User:Dcljr/Math

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What is mathematics?[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

To be merged and/or expanded upon.

?

  • 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

?

  • 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

?

  • 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

?

  • Fermat's Last Theorem

References[edit]

  • 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.