The Math Book
Page 1
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
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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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 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
INTRODUCTION
The history of mathematics reaches back to prehistory, when early humans found ways to count and quantify things. In doing so, they began to identify certain patterns and rules in the concepts of numbers, sizes, and shapes. They discovered the basic principles of addition and subtraction—for example, that two things (whether pebbles, berries, or mammoths) when added to another two invariably resulted in four things. While such ideas may seem obvious to us today, they were profound insights for their time. They also demonstrate that the history of mathematics is above all a story of discovery rather than invention. Although it was human curiosity and intuition that recognized the underlying principles of mathematics, and human ingenuity that later provided various means of recording and notating them, those principles themselves are not a human invention. The fact that 2 + 2 = 4 is true, independent of human existence; the rules of mathematics, like the laws of physics, are universal, eternal, and unchanging. When mathematicians first showed that the angles of any triangle in a flat plane when added together come to 180°, a straight line, this was not their invention: they had simply discovered a fact that had always been (and will always be) true.
Early applications
The process of mathematical discovery began in prehistoric times, with the development of ways of counting things people needed to quantify. At its simplest, this was done by cutting tally marks in a bone or stick, a rudimentary but reliable means of recording numbers of things. In time, words and symbols were assigned to the numbers and the first systems of numerals began to evolve, a means of expressing operations such as acquisition of additional items, or depletion of a stock, the basic operations of arithmetic.
As hunter-gatherers turned to trade and farming, and societies became more sophisticated, arithmetical operations and a numeral system became essential tools in all kinds of transactions. To enable trade, stocktaking, and taxes in uncountable goods such as oil, flour, or plots of land, systems of measurement were developed, putting a numerical value on dimensions such as weight and length. Calculations also became more complex, developing the concepts of multiplication and division from addition and subtraction—allowing the area of land to be calculated, for example.
In the early civilizations, these new discoveries in mathematics, and specifically the measurement of objects in space, became the foundation of the field of geometry, knowledge that could be used in building and toolmaking. In using these measurements for practical purposes, people found that certain patterns were emerging, which could in turn prove useful. A simple but accurate carpenter’s square can be made from a triangle with sides of three, four, and five units. Without that accurate tool and knowledge, the roads, canals, ziggurats, and pyramids of ancient Mesopotamia and Egypt could not have been built. As new applications for these mathematical discoveries were found—in astronomy, navigation, engineering, bookkeeping, taxation, and so on—further patterns and ideas emerged. The ancient civilizations each established the foundations of mathematics through this interdependent process of application and discovery, but also developed a fascination with mathematics for its own sake, so-called pure mathematics. From the middle of the first millennium BCE, the first pure mathematicians began to appear in Greece, and slightly later in India and China, building on the legacy of the practical pioneers of the subject—the engineers, astronomers, and explorers of earlier civilizations.
Although these early mathematicians were not so concerned with the practical applications of their discoveries, they did not restrict their studies to mathematics alone. In their exploration of the properties of numbers, shapes, and processes, they discovered universal rules and patterns that raised metaphysical questions about the nature of the cosmos, and even suggested that these patterns had mystical properties. Often mathematics was therefore seen as a complementary discipline to philosophy—many of the greatest mathematicians th
rough the ages have also been philosophers, and vice versa—and the links between the two subjects have persisted to the present day.
It is impossible to be a mathematician without being a poet of the soul.
Sofya Kovalevskaya
Russian mathematician
Arithmetic and algebra
So began the history of mathematics as we understand it today—the discoveries, conjectures, and insights of mathematicians that form the bulk of this book. As well as the individual thinkers and their ideas, it is a story of societies and cultures, a continuously developing thread of thought from the ancient civilizations of Mesopotamia and Egypt, through Greece, China, India, and the Islamic empire to Renaissance Europe and into the modern world. As it evolved, mathematics was also seen to comprise several distinct but interconnected fields of study.
The first field to emerge, and in many ways the most fundamental, is the study of numbers and quantities, which we now call arithmetic, from the Greek word arithmos (“number”). At its most basic, it is concerned with counting and assigning numerical values to things, but also the operations, such as addition, subtraction, multiplication, and division, that can be applied to numbers. From the simple concept of a system of numbers comes the study of the properties of numbers, and even the study of the very concept itself. Certain numbers—such as the constants π, e, or the prime and irrational numbers—hold a special fascination and have become the subject of considerable study.
Another major field in mathematics is algebra, which is the study of structure, the way that mathematics is organized, and therefore has some relevance in every other field. What marks algebra from arithmetic is the use of symbols, such as letters, to represent variables (unknown numbers). In its basic form, algebra is the study of the underlying rules of how those symbols are used in mathematics—in equations, for example. Methods of solving equations, even quite complex quadratic equations, had been discovered as early as the ancient Babylonians, but it was medieval mathematicians of the Islamic Golden Age who pioneered the use of symbols to simplify the process, giving us the word “algebra,” which is derived from the Arabic al-jabr. More recent developments in algebra have extended the idea of abstraction into the study of algebraic structure, known as abstract algebra.