# 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?
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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"
• 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.

?

• 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

?

• Solving equations containing two or more different variables

?

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

?

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

?

• Fermat's Last Theorem