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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.

Chapter 1, Introduction
- Motivation
- Potential scope of the book
- What background knowledge you need.
  - ideally this would just be fluency in english, and elementary school math.
- What is math?
- Why are we interested in it?

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

Chapter 2, Boolean Algebra
- Introduction
- Basic Logic
  - True and False
    - \land, \lor, and \lnot
  - Logic notation
    - \implies, \impliedby, \iff
  - Exercises
- Mildly more complicated Logic
  - Combining All of them
    - \lnot\lnot, \lnot\land
  - More logic notation
    - \notimplies, \notimpliedby, \notiff
  - Exercises
    - \lnot\lor
- Idris stuff
  - Do all the previous stuff in Idris.
Chapter 3, Sets
- Lists and ordered pairs
- Sets
  - ElementOf
  - ImproperSubset
  - ProperSubset
  - Exercises
- Operators on Sets
  - Unary opearators
  - Binary operators
  - The set of booleans
    - Unary ~ operator
    - Binary V and ^ operators
    - Binary V and ^ operators
  - Exercises
The set of natural numbers
The set of integers
The set of real numbers
Chapter 3, Proofs
- What are proofs?
- Proof-based approach to groups, rings, fields.
- Peano axioms
  - Basically go through Landau's Foundations of Analysis
Chapter 4, Special sets
- Magmas
- Semigroups
- Categories
- Monoids
- Groups
- Rings
- Fields
Chapter 5, fancy functions
- Homomorphisms
- Isomorphisms
- Endomorphisms
- Injective functions
- Surjective functions
- Bijective functions
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

Let us make this our goal for now, then we will move on.

Chapter 8, exponential functions
Chapter 9, logarithms
Chapter 10, trig functions

This is a good segue to talk about Complex numbers

Chapter 11, complex and imaginary numbers
Chapter 12, Complex functions
Chapter 13, Complex algorithms

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 F^n to F, where F is a field.
Chapter 18, functions that go from F to F^n, where F is a field.
Chapter 19, functions that go from F^n to F^m, where 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

Chapter 35, calculus

... Go through calculus and differential equations

Chapter 52, Statistics
Appendix B - boring stuff
- Introduction of the primary authors (Peter Harpending, Randy Brown).
- Book license
- How to contribute

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Outline

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パーマリンク履歴 Raw