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 marvellous 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 colours but no more • The four-colour theorem
Securing data with a one-way calculation • Cryptography
Jewels strung on an as-yet invisible thread • Finite simple groups
A truly marvellous proof • Proving Fermat’s last theorem
No other recognition is needed • Proving the Poincaré conjecture
DIRECTORY
GLOSSARY
CONTRIBUTORS
QUOTATIONS
ACKNOWLEDGEMENTS
COPYRIGHT