186 // The Most Beautiful Formula

Multiplication was more complicated. The main point was that multiplying a number by i rotated it around the origin O through a right angle, anticlockwise. For instance, 3 multiplied by i is 3i, and that's what you get when you rotate the point labelled 3 through 908.

The new numbers extended the familiar real number line to a larger space, a number plane. Three mathematicians discovered this idea independently: the Norwegian Caspar Wessel, the Frenchman Jean-Robert Argand and the German Carl Friedrich Gauss.

Complex numbers don't turn up in everyday situations, such as checking the supermarket bill or measuring someone for a suit. Their applications are in things like electrical engineering and aircraft design, which lead to technology that we can use without having to know the underlying mathematics.

...........................................

The engineers and designers need to know it, though.

The Most Beautiful Formula Occasionally people hold polls for the most beautiful mathe- matical formula of all time ­ yes, they really do, I'm not making this up, honest ­ and nearly always the winner is a famous formula discovered by Euler, which uses complex numbers to link the two famous constants e and p. The formula is

eip ¼ À1

and it is extremely influential in a branch of maths known as

........................................... complex analysis.

Why is Euler's Beautiful Formula True? I often get asked whether there is a simple way to explain why Euler's formula eip ¼ À1 is true. It turns out that there is, but Why is Euler's Beautiful Formula True? // 187

some preparation is needed ­ about two years of undergraduate mathematics.

This is uncomfortably like the joke about the professor who says in a lecture that some fact is obvious, and when challenged goes away for half an hour and returns to say `yes, it is obvious,' and then continues lecturing without further explanation. It just takes two years instead of half an hour. So I'm going to give you the explanation. Skip this bit if it doesn't make sense ­ but it does illustrate how higher mathematics sometimes gains new insights by putting different ideas together in unexpected ways. The necessary ingredients are some geometry, some differential equations and a bit of complex analysis.

The main idea is to solve the differential equation

dz

¼ iz

dt where z is a complex function of time t, with the initial condition zð0Þ ¼ 1. It is standard in differential equations courses that the solution is

zðtÞ ¼ eit

Indeed, you can define the exponential function ew this way.

Geometry of the

differential equation.

Now let's interpret the equation geometrically. Multiplication by i is the same as rotation through a right angle, so iz is at right angles to z. Therefore the tangent vector iz(t) to the solution at any point z(t) is always at right angles to the

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