The Math Book: Big Ideas Simply Explained

HOW TO USE THIS EBOOK
INTRODUCTION
ANCIENT AND CLASSICAL PERIODS 6000 BCE–500 CE
	Numerals take their places • Positional numbers
	The square as the highest power • Quadratic equations
	The accurate reckoning for inquiring into all things • The Rhind papyrus
	The sum is the same in every direction • Magic squares
	Number is the cause of gods and daemons • Pythagoras
	A real number that is not rational • Irrational numbers
	The quickest runner can never overtake the slowest • Zeno’s paradoxes of motion
	Their combinations give rise to endless complexities • The Platonic solids
	Demonstrative knowledge must rest on necessary basic truths • Syllogistic logic
	The whole is greater than the part • Euclid’s Elements
	Counting without numbers • The abacus
	Exploring pi is like exploring the Universe • Calculating pi
	We separate the numbers as if by some sieve • Eratosthenes’ sieve
	A geometrical tour de force • Conic sections
	The art of measuring triangles • Trigonometry
	Numbers can be less than nothing • Negative numbers
	The very flower of arithmetic • Diophantine equations
	An incomparable star in the firmament of wisdom • Hypatia
	The closest approximation of pi for a millennium • Zu Chongzhi
THE MIDDLE AGES 500–1500
	A fortune subtracted from zero is a debt • Zero
	Algebra is a scientific art • Algebra
	Freeing algebra from the constraints of geometry • The binomial theorem
	Fourteen forms with all their branches and cases • Cubic equations
	The ubiquitous music of the spheres • The Fibonacci sequence
	The power of doubling • Wheat on a chessboard
THE RENAISSANCE 1500–1680
	The geometry of art and life • The golden ratio
	Like a large diamond • Mersenne primes
	Sailing on a rhumb • Rhumb lines
	A pair of equal-length lines • The equals sign and other symbology
	Plus of minus times plus of minus makes minus • Imaginary and complex numbers
	The art of tenths • Decimals
	Transforming multiplication into addition • Logarithms
	Nature uses as little as possible of anything • The problem of maxima
	The fly on the ceiling • Coordinates
	A device of marvelous invention • The area under a cycloid
	Three dimensions made by two • Projective geometry
	Symmetry is what we see at a glance • Pascal’s triangle
	Chance is bridled and governed by law • Probability
	The sum of the distance equals the altitude • Viviani’s triangle theorem
	The swing of a pendulum • Huygens’s tautochrone curve
	With calculus I can predict the future • Calculus
	The perfection of the science of numbers • Binary numbers
THE ENLIGHTENMENT 1680–1800
	To every action there is an equal and opposite reaction • Newton’s laws of motion
	Empirical and expected results are the same • The law of large numbers
	One of those strange numbers that are creatures of their own • Euler’s number
	Random variation makes a pattern • Normal distribution
	The seven bridges of Königsberg • Graph theory
	Every even integer is the sum of two primes • The Goldbach conjecture
	The most beautiful equation • Euler’s identity
	No theory is perfect • Bayes’ theorem
	Simply a question of algebra • The algebraic resolution of equations
	Let us gather facts • Buffon’s needle experiment
	Algebra often gives more than is asked of her • The fundamental theorem of algebra
THE 19TH CENTURY 1800–1900
	Complex numbers are coordinates on a plane • The complex plane
	Nature is the most fertile source of mathematical discoveries • Fourier analysis
	The imp that knows the positions of every particle in the Universe • Laplace’s demon
	What are the chances? • The Poisson distribution
	An indispensable tool in applied mathematics • Bessel functions
	It will guide the future course of science • The mechanical computer
	A new kind of function • Elliptic functions
	I have created another world out of nothing • Non-Euclidean geometries
	Algebraic structures have symmetries • Group theory
	Just like a pocket map • Quaternions
	Powers of natural numbers are almost never consecutive • Catalan’s conjecture
	The matrix is everywhere • Matrices
	An investigation into the laws of thought • Boolean algebra
	A shape with just one side • The Möbius strip
	The music of the primes • The Riemann hypothesis
	Some infinities are bigger than others • Transfinite numbers
	A diagrammatic representation of reasonings • Venn diagrams
	The tower will fall and the world will end • The Tower of Hanoi
	Size and shape do not matter, only connections • Topology
	Lost in that silent, measured space • The prime number theorem
MODERN MATHEMATICS 1900–PRESENT
	The veil behind which the future lies hidden • 23 problems for the 20th century
	Statistics is the grammar of science • The birth of modern statistics
	A freer logic emancipates us • The logic of mathematics
	The Universe is four-dimensional • Minkowski space
	Rather a dull number • Taxicab numbers
	A million monkeys banging on a million typewriters • The infinite monkey theorem
	She changed the face of algebra • Emmy Noether and abstract algebra
	Structures are the weapons of the mathematician • The Bourbaki group
	A single machine to compute any computable sequence • The Turing machine
	Small things are more numerous than large things • Benford’s law
	A blueprint for the digital age • Information theory
	We are all just six steps away from each other • Six degrees of separation
	A small positive vibration can change the entire cosmos • The butterfly effect
	Logically things can only partly be true • Fuzzy logic
	A grand unifying theory of mathematics • The Langlands Program
	Another roof, another proof • Social mathematics
	Pentagons are just nice to look at • The Penrose tile
	Endless variety and unlimited complication • Fractals
	Four colors but no more • The four-color theorem
	Securing data with a one-way calculation • Cryptography
	Jewels strung on an as-yet invisible thread • Finite simple groups
	A truly marvelous proof • Proving Fermat’s last theorem
	No other recognition is needed • Proving the Poincaré conjecture
DIRECTORY
GLOSSARY
CONTRIBUTORS
QUOTATIONS
ACKNOWLEDGMENTS
COPYRIGHT
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