Ian Stewart - Seventeen Equations That Changed the World

SEVENTEEN EQUATIONS

THAT CHANGED THE WORLD

By the same author

Concepts of Modern Mathematics

Game, Set, and Math

The Problems of Mathematics

Does God Play Dice?

Another Fine Math Youve Got Me into

Fearful Symmetry (with Martin Golubitsky)

Natures Numbers

From Here to Infinity

The Magical Maze

Lifes Other Secret

Flatterland

What Shape Is a Snowflake?

The Annotated Flatland

Math Hysteria

The Mayor of Uglyvilles Dilemma

Letters to a Young Mathematician

Why Beauty Is Truth

How to Cut a Cake

Taming the Infinite/The Story of Mathematics

Professor Stewarts Cabinet of Mathematical Curiosities

Professor Stewarts Hoard of Mathematical Treasures

Cows in the Maze

Mathematics of Life

with Terry Pratchett and Jack Cohen

The Science of Discworld

The Science of Discworld II: the Globe

The Science of Discworld III: Darwins Watch

with Jack Cohen

The Collapse of Chaos

Figments of Reality

Evolving the Alien/What Does a Martian Look Like?

Wheelers (science fiction)

Heaven (science fiction)

THAT CHANGED THE WORLD

IAN STEWART

First published in Great Britain in 2012 by
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To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: , bicause noe .2. thynges, can be moare equalle.

Robert Recorde, The Whetstone of Witte, 1557

Equations are the lifeblood of mathematics, science, and technology. Without them, our world would not exist in its present form. However, equations have a reputation for being scary: Stephen Hawkings publishers told him that every equation would halve the sales of A Brief History of Time, but then they ignored their own advice and allowed him to include E = mc2 when cutting it out would allegedly have sold another 10 million copies. Im on Hawkings side. Equations are too important to be hidden away. But his publishers had a point too: equations are formal and austere, they look complicated, and even those of us who love equations can be put off if we are bombarded with them.

In this book, I have an excuse. Since its about equations, I can no more avoid including them than I could write a book about mountaineering without using the word mountain. I want to convince you that equations have played a vital part in creating todays world, from mapmaking to satnav, from music to television, from discovering America to exploring the moons of Jupiter. Fortunately, you dont need to be a rocket scientist to appreciate the poetry and beauty of a good, significant equation.

There are two kinds of equations in mathematics, which on the surface look very similar. One kind presents relations between various mathematical quantities: the task is to prove the equation is true. The other kind provides information about an unknown quantity, and the mathematicians task is to solve it to make the unknown known. The distinction is not clear-cut, because sometimes the same equation can be used in both ways, but its a useful guideline. You will find both kinds here.

Equations in pure mathematics are generally of the first kind: they reveal deep and beautiful patterns and regularities. They are valid because, given our basic assumptions about the logical structure of mathematics, there is no alternative. Pythagorass theorem, which is an equation expressed in the language of geometry, is an example. If you accept Euclids basic assumptions about geometry, then Pythagorass theorem is true.

Equations in applied mathematics and mathematical physics are usually of the second kind. They encode information about the real world; they express properties of the universe that could in principle have been very different. Newtons law of gravity is a good example. It tells us how the attractive force between two bodies depends on their masses, and how far apart they are. Solving the resulting equations tells us how the planets orbit the Sun, or how to design a trajectory for a space probe. But Newtons law isnt a mathematical theorem; its true for physical reasons, it fits observations. The law of gravity might have been different. Indeed, it is different: Einsteins general theory of relativity improves on Newton by fitting some observations better, while not messing up those where we already know Newtons law does a good job.

The course of human history has been redirected, time and time again, by an equation. Equations have hidden powers. They reveal the innermost secrets of nature. This is not the traditional way for historians to organise the rise and fall of civilisations. Kings and queens and wars and natural disasters abound in the history books, but equations are thin on the ground. This is unfair. In Victorian times, Michael Faraday was demonstrating connections between magnetism and electricity to audiences at the Royal Institution in London. Allegedly, Prime Minister William Gladstone asked whether anything of practical consequence would come from it. It is said (on the basis of very little actual evidence, but why ruin a nice story?) that Faraday replied: Yes, sir. One day you will tax it. If he did say that, he was right. James Clerk Maxwell transformed early experimental observations and empirical laws about magnetism and electricity into a system of equations for electromagnetism. Among the many consequences were radio, radar, and television.

An equation derives its power from a simple source. It tells us that two calculations, which appear different, have the same answer. The key symbol is the equals sign, =. The origins of most mathematical symbols are either lost in the mists of antiquity, or are so recent that there is no doubt where they came from. The equals sign is unusual because it dates back more than 450 years, yet we not only know who invented it, we even know why. The inventor was Robert Recorde, in 1557, in The Whetstone of Witte