## Contents

## 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

- Physical space and geometrical objects
- 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 systemsEdit

- 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

- Multiplication as repeated addition (or subtraction)
- 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?

- Division as apportionment
- 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

- Continuum hypothesis

## Measurement and elementary geometryEdit

- 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

- Six cardinal directions
- Relationships between elementary geometrical objects and dimensions
- Points on a line
- Lines in a plane
- Planes in space

- More than three dimensions?

- Why three dimensions?

- Two dimensions on a flat surface
- 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

- Triangles
- 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

- Relation to square numbers
- 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

- Perimeter of a triangle

- Area
- 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

- Lines
- 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

- Volume of a cube
- Surface area
- Surface area of a polyhedron

- Volume
- 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

- Cylinders and cones

- Points, lines and planes in space
- 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

- Right circular cones revisited: 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

- Projections and shadows
- 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

- Time as the fourth dimension?

- Visualizing higher dimensions in lower ones

## Elementary algebraEdit

- 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

- Percent problems

- Review of arithmetic from an algebraic perspective
- 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

- Monomials
- 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

- Addition and subtraction 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

- The algebra of monomials

## Algebra and geometry unitedEdit

- 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?

- The Euclidean plane

- 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

- Characterizing a line in the plane

## Other stuffEdit

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

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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

?

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

?

- Fermat's Last Theorem

## ReferencesEdit

- 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.