An equation is a bridge. No matter how great the chasm between two ideas on the opposite sides of an "=" symbol might be, an equals sign yokes them together, turning them into something greater. At its heart, an equation is the distillate of humanity's desire to forge links between realms, links that make up the framework that we use to try to make sense of the Universe.

So when Ian Stewart, professor of mathematics at the University of Warwick and veteran maths populariser, claims that the 17 equations in his volume have changed the world, it's no hyperbole (even when he discusses the hyperbolic plane). With his handful of equations, Stewart's list is nothing less than a quick tour of the history of scientific thought, from ancient Greece to quantum mechanics and beyond.

Take, for example, that most famous of equations - and a shoo-in for anybody's list: *E = mc*^{2}. It doesn't take long to grasp what the elements in that equation represent: *E* stands for the energy of an object, *m* for its mass, and *c* represents the speed of light, roughly 300,000 kilometres per second. The form of the equation is childishly simple. However, that simplicity is deceptive. The expression encodes Einstein's realisation that mass and energy are not independent, but are in fact intimately related to each other - a realisation that has tremendous consequences. The moment when we first figured out a controlled manner by which we could take matter on one side of the equation and convert it to energy on the other side of the equation was the moment when humanity entered the nuclear age. And the equation sits at the heart of the special theory of relativity, which shows that space and time - again, seemingly unrelated ideas - are tied together in completely unexpected ways.

Stewart takes pains to keep his prose understandable for non-mathematicians, in part, by stressing everyday applications of the great equations. For example, when explaining the Fourier transform - a family of equations that allows mathematicians to break a complicated formula down into a set of simple components - he demonstrates how those equations are responsible for the primary scheme that our computers use to render images. Yet it's when he strays away from the concrete that Stewart really gives a glimpse of how deep the rabbit hole goes. In the case of Fourier transforms, the equations provide not just a single way of breaking down mathematical formulae into components but an infinite variety of ways. The mechanics of the Fourier transform allowed later mathematicians to view those formulae as residents of an infinite-dimensional space populated by an uncountable bestiary of exotic functions.

A measure of Stewart's success is that he's able to give the reader a clear path through some very hairy mathematics, passing easily from ordinary equations of the *E = mc*^{2} variety to differential equations (which, roughly speaking, express relationships among ordinary equations, just as ordinary equations express relationships among numbers). As a result, the reader who isn't scared off by exotic-looking mathematical operators will come out the other end with at least a hint of the beautiful symmetries of Maxwell's equations or of the importance of the Navier-Stokes equation.

*17 Equations that Changed the World* isn't the only book of its kind, as anyone who attempts to generate a list of great equations will, of necessity, walk on some well-trodden turf. Nevertheless, Stewart's new volume is worth a read, for when you cross a great bridge, you're all but guaranteed a memorable view.

## 17 Equations that Changed the World

By Ian Stewart

Profile Books, 352pp, £15.99

ISBN 9781846685316 Published 9 February 2012

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