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