Table Of Content3
CONTENTS
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
4
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
5
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
6
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
7
How to use this eBook
Preferred application settings
For the best reading experience, the following application settings are
recommended:
Color theme: White background
Font size: At the smallest point size
Orientation: Landscape (for screen sizes over 9”/23cm), Portrait (for screen
sizes below 9”/23cm)
Scrolling view: [OFF]
Text alignment: Auto-justification [OFF] (if the eBook reader has this
feature)
Auto-hyphenation: [OFF] (if the eBook reader has this feature)
Font style: Publisher default setting [ON] (if the eBook reader has this
feature)
Images: Double tap on the images to see them in full screen and be able to
zoom in on them
8
FOREWORD
Summarizing all of mathematics in one book is a daunting and indeed impossible
task. Humankind has been exploring and discovering mathematics for millennia.
Practically, we have relied on math to advance our species, with early arithmetic
and geometry providing the foundations for the first cities and civilizations. And
philosophically, we have used mathematics as an exercise in pure thought to
explore patterns and logic.
As a subject, mathematics is surprisingly hard to pin down with one catch-all
definition. “Mathematics” is not simply, as many people think, “stuff to do with
numbers.” That would exclude a huge range of mathematical topics, including
much of the geometry and topology covered in this book. Of course, numbers are
still very useful tools to understand even the most esoteric areas of mathematics,
but the point is that they are not the most interesting aspect of it. Focusing just on
numbers misses the forest for the threes.
For the record, my own definition of math as “the sort of things that
mathematicians enjoy doing,” while delightfully circular, is largely unhelpful. Big
Ideas Simply Explained is actually not a bad definition. Mathematics could be
seen as the attempt to find the simplest explanations for the biggest ideas. It is the
endeavor of finding and summarizing patterns. Some of those patterns involve the
practical triangles required to build pyramids and divide land; other patterns
attempt to classify all of the 26 sporadic groups of abstract algebra. These are
very different problems in terms of both usefulness and complexity, but both
types of pattern have become the obsession of mathematicians throughout the
ages.
There is no definitive way to organize all of mathematics, but looking at it
chronologically is not a bad way to go. This book uses the historical journey of
humans discovering math as a way to classify it and wrangle it into a linear
progression, which is a valiant but difficult effort. Our current mathematical body
of knowledge has been built up by a haphazard and diverse group of people
across time and cultures.
So something like the short section on magic squares covers thousands of years
and the span of the globe. Magic squares—arrangements of numbers where the
sum in each row, column, and diagonal is always the same—are one of the oldest
9
areas of recreational mathematics. Starting in the 9th century BCE in China, the
story then bounces around via Indian texts from 100 CE, Arab scholars in the
Middle Ages, Europe during the Renaissance, and finally modern Sudoku-style
puzzles. Across a mere two pages this book has to cover 3,000 years of history
ending with geomagic squares in 2001. And even in this small niche of
mathematics, there are many magic square developments that there was simply
not enough room to include. The whole book should be viewed as a curated tour
of mathematical highlights.
Studying even just a sample of mathematics is a great reminder of how much
humans have achieved. But it also highlights where mathematics could do better;
things like the glaring omission of women from the history of mathematics cannot
be ignored. A lot of talent has been squandered over the centuries, and a lot of
credit has not been appropriately given. But I hope that we are now improving the
diversity of mathematicians and encouraging all humans to discover and learn
about mathematics.
Because going forward, the body of mathematics will continue to grow. Had this
book been written a century earlier it would have been much the same up until
about page 280. And then it would have ended. No ring theory from Emmy
Noether, no computing from Alan Turing, and no six degrees of separation from
Kevin Bacon. And no doubt that will be true again 100 years from now. The
edition printed a century from now will carry on past page 325, covering patterns
totally alien to us. And because anyone can do math, there is no telling who will
discover this new math, and where or when. To make the biggest advancement in
mathematics during the 21st century, we need to include all people. I hope this
book helps inspire everyone to get involved.
Matt Parker
10
