The Math Book: Big Ideas Simply Explained – Book Sample

## CONTENTS – The Math Book: Big Ideas Simply Explained

### 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 Math Book: Big Ideas Simply Explained

• 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
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### THE ENLIGHTENMENT 1680–1800 – The Math Book: Big Ideas Simply Explained

• 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 Math Book: Big Ideas Simply Explained

• 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