The Foundations of Math introduces students not only to the theories and formulas that form the basis of each field of mathematics, but also to the individuals responsible for major breakthroughs. Detailed diagrams provide visual summaries of complex concepts and make this an asset to both to lovers of math and to those who may find math unapproachable.

Table of Contents

• Algebra and Trigonometry
• Introduction
• Algebra
• The Emergence of Formal Equations
• Classical Algebra
• Structural Algebra
• Elementary Algebra
• Linear Algebra
• Modern Algebra
• Trigonometry
• Ancient Egypt and the Mediterranean World
• India and the Islamic World
• Passage to Europe
• From Geometric to Analytic Trigonometry
• Trigonometric Functions
• Plane Trigonometry
• Spherical Trigonometry
• Analytic Trigonometry
• Coordinates and Transformation of Coordinates
• Great Algebraists
• Early Algebraists (Through The 16th Century)
• Classical Algebraists (17th–19th Centuries)
• Algebraists of the Structural Period (20th–21st Centuries)
• Great Trigonometricians
• Aryabhata I
• Al-Battani
• Abraham De Moivre
• Leonhard Euler
• James Gregory
• Hipparchus
• Menelaus of Alexandria
• Ptolemy
• Regiomontanus
• Naṣīr al-Dīn al-Ṭūsī
• Algebraic and Trigonometric Terms and Concepts
• Alfonsine Tables
• Algebraic Equation
• Algebraic Number
• Almagest
• Associative Law
• Automorphism
• Binomial Theorem
• Commutative Law
• Complex Number
• Cramer's Rule
• Degree of Freedom
• Determinant
• Discriminant
• Distributive Law
• Eigenvalue
• Equation
• Factor
• Fourier Series
• Fundamental Theorem of Algebra
• Gauss Elimination
• Group
• Group Theory
• Hodge Conjecture
• Homomorphism
• Hyperbolic Function
• Ideal
• Imaginary Number
• Injection
• Irrational Number
• Law of Cosines
• Law of Sines
• Linear Equation
• Liouville Number
• Matrix
• Multinomial Theorem
• Parameter
• Pascal's Triangle
• Polynomial
• Quadratic Equation
• Quaternion
• Rational Number
• Ring
• Root
• Square Root
• Surjection
• Synthetic Division
• System of Equations
• Triangulation
• Trigonometric Function
• Trigonometry Table
• Variable
• Vector
• Vector Operations
• Vector Space
• Glossary
• Bibliography
• Analysis and Calculus
• Introduction
• Measuring Continuous Change
• Bridging the Gap Between Arithmetic and Geometry
• Numbers and Functions
• The Problem of Continuity
• Properties of the Real Numbers
• Differential Equations
• Ordinary Differential Equations
• Partial Differential Equations
• Other Areas of Analysis
• Complex Analysis
• Measure Theory
• Other Areas of Analysis
• History of Analysis
• The Greeks Encounter Continuous Magnitudes
• Models of Motion in Medieval Europe
• Analytic Geometry
• The Fundamental Theorem of Calculus
• Rebuilding the Foundations
• Calculus
• Differentiation
• Integration
• Concepts in Analysis and Calculus
• Argand Diagram
• Bessel Function
• Boundary Value
• Calculus of Variations
• Chaos Theory
• Continuity
• Convergence
• Curvature
• Derivative
• Difference Equation
• Differential
• Differential Equation
• Differentiation
• Direction Field
• Dirichlet Problem
• Elliptic Equation
• Exact Equation
• Exponential Function
• Extremum
• Fluxion
• Fourier Transform
• Function
• Harmonic Analysis
• Harmonic Function
• Infinite Series
• Infinitesimals
• Infinity
• Integral
• Integral Equation
• Integral Transform
• Integraph
• Integration
• Integrator
• Isoperimetric Problem
• Kernel
• Lagrangian Function
• Laplace's Equation
• Laplace Transform
• Lebesgue Integral
• Limit
• Line Integral
• Mean-Value Theorem
• Measure
• Minimum
• Newton and Infinite Series
• Orthogonal Trajectory
• Parabolic Equation
• Partial Differential Equation
• Planimeter
• Power Series
• Quadrature
• Separation of Variables
• Singular Solution
• Singularity
• Special Function
• Spiral
• Stability
• Sturm-Liouville Problem
• Taylor Series
• Variation of Parameters
• Great Figures in the History of Analysis
• The Ancient and Medieval Period
• The 17th and 18th Centuries
• The 19th and 20th Centuries
• Glossary
• Bibliography
• Geometry
• Introduction
• History of Geometry
• Ancient Geometry: Practical and Empirical
• Ancient Geometry: Abstract and Applied
• Ancient Geometry: Cosmological and Metaphysical
• The Post-Classical Period
• Transformation
• Relaxation and Rigour
• Branches of Geometry
• Euclidean Geometry
• Solid Geometry
• Analytic Geometry
• Projective Geometry
• Differential Geometry
• Non-Euclidean Geometry
• Topology
• Algebraic Topology
• Graph Theory
• Geometric Terms and Concepts
• Algebraic Surface
• Angle Trisection: Archimedes’ Method
• Trisecting The Angle: The Quadratrix of Hippias
• Axiom
• Axiomatic Method
• Brachistochrone
• Bridge of Asses
• Brouwer's Fixed Point Theorem
• Catenary
• Ceva's Theorem
• Circle
• Compactness
• Cone
• Coordinate Systems
• Cross Ratio
• Curve
• Cycloid
• Cylinder
• Desargues's Theorem
• Dimension
• Duality
• Ellipse
• Ellipsoid
• Envelope
• Euclid's Windmill
• Euclidean Space
• Method of Exhaustion
• Fractal
• Golden Ratio
• Graph
• Harmonic Construction
• Hausdorff Space
• Hilbert Space
• Hippocrates’ Quadrature of the Lune
• Hyperbola
• Hyperboloid
• Incommensurables
• Isometric Drawing
• Königsberg Bridge Problem
• Line
• Measuring the Earth, Classical and Arabic
• Measuring The Earth, Modern
• Metric Space
• Parabola
• Paraboloid
• Parallel Postulate
• Pencil
• Pi
• Platonic Solid
• Polygon
• Projection
• Pythagorean Theorem
• Space-Time
• Spiral
• Square
• Thales’ Rectangle
• Topological Space
• Biographies of Great Geometers
• Ancient Greek and Islamic Geometers
• Pre-Modern (Pre-1800) Geometers
• Modern Geometers
• Glossary
• Bibliography
• The History of Mathematics
• Introduction
• The Foundations Of Mathematics
• Arithmetic or Geometry
• Being Versus Becoming
• Universals
• The Axiomatic Method
• Number Systems
• Calculus Reopens Foundational Questions
• Non-Euclidean Geometries
• Cantor
• The Quest for Rigour
• Category Theory
• Ancient Western Mathematics
• Mathematics in Ancient Mesopotamia
• Mathematics in Ancient Egypt
• Greek Mathematics
• Geometry in the 3rd Century BCE
• Later Trends in Geometry and Arithmetic
• Mathematics in the Islamic World (8th–15th Century)
• European Mathematics Since The Middle Ages
• European Mathematics During the Middle Ages and Renaissance
• Mathematics in the 17th and 18th Centuries
• Other Developments
• Mathematics in the 19th Century
• Mathematics in the 20th and 21st Centuries
• Indian And East Asian Mathematics
• Vedic Number Words and Geometry
• The Post-Vedic Context
• Indian Numerals and the Decimal Place-Value System
• The Role of Astronomy and Astrology
• Classical Mathematical Literature
• The Changing Structure of Mathematical Knowledge
• Mahavira and Bhaskara II
• Teachers and Learners
• The School of Madhava in Kerala
• Mathematics in China
• Mathematics in Japan
• The Philosophy Of Mathematics
• Mathematical Platonism
• Mathematical Anti-Platonism
• Logicism, Intuitionism, and Formalism
• Mathematical Platonism: For and Against
• Glossary
• Bibliography
• Numbers and Measurements
• Introduction
• Numbers
• Numeral Systems
• Development of Modern Numerals And Numeral Systems
• Arithmetic
• Number Theory
• Set Theory
• Measurement Systems
• Great Arithmeticians And Number Theorists
• The Ancient and Medieval World
• Renaissance to the 19th Century
• The Modern Era
• Measurement Pioneers
• Norman Robert Campbell
• Anders Celsius
• Giovanni Giorgi
• Edmund Gunter
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Pierre Mechain
• Jesse Ramsden
• Numerical Terms And Concepts
• Algorithm
• Arithmetic Function
• Associative Law
• Axiom of Choice
• Binary Code
• Binary Number System
• Cantor's Theorem
• Chinese Remainder Theorem
• Church's Thesis
• Commutative Laws
• Complex Number
• Continuum Hypothesis
• Decimal Number System
• Decision Problem
• Dedekind Cut
• Diophantine Equation
• Dirichlet's Theorem
• Distributive Law
• Equivalence Relation
• Euclidean Algorithm
• Euclid's Twin Prime Conjecture
• Factor
• Factorial
• Fermat Prime
• Fermat's Last Theorem
• Fermat's Little Theorem

• Fibonacci Numbers
• Fraction
• Fundamental Theorem of Arithmetic
• Geometric Series
• Goldbach Conjecture
• Harmonic Sequence
• Imaginary Number
• Incompleteness Theorem
• Inequality
• Infinite Series
• Infinity
• Integer
• Lagrange's Four-Square Theorem
• Mersenne Prime
• Number
• Perfect Number
• Prime
• Prime Number Theorem
• Pseudoprime
• Rational Number
• Real Number
• Riemann Zeta Function
• Root
• Russell's Paradox
• Set
• Square Root
• Transfinite Number
• Transitive Law
• Turing Machine
• Vinogradov's Theorem
• Waring's Problem
• Wilson's Theorem
• Zorn's Lemma
• Measurement Terms And Concepts
• Acre
• Amphora
• Angstrom (Å)
• Apothecaries’ Weight
• Are
• Avoirdupois Weight
• Barrel
• Bat
• British Imperial System
• Bushel
• Centimetre (cm)
• Cord
• Cubit
• Cup
• Dram
• Fathom
• Finger
• Foot
• Furlong
• Gal
• Gill
• Grain
• Gram (gm or g)
• Gunter's Chain
• Hand
• Hectare
• Inch
• International System of Units
• International Unit (IU)
• Kilometre (km)
• Knot
• League
• Libra
• Litre (l)
• Log
• Metre
• Metrētēs
• Metric System
• Micrometre
• Mile
• Millimetre (mm)
• Mina
• Mou
• Ounce
• Peck
• Pint
• Pound
• Qa
• Quart
• Rod
• Scruple
• Shi
• Steradian
• Stere
• Stone
• Talent
• Ton
• Troy Weight
• Zhang
• Glossary
• Bibliography
• Statistics and Probability
• Introduction
• History of Statistics and Probability
• Early Probability
• The Rise of Statistics
• The Spread of Statistical Mathematics
• Statistical Theories in the Sciences
• Statistics
• Descriptive Statistics
• Probability
• Estimation
• Estimation Procedures for Two Populations
• Hypothesis Testing
• Bayesian Methods
• Experimental Design
• Time Series and Forecasting
• Nonparametric Methods
• Statistical Quality Control
• Acceptance Sampling
• Decision Analysis
• Probability
• Experiments, Sample Space, Events, and Equally Likely Probabilities
• Conditional Probability
• Random Variables, Distributions, Expectation, and Variance
• An Alternative Interpretation of Probability
• The Law of Large Numbers, The Central Limit Theorem, and the Poisson Approximation
• Infinite Sample Spaces and Axiomatic Probability
• Conditional Expectation and Least Squares Prediction
• The Poisson Process and the Brownian Motion Process
• Stochastic Processes
• Combinatorics
• History
• Problems of Enumeration
• Problems of Choice
• Design Theory
• Latin Squares and the Packing Problem
• Graph Theory
• Applications of Graph Theory
• Combinatorial Geometry
• Game Theory
• Classification of Games
• One-Person Games
• Two-Person Constant-Sum Games
• Games of Imperfect Information
• Mixed Strategies and the Minimax Theorem
• Two-Person Variable-Sum Games
• Cooperative Versus Noncooperative Games
• N-Person Games
• Biographies
• Jean Le Rond d'Alembert
• Thomas Bayes
• Daniel Bernoulli
• Jakob Bernoulli
• Bhāskara II
• Ludwig Eduard Boltzmann
• George Boole
• Girolamo Cardano
• Arthur Cayley
• Francis Ysidro Edgeworth
• Pierre de Fermat
• Sir Ronald Aylmer Fisher
• John Graunt
• Pierre-Simon, marquis de Laplace
• Adrien-Marie Legendre
• Abraham de Moivre
• John F. Nash, Jr.
• Jerzy Neyman
• Karl Pearson
• Sir William Petty
• Siméon-Denis Poisson
• Adolphe Quetelet
• Jakob Steiner
• James Joseph Sylvester
• John von Neumann
• Special Topics
• Bayes's Theorem
• Central Limit Theorem
• Chebyshev's Inequality
• Decision Theory
• Distribution Function
• Error
• Estimation
• Indifference
• Inference
• Interval Estimation
• Law of Large Numbers
• Least Squares Approximation
• Markov Process
• Mean
• Normal Distribution
• Permutations and Combinations
• Point Estimation
• Poisson Distribution
• Queuing Theory
• Random Walk
• Sampling
• Standard Deviation
• Stochastic Process
• Student's T-Test
• Glossary
• Bibliography