It may be square, but it's wondrous

The original seven wonders were built without it. But the technological world would not exist without the square root of -1, says Charles Arthur

What made the original "seven wonders of the world" wonderful? Their elegance, perhaps; their locations (in the centres of old civilisations, such as Rhodes and Olympia), maybe. But more than anything, it was their size. They inspired wonder because they were fantastic feats of engineering.

Well, we are toolmaking apes; no wonder we felt proud of our artefacts. You might argue that the Hanging Gardens of Babylon were amazing for their vegetation, but putting plants on a terrace isn't hard. The trick is keeping them alive. (That was done using Archimedes Screws - which look like stationary drills inside a closed cylinder - to lift the water from the Euphrates river, up 75 feet to the trees and shrubs.)

But the original wonders have not survived very well. Of the original seven, only the pyramids are still standing. (The Taj Mahal was drafted in as a substitute wonder in modern times.) So what should we find wonderful in the modern world?

On Monday the BBC starts a new TV series, Seven Wonders of the World, in which it asks seven scientists to nominate their seven "modern wonders". Luminaries such as Richard Dawkins and Steven Pinker offer ideas such as the digitisation of our voices by the phone network, the "ultimate mystery" of consciousness, ultrasound pictures of babies in the womb, genes that control limb development, the songs of whales.

All very worthy; but none of those, on its own, gives a clue to how we have advanced in the few thousand years since the first of the original seven wonders was built. Yet we know that we have; even on a visceral level, we know that we're more advanced than those ancient peoples. Not physically, maybe, but mentally. We know something they didn't. But what? What is the tool we've made that sets us apart?

There is one overwhelming candidate, which appears in the first programme of the series about Aubrey Manning, professor of natural history at Edinburgh University. He calls it the result of "the human intellect firing on all cylinders". All the original seven wonders of the world were built without it. But almost nothing in the modern, technological world could be.

It's a tool you can't touch. Yet without it many of those things described above - the digital phone network, ultrasound monitors, the microphones able to pick up whale songs, autofocus cameras, the machines that sequence genes so that scientists can identify them - would be a dream. How appropriate then that the tool is itself imaginary: i, the square root of -1.

Now, all numbers are in one sense imaginary; they're just ways of clumping together objects and concepts. But we can easily understand the idea of dropping one brick, or half a brick, on our feet. We can follow that the circumference of a circle is always twice its radius times a constant called pi. We can understand that you need square roots to work out the length of the longest side of a right-angled triangle. The ancients had all those mathematical tools.

However, according to the "rules" of everyday maths, there is no such thing as a number that, multiplied by itself, gives you a negative number. Yet you can easily create equations in which that would be required: for example, when x2+1 = 0. What values of x make this true? Clearly, when x equals the square root of -1.

You could grapple with this problem for a while before deciding it was insoluble. And for centuries that's what mathematicians did. But they also recognised that it would be really useful if -1 did have a square root.

The concept of i and of solving equations such as x2-2x+2 = 0 (which is true when x = 1+i or x = 1-i) was first formally introduced by the Italian mathematician Rafael Bombelli in the 16th century, though this tool didn't begin to be used properly for another 100 years, by Jean Robert Argand, a French mathematician. Such solutions to equations, in which i is mixed in with "real" numbers, are known as "complex numbers".

Of course you wonder what use such equations are in the modern world, beyond giving maths-phobic schoolchildren (and parents) headaches. Ironically, many schoolchildren harbour dreams that couldn't be fulfilled without complex numbers.

Take a simple electrical circuit with a resistor consisting of a wound wire. Put an alternating current in. To understand and predict how the output of that circuit varies as you change its characteristics (the frequency of the alternating current, the thickness of the resistor wire), you'll have to use complex numbers.

Sound abstruse? Actually, that wound-wire circuit is an exact description of an electric guitar pickup, or a microphone. (The movement of the metal guitar string or microphone diaphragm disturbs a field generated by a magnet inside the instrument: that induces a current in a wire wrapped around the magnet.) Everyone from Hank Marvin to Noel Gallagher to Damon Albarn has been dabbling in i without realising it.

Pretty soon you find that complex numbers are necessary to manipulate anything involving electromagnetism. Then an engineer will tell you that they're remarkably useful for something called "fast Fourier transforms" - a means of finding out what frequencies are contained in an electric or sound waveform. In turn that helps you digitise the waveform and then reconstitute it - from a compact disc, say.

A radio engineer will helpfully point out that complex numbers are essential for understanding the behaviour of transmitting and receiving aerials, and also help you design working fibre-optic cables. The tools that build computers and the computers themselves all rely on an understanding of the electronic circuits inside them: more complex numbers. The whole phone network - from the handset to the cable to the transmission - relies on the damn things. The machines that sequence genes do, too. Anywhere you look in a modern room, you'll see something that either uses them or was made using electricity whose generation required their manipulation.

Yet despite permeating our lives, complex numbers and i retain their ethereal property: you can't point to them, and you can't draw them on a piece of paper. Yet they are the most useful tool that the human race has developed in the past 2,000 years. If you want to imagine a world without imaginary numbers, just go back a millennium or so. That really is something to wonder about.