Outline

Here's my (pharpend) basic outline for the book. It's extremely rough at this point and will probably be gutted and slaughtered in its entirety.

All of the chapters beyond this point will be assumed to have a multitude of exercises, graphs, examples, applications, etc.

Chapter 2, Groups, Rings, and Fields
- Introduction to Groups, Rings, and Fields
- \(\mathbb{N}\) is a group, \(\mathbb{Z}\) is a ring, \(\mathbb{R}\) is a field.
- How this applies to everyday life (Boring!)
Chapter 3, Proofs
- What are proofs?
- Proof-based approach to groups, rings, fields.
Chapter 4, Functions
- What are functions?
- Some examples
Chapter 5, Algorithms
- What are algorithms?
- Examples
Chapter 6, monomials
- Examples
- How to manipulate them algebraically
- Graphs of lines
Chapter 7, polynomials
- Examples
- How to manipulate them algebraically
- Graphs of lines
- Quadratic formula
- Proof that there is no quintic formula?
Chapter 8, exponential functions
Chapter 9, logarithms
Chapter 10, trig functions

This is a good segue to talk about Complex numbers

Good segue to talk about the concept of dimensions

Chapter 14, Dimensions
Chapter 15, Parametric functions
Chapter 16, Complex parametric functions
Chapter 17, functions that go from \(\mathbb{F}^n\) to \(\mathbb{F}\), where \(\mathbb{F}\) is a field.
Chapter 18, functions that go from \(\mathbb{F}\) to \(\mathbb{F}^n\), where \(\mathbb{F}\) is a field.
Chapter 19, functions that go from \(\mathbb{F}^n\) to \(\mathbb{F}^m\), where \(\mathbb{F}\) is a field.

We'll next want to approach systems of equations. first matrices

Chapter 20, Matrices
- Matrix addition, multiplication, etc
- Matrices as linear functions
Chapter 21, Systems of equations
- What is a system of equations
- using matrices to solve for them
Chapter 22, Vector spaces

... Basically go through linear algebra

... Go through calculus and differential equations