Ian Stewart is a mathematician at the University of Warwick in the United Kingdom. His new book "In Pursuit of the Unknown" is published by Basic Books in the United States in March. In the United Kingdom it is available from Profile Books with the title "Seventeen Equations That Changed the World."
I was one of those annoying kids who actually liked equations. I collected them in a notebook. I loved the way you could plug a few numbers into an equation and find out how bright the Sun would be if you were standing on Pluto. Or work out how big a rainbow looks from the refractive index of water and the time of day.
I realize I am a rarity in that respect. Stephen Hawking’s publishers allegedly told him that every equation he put into his runaway bestseller "A Brief History of Time" would halve its sales. So, if he’d left out Einstein’s E=mc2, he would have sold another 10 million copies. But his publishers had a point. Although the great equations have had more impact on humanity than all the kings and queens in the history books put together, they can look very off-putting.
That’s why I wrote "In Pursuit of the Unknown: Seventeen Equations That Changed the World." We need to stop being put off, and learn to value our equations. It’s hard to write a book about equations without including any, so I decided to follow the age-old theatrical advice: ‘if you’ve got a wooden leg, wave it.’ Make equations the main characters in a story of the rise — and occasional fall — of humanity.
Let’s not overdo it, though. I’m a mathematician; it’s my job to understand the nuts and bolts of equations. You don’t have to. But I reckon it may well be worth your while to appreciate where equations came from, what they say, and what they’ve done for us. Without actually doing any of the sums. It’s like enjoying music without knowing how to compose it or how to read a score. All gain and no pain.
The book came about by accident and serendipity. A Dutch publisher asked my English publisher if they had a book in the pipeline about equations for non-specialists. They didn’t, but that can always be fixed. The more we thought about the idea, the better we liked it. We moved a couple of my other projected books up a year to make room to write it.
The first task was to choose the equations. I decided that unless they really had changed the world, in a big way, then they didn’t go in. However pretty they were, however fascinating they were to mathematicians... no major impact on humanity meant they got the chop. I managed to cut my original list down to about 20. To avoid unnecessary overlaps, I started combining closely related equations into a single chapter. So Einstein’s equation could be a placeholder for the whole of special and general relativity, and a calculus equation would take care of Newton’s laws of motion as well. That whittled the number down to 17—a nice, mysterious, seductively precise yet totally arbitrary figure.
As the book evolved—no book of mine ever follows the original plan exactly, that’s more of a negotiating platform—I found myself spending more and more time trying to picture what the world had been like before the equation was invented or discovered. Only then could I explain what it had changed. Before James Clerk Maxwell wrote down his equations for electromagnetism, streets were lit by burning gas and vital messages went on horseback. Afterwards we had radio, and television soon followed. In between came electric lighting and the telegraph. I learned a lot of history and managed to avoid a few myths.
Most of the great equations emerged from the needs of science. Relativity and quantum mechanics encompass almost the whole of modern physics. Newton’s law of gravity explains most of what we know about the movements of the stars and planets. General relativity plugs a few gaps where Newton’s law isn’t accurate enough. But some came from elsewhere—the Black-Scholes equation for pricing options originated in the needs of global finance. The equation for information came from code-breaking and communications. The equations of statistics began as aids for gamblers.
A key theme in the book is: What do equations do for us—today, in our homes, in daily life? Maxwell and the wave equation gave us radio, TV, cellphones and wireless computer peripherals. The Fourier transform, introduced to understand the flow of heat, makes digital cameras feasible by compressing image data. Engineers use Newton’s law of gravity to design orbits for spacecraft and to launch satellites, giving us thousands of TV channels and global communications. Without both special and general relativity, satnav would never work. Without quantum mechanics, leading to memory chips and most modern electronics, there would be no laptops, iPads, or Amazon Kindles.
I now find that as I go about my everyday activities, it’s as if little labels keep popping up in my head: Equations Inside. I sense them in the supermarket—a pack of carrots pops up a label with equations from statistics, used to develop new breeds of vegetables suitable for storage and transportation. A passenger jet passes overhead, and up pops the Navier-Stokes equation for fluid flow, implemented through computational fluid dynamics, whose applications also include the design of stents to keep arteries open and prevent heart attacks. And I realize that the title of my book is no exaggeration. Equations really have changed the world, repeatedly altering the course of history. And they’ll do it again.