User:Dcljr/Math

What is mathematics?

The study of quantity
- Counting discrete things: how many?
- Measuring on a continuum: how much?
The study of space
- Physical space and geometrical objects
  - Directions and dimensions
  - Figures and their measurement
  - Perspective
- Mathematical space
  - Coordinate systems
The study of structure
- Patterns
- Abstraction and formalism
- Logical relationships
The study of change
- Discrete change: arithmetic and algebra
- Continuous change: functions and analysis

Arithmetic and number systems

Counting and natural numbers
- Where does counting come from?
- Can animals other than humans count?
Addition and whole numbers
- Addition as accumulation
- Reversing addition
- The meaning of adding zero
- Why numbers other than whole numbers are needed
Subtraction and negative numbers
- Subtraction as the inverse of addition
- Negative numbers
- Subtraction as addition with negative numbers
Integers
Multiplication
- Multiplication as repeated addition (or subtraction)
  - "Times"
- Reversing multiplication
- The meaning of multiplication by one
- The meaning of multiplication by zero
- Why numbers other than integers are needed
Division and rational numbers
- Division as apportionment
  - Divisibility
  - Even and odd numbers
  - Prime and composite numbers
- Division as the inverse of multiplication
- The meaning of division by one
- Rational numbers
  - Fractions
  - Decimals
  - Reciprocals
- Division as multiplication by a rational number
- Why is division by zero not allowed?
Digression: numerals and numeration systems
- Numbers vs. numerals
- Additive numeration systems
  - Egyptian numerals
  - Roman numerals
- Multiplicative numeration systems
  - Chinese numerals
- Ciphered numeration systems
  - Greek numerals
- Positional numeration systems
  - Babylonian numerals
  - Aztec numerals
  - Hindu-Arabic numerals
    - Zero finally becomes a number
Exponentiation
- Exponentiation as repeated multiplication (or division)
- Terminology: power, base, exponent
  - Powers of 2 and 3
  - Squares and cubes
- The meaning of a zero exponent
- Powers of 10 and place value in our base-ten numeration system
  - Scientific notation
- Reversing exponentiation in two different ways
  - Finding the right base
  - Finding the right exponent
- Reciprocals as powers with negative exponents
- Why numbers other than rational numbers are needed
Roots and irrational numbers
- Roots as inverses of powers
- Radical notation
- Roots as powers with reciprocal exponents
- Expressing powers with arbitrary rational exponents as roots
- Why are there two ways of reversing exponentiation but not addition or multiplication?
- Why are some roots irrational?
  - Proof that the square root of 2 is irrational
  - Proof that the square root of 3 is irrational
  - Why wouldn't the same argument prove that the square root of 4 is irrational?
- Irrational numbers in radical form
- Irrational numbers in decimal form
  - Square root of 2 in decimal form
- Are all irrational numbers expressible using roots and/or rational exponents?
- Are all irrational numbers expressible using decimal numbers?
- Real numbers as all numbers with a decimal representation
  - Reals contain all rationals and all irrationals
  - Are there other types of numbers?
Rules of arithmetic
- Adding and subtracting integers
- Multiplying integers
- Dividing integers
  - Long division
- Divisibility rules
- Factoring an integer
  - Prime factorization
  - Greatest common factor
  - Least common multiple
- Order of operations
Simplifying arithmetic expressions
- Reducing fractions
- Multiplying and dividing fractions
- Adding and subtracting fractions with the same denominator
- Adding and subtracting fractions with different denominators
  - Least or lowest common denominator
- Simplifying fractions containing other fractions
- Adding and subtracting decimal numbers
- Multiplying and dividing decimal numbers
- Converting between fractions and decimals
- Multiplying and dividing powers
- Multiplication and division in scientific notation
- Reducing radicals
- Multiplying and dividing radicals
Counting revisited
- How high can you count?
- Infinity
  - Some strange properties of infinity
- Are there more rational numbers than natural numbers?
  - Countability
- Are there more irrational numbers than rational numbers?
  - Uncountability and Cantor's diagonal argument
- Cardinality and orders of infinity
  - Continuum hypothesis
    - Sets and subsets
    - Power set
  - Transfinite numbers

Measurement and elementary geometry

Measurement
- How measurement is different from counting
- How measurement is related to counting
  - Units of measurement
Basics of Euclidean geometry
- Two dimensions on a flat surface
  - Maps and compass directions (N, S, E, W)
- Three dimensions of physical space
  - Why three dimensions?
    - Six cardinal directions
      - Up and down, forward and backward, left and right
  - Relationships between elementary geometrical objects and dimensions
    - Points on a line
    - Lines in a plane
    - Planes in space
  - More than three dimensions?
Euclidean geometry in two dimensions
- Lines
  - Lines and line segments
  - Parallel and intersecting lines
    - Euclid's fifth postulate
    - Non-Euclidean geometries
  - Measuring lengths along a line
    - Lengths as multiples of a unit
    - Rational magnitudes
    - Are all lengths rational with respect to a given unit?
- Angles
  - Opposite and adjacent angles
  - Right angles
  - Acute and obtuse angles
  - Complementary and supplementary angles
  - Relationship between angles when two parallel lines are cut by a transversal
  - Measuring angles
    - What unit should be used to measure angles?
    - Multiples and fractions of a right angle
    - Degrees
      - Why 360 degrees?
    - Are all angles rational?
- Polygons
  - Triangles
    - Equilateral triangles
    - Isosceles triangles
    - Scalene triangles
    - Right triangles
      - Hypotenuse
  - Quadrilaterals
    - Squares
    - Rectangles
    - Parallelograms
    - Trapezoids
  - Convex and concave polygons
  - Regular polygons
- Measuring polygons
  - Area
    - What unit should be used to measure area?
    - Area of a square
      - Relation to square numbers
        Digression: other figurate numbers
      - Square root
    - Area of a rectangle
    - Area of a triangle
    - Area of a parallelogram
    - Area of a polygon
    - Are all areas rational?
  - Pythagorean theorem
    - Pythagorean triples
    - Not all lengths are rational!
      - Square root of two and the Pythagoreans
  - Perimeter
    - Perimeter of a triangle
      - Heron's formula for the area of a triangle
    - Perimeter of a general polygon
- Circles
  - Center and radius of a circle
  - Diameter of a circle
  - Circumference of a circle
    - Measuring a non-linear length
    - Pi
      - Is pi a rational number?
      - How can pi be computed?
        Method of exhaustion
  - The radius as a unit of angle measurement
    - Radians
    - Converting between degrees and radians
      - Not all angles are rational!
  - Area of a circle
- Triangles revisited
  - The many "centers" of a triangle
  - Congruence of triangles
    - SSS, SAS, ASA, AAS
    - SSA and the ambiguous case
Euclidean geometry in three dimensions
- Points, lines and planes in space
  - Parallel lines in space
  - Angles formed by intersecting lines in space
  - Skew lines
  - Parallel planes
  - Angles formed by intersecting planes
- Solids
  - Faces, edges, and vertices
  - Pyramids
  - Prisms
  - Parallelepipeds
  - The five regular solids
    - Tetrahedron
    - Cube
    - Octahedron
    - Dodecahedron
    - Icosahedron
    - Why only five regular solids?
  - Other polyhedra
    - Polyhedron duals
    - Stellations
- Measuring polyhedra
  - Volume
    - Volume of a cube
      - Relation to cubic numbers
      - Cube root
    - Volume of a parallelepiped
    - Volume of a prism
    - Volume of a pyramid
    - Volume of a general polyhedron
  - Surface area
    - Surface area of a polyhedron
- Non-polyhedral solids
  - Cylinders and cones
    - Right circular cylinders
    - Cylindrical solids in general
      - Volume of a cylinder
      - Surface area of a cylinder
    - Conical solids in general
      - Volume of a cone
      - Surface area of a cone
  - Spheres
    - Center and radius of a sphere
    - Volume of a sphere
    - Surface area of a sphere
  - Solid angles
Mixing dimensions
- Zero-dimensional points on a line
- One-dimensional curves in a plane
- Two-dimensional surfaces in space
  - The Möbius strip
- One-dimensional curves on two-dimensional surfaces
  - Right circular cones revisited: conic sections
    - Parabolas
    - Ellipses
    - Hyperbolas
    - Degenerate conic sections
Higher dimensions
- Visualizing higher dimensions in lower ones
  - Projections and shadows
    - Two-dimensional figures projected onto a one-dimensional line
    - Three-dimensional solids projected onto a two-dimensional plane
    - Flatland and the weirdness involved in crossing dimensions
- Which way is the fourth dimension?
  - Time as the fourth dimension?
    - Relativity, spacetime, and Minkowski space
  - Real four-dimensional Euclidean space
    - Tesseracts and hypercubes
    - Measurement in higher dimensions

Elementary algebra

Variables and equations
- Review of arithmetic from an algebraic perspective
  - Properties of equality
  - Properties of arithmetic operations
    - Commutativity
    - Associativity
    - Distributivity
- Solving simple equations
  - Percent problems
    - Finding a certain percent of a given number
    - Finding what number is a certain percent of a given number
    - Finding what percent of one number another number is
    - Calculating percent growth and percent reduction
Some history and context
- The origins of algebra
- The origin of the word "algebra"
- Some different meanings of the word "algebra"
- Some different meanings of the word "algebraic"
Algebraic expressions
- Monomials
  - Geometric interpretation of monomials
- Binomials
- Polynomials
  - Numerical interpretation of polynomials
Arithmetic with algebraic expressions
- The algebra of monomials
  - Addition and subtraction of monomials
    - Combining like terms
  - Multiplication of monomials
  - Division of monomials
  - Powers of monomials
  - Roots of monomials
  - Rules of exponents and radicals revisited
  - Greatest common factor of monomials
  - Least common multiple of monomials
- The algebra of polynomials
  - Addition and subtraction of polynomials
  - Multiplication of binomials
    - "FOILing"
  - Multiplication of polynomials
    - Multiplication of polynomials by distributing
    - Multiplication of polynomials by the table method
  - Factoring polynomials
    - Factoring out a common monomial
    - Factoring trinomials
      - Recognizing a perfect-square trinomial
      - Factoring a trinomial by guess-and-check
      - Factoring a trinomial by the "diamond method"
    - Factoring quadrinomials by grouping
    - Application of factoring: solving polynomial equations
  - Division of polynomials
    - Rational expressions
    - Polynomial long division
    - Synthetic division
    - Remainder theorem
    - Factoring polynomials of arbitrary degree
      - Rational zeros theorem
  - Powers of binomials
    - Pascal's triangle
    - The binomial theorem
    - How binomial coefficients are related to counting
      - Factorials
      - Permutations
      - Combinations
  - Powers of polynomials
    - Multinomial coefficients
    - The multinomial theorem
- The algebra of rational expressions
  - Reducing rational expressions
  - Multiplying and dividing rational expressions
  - Adding and subtracting rational expressions
  - Simplifying rational expressions within other rational expressions
    - Digression: continued fractions
  - Solving rate and ratio problems
    - Distance-rate-time problems
    - Proportion problems
      - Similar triangles
      - Similar figures in general
      - Scaling and its effect on length, area, and volume
    - Work problems
    - Mixture problems

Algebra and geometry united

Linking geometry to arithmetic: the real number line
- Zeno's paradox and the idea of the limit of a sequence (??)
- Irrational numbers as limits of sequences of rational numbers (??)
- The real number line
- Coordinate systems
  - The Euclidean plane
    - Axes and coordinates
    - Characterizing a point in the plane
    - Midpoint and distance between two points in the plane
  - Euclidean space
    - Points in space
    - Midpoint and distance between two points in space
  - Are other two- and three-dimensional coordinate systems possible?
  - Can coordinate systems involve numbers that aren't real?
Relations and equations
- Characterizing a line in the plane
  - Vertical and horizontal lines
  - The slope of a line
  - The equation of a line
    - Slope-intercept form
    - Point-slope form
    - Two-intercept form
    - General form
- Characterizing a line in space
- Characterizing a plane in space

Other stuff

To be merged and/or expanded upon.

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Sequences and series
- Arithmetic means and progressions
- Geometric means and progressions
- Sum of a finite arithmetic series
- Sum of a finite geometric series
- Sum of an infinite geometric series

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Classical Euclidean geometry as the first formal mathematical system
- Actual vs. idealized figures
- Constructions using compass and straightedge
  - The five basic constructions
  - Derived constructions
  - Impossible constructions

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Functions
Graphs

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Solving equations containing two or more different variables

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Inequalities
- Solving inequalities in one variable
- Solving inequalities in more than one variable

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Logarithms and transcendental numbers
- Powers vs. exponentials
- Logarithms as inverses of exponentials
- Transcendental and algebraic numbers

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Fermat's Last Theorem

References

Ifrah, Georges - The Universal History of Numbers (Wiley)
Ancient counting methods, number names and numerals for many different civilizations.
Cajori, Florian - A History of Mathematical Notations (Dover)
Numerals and other symbols used for arithmetic, algebra, and higher mathematics throughout history.
Boyer, Carl & Uta Merzbach - A History of Mathematics, 2nd ed. (Wiley)
A standard history of mathematics through the 19th century, written at a popular level.