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 Kưnigsberg • 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 Mưbius 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 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 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 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 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 Theorem A significant proven result on a mathematical topic, especially one that is not self-evident An unproved statement is called a conjecture Topology The branch of mathematics that studies surfaces and objects by examining how their parts are connected rather than according to their exact geometrical shapes For example, a doughnut and a teacup are topologically similar because they are both shapes that have one hole going through them (going through the handle, in the case of the teacup) Transcendental number Any irrational number that is not an algebraic number The number pi (π) and Euler’s number e are both transcendental numbers Transfinite number Another term for an infinite number It is used particularly when infinities of different sizes or infinite collections of objects are compared Transformation The conversion of a given shape or mathematical expression into another related one, using a particular rule Translation A function that moves an object a certain distance in a direction without affecting its shape, size, or orientation Trigonometry Originally, the study of the way the ratios between different sides of a right-angled triangle change when other angles in the triangle change, and later extended to all triangles The way the ratios change is described by trigonometric functions, which are now fundamental to many branches of mathematics Variable A mathematical quantity that can take on different values, often symbolized by a letter such as x or y Vector A mathematical or physical quantity that has both magnitude and direction In diagrams, vectors are often represented by bold arrows Vector space A complex abstract mathematical structure that involves the multiplication of vectors by each other and by scalars Venn diagram A diagram that shows sets of data as overlapping circles The overlaps show what the sets have in common Vertex (plural vertices) A corner or angle, where two or more lines, curves, or edges meet Volume The amount of space inside a 3-D object Whole number Any of the negative and positive counting numbers For example, –1, 0, 19, 55, and so on It is another term for integer CONTRIBUTORS KARL WARSI, CONSULTANT EDITOR Karl Warsi taught mathematics in UK schools and colleges for many years In 2000, he began publishing books on mathematics, creating bestselling textbook series for secondary-level students, both in the UK and worldwide He is committed to inclusion in education, and the idea that people of all ages learn in different ways JAN DANGERFIELD A lecturer and senior examiner in Further Mathematics, Jan Dangerfield is also a fellow of the UK’s Chartered Institute of Educational Assessors and a Fellow of the Royal Statistical Society She has been a member of the British Society for the History of Mathematics for more than 30 years HEATHER DAVIS British author and educator Heather Davis has taught mathematics for 30 years She has published textbooks for Hodder Education and managed publications for the UK’s Association of Teachers of Mathematics She presents courses for examination boards both in the UK and internationally and writes and presents enrichment activities for students JOHN FARNDON A widely published author of popular books on science and nature, John Farndon has been shortlisted five times for the Royal Society’s Young People’s Science Book Prize, among other awards He has written around 1,000 books on a range of subjects, including internationally acclaimed titles such as The Oceans Atlas, Do You Think You’re Clever? and Do Not Open, and contributed to major books such as Science and Science Year by Year JONNY GRIFFITHS After studying mathematics and education at Cambridge University, the Open University, and the University of East Anglia, Jonny Griffiths taught math at Paston Sixth Form College in Norfolk, UK, for over 20 years In 2005–06, he was made a Gatsby Teacher Fellow for creating the popular mathematics website Risps In 2016, he founded the competition Ritangle for students of mathematics TOM JACKSON A writer for 25 years, Tom Jackson has written about 200 non-fiction books for adults and children and contributed to many more on a wide range of science and technology topics They include Numbers: How Counting Changed the World; Everything is Mathematical, a book series with Marcus du Sautoy; and Help Your Kids with Science with Carol Vorderman MUKUL PATEL Mukul Patel, who studied mathematics at Imperial College, London, writes and collaborates across many disciplines He is the author of We’ve Got Your Number, a book on mathematics for children, and film scripts voiced by Tilda Swinton He has also composed extensively for contemporary choreographers and designed sound installations for architects He is currently investigating ethical issues in AI SUE POPE A mathematics educator, Sue Pope is a long-standing member of the Association of Teachers of Mathematics and co-runs workshops on the history of mathematics in teaching at their conferences Published widely, she recently co-edited Enriching Mathematics in the Primary Curriculum MATT PARKER, FOREWORD Originally a math teacher from Australia, Matt Parker is a now a stand-up comedian, mathematics communicator, and a prominent math YouTuber on the Numberphile and Stand-up Maths channels, where his videos have had more than 100 million views Matt performs live comedy with Festival of the Spoken Nerd and once calculated pi live in front of a sold-out Royal Albert Hall He also presents television and radio programs for Discovery Channel and the BBC, and his 2019 book Humble Pi: A Comedy of Maths Errors topped the Sunday Times best-seller chart QUOTATIONS The following primary quotations are attributed to people who are not the key figure for the relevant topic ANCIENT AND CLASSICAL PERIODS Exploring pi is like exploring the Universe David Chudnovsky, Ukrainian–American mathematician The art of measuring triangles Samuel Johnson, English writer The very flower of arithmetic Regiomontanus, German mathematician and astronomer An incomparable star in the firmament of wisdom Martin Cohen, British philosopher THE MIDDLE AGES Algebra is a scientific art Omar Khayyam, Persian mathematician and poet The ubiquitous music of the spheres Guy Murchie, American writer The power of doubling Ibn Khallikan, Islamic scholar and biographer THE RENAISSANCE The geometry of art and life Matila Ghyka, Romanian novelist and mathematician Like a large diamond Chris Caldwell, American mathematician A device of marvelous invention Evangelista Torricelli, Italian physicist and mathematician Chance is bridled and governed by law Boëthius, Roman senator With calculus I can predict the future Steven Strogatz, American mathematician THE ENLIGHTENMENT One of those strange numbers that are creatures of their own Ian Stewart, British mathematician The most beautiful equation Keith Devlin, British mathematician No theory is perfect Nate Silver, American statistician Simply a question of algebra Robert Simpson Woodward, American engineer, physicist, and mathematician Algebra often gives more than is asked of her Jean d’Alembert, French mathematician and philosopher THE 19TH CENTURY The imp that knows the positions of every particle in the Universe Steven Pinker, Canadian psychologist An indispensable tool in applied mathematics Walter Fricke, German astronomer and mathematician A new kind of function W W Rouse Ball, British mathematician and lawyer Just like a pocket map attributed to Peter Tait, British physicist and mathematician, by Silvanus Phillips Thompson, British physicist and engineer The matrix is everywhere from the film The Matrix The music of the primes Marcus du Sautoy, British mathematician and author Some infinities are bigger than others John Green, American author Lost in that silent, measured space Paolo Giordano, Italian author MODERN MATHEMATICS Statistics is the grammar of science Karl Pearson, British mathematician and statistician Rather a dull number G H Hardy, English mathematician A million monkeys banging on a million typewriters Robert Wilensky, American computer scientist She changed the face of algebra Hermann Weyl, German mathematician A blueprint for the digital age Robert Gallagher, American engineer A small positive vibration can change the entire cosmos Amit Ray, Indian author A grand unifying theory of mathematics Edward Frenkel, Russian–American mathematician Endless variety and unlimited complication Roger Penrose, British mathematician Jewels strung on an as-yet invisible thread Ronald Solomon, American mathematician A truly marvelous proof Pierre de Fermat, French lawyer and mathematician ACKNOWLEDGMENTS Dorling Kindersley would like to thank Gadi Farfour, Meenal Goel, Debjyoti Mukherjee, Sonali Rawat, and Garima Agarwal for design assistance; Rose Blackett-Ord, Daniel Byrne, Kathryn Hennessy, Mark Silas, and Shreya Iyengar for editorial assistance; and Gillian Reid, Amy Knight, Jacqueline StreetElkayam, and Anita Yadav for production assistance PICTURE CREDITS Disclaimer: Page numbers and positions correspond to the print book The publisher would like to thank the following for their kind permission to reproduce their photographs: (Key: a-above; b-below/bottom; c-center; f-far; l-left; r-right; t-top) 25 Getty Images: Universal History Archive / Universal Images Group (crb) Science Photo Library: New York Public Library (bl) Alamy Stock Photo: Artokoloro Quint Lox Limited (tr); NMUIM 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