User:Ta bu shi da yu/Math structure

Some thoughts on the structure of learning about Mathematics. May go into a Wikireader.

History

1. History of Mathematics

Number

1. Zero
2. One
3. Quantity
4. Natural numbers
   1. Whole numbers
   2. Negative and non-negative numbers
   3. Number line
   4. Decimal
5. Real numbers
   1. Rational number
   2. Irrational number
6. Fraction
   1. Vulgar fraction
   2. Irreducible fraction
   3. Proper fraction
   4. Improper fraction
   5. Unit fraction
   6. Egyptian fraction
   7. Decimal fraction
   8. Dyadic rational
   9. Mixed fraction
   10. Continued fraction
   11. Compound fraction
   12. Partial fraction

Arithmetic

1. Addition
   1. Counting
   2. Addition of natural numbers
   3. Commutative operation
   4. Associativity
   5. Absolute value
2. Subtraction
3. Plus and minus signs
4. Multiplication
5. Division
   1. Long division
   2. Division by zero
6. Percent
7. Square root
8. Exponentiation
9. Logarithm
10. Order of operations

Algebra

Elementary algebra

1. Variable
2. Equation
3. Function (mathematics)
4. Polynomial
   1. Linear equation / Linear function
   2. Quadratic equation
   3. Cubic equation

Set theory

Naive set theory

1. Set
2. Element
3. Empty set
4. Cardinal number
5. Subset
6. Union
7. Intersection
8. Complement
9. Russell's paradox
10. Algebra of sets
11. Relation

Functions

1. Domain
2. Codomain
3. Range
4. Multivalued function
5. Partial function
6. Types:
   1. Injective function
   2. Surjective function
   3. Bijective function
7. Image & Preimage or Inverse image

Axiomatic set theory

1. Axiom of extensionality
2. Axiom of empty set
3. Axiom of pairing
4. Axiom of union
5. Axiom of infinity
6. Axiom schema of specification
7. Axiom schema of replacement
8. Axiom of power set
9. Axiom of regularity
10. Axiom of choice

Abstract algebra

1. Algebraic structure

Graphing

1. Coordinate system
2. Coordinates (elementary mathematics)
   1. Cartesian coordinate system
   2. Polar coordinates
   3. Circular coordinates
   4. Cylindrical coordinates
   5. Spherical coordinates
3. Graph of a function

Geometry

Euclidean geometry

1. Point
2. Line
   1. Ray
   2. Line segment
   3. Curve
3. Plane
4. Angle
5. Parallel
6. Perpendicular
7. Congruency
8. Similarity
9. Polygon
   1. Triangle
      1. Congruent triangles
      2. Similar triangles
   2. Pentagon
   3. Hexagon
   4. Heptagon
   5. Octagon
   6. Enneagon
   7. Decagon
   8. Hendecagon
   9. Dodecagon
   10. Quadrilateral
       1. Trapezoid
       2. Isosceles trapezium
       3. Parallelogram
          1. Rhombus
             1. Kite
          2. Rectangle
          3. Square
10. Circle
    1. Circumference
    2. Diameter
    3. Radius
    4. Secant
    5. Chord
    6. Arc
    7. Tangent
    8. Segment

Two and three-dimensional geometry

1. Perimeter
2. Arclength
3. Area
4. Volume
5. Pythagorean Theorem
6. Ellipse
7. Polyhedron
8. Ellipsoid
9. Sphere

Traditional geometry

1. Topology
2. Erlangen program
3. Hyperbolic geometry
4. Elliptic geometry
5. Affine geometry
6. Projective geometry
7. Inversive geometry

Modern geometry

1. Algebraic geometry
2. Differential geometry
3. Noncommutative geometry

Trigonometry

Basic concepts

1. Radian
2. Degree
3. Unit circle
4. Trigonometric functions
5. Sine wave
6. Simple harmonic motion
7. Exact trigonometric constants
8. Generating trigonometric tables

Analytical trigonometry

1. Trigonometric identity
2. Sum and difference formula (trigonometry)

Other

1. Law of sines
2. Law of cosines