.

This course, Part 2: Gauss has the intention to initiate the student into the Modern Math paradigm of "structuralism" and witness the separation of syntax and semantics as we developed in Part 1: Euclid. This course is also a fully developed and self-contained introduction to Number Theory and Abstract Algebra which introduces modular arithmetic and culminates in the  second-most proved theorem of all time, Gauss' "Golden Theorem" called the Law of Quadratic Reciprocity. There are many worksheets and articles giving extra examples and problems for students to get hands-on with this very esoteric and abstract way of thinking.

               Core Objectives:

1. Learn basic facts about the "Classical" system of numbers: Integers, rational numbers, real and complex.
2. Define a new "world of arithmetic" called Modular Arithmetic and investigate it's algebraic properties with respect to addition and multiplication.
3. Learn the algebraic properties of commutative rings (units, zero divisors, fields).
4. Learn about properties of abelian groups (cyclic and non-cyclic groups, subgroups, multiplicative group of units, orders of elements, Lagrange's Theorem.)
5. Look over the actual text of Gauss and comment on his personal style and influence on "Modern Mathematics"
6. Learn how Gauss solved linear equations in modular arithmetic.
7. Understand the statement of Quadratic Reciprocity and how it is used to solve problems.