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Putting the ‘i’ in iPods by Michael Brooks

They exasperated their 16th-century discoverer, but imaginary numbers have given us everything from quantum mechanics to portable music

WHEN students encounter imaginary numbers, a common response is: what’s the point? Well, quite a lot as it happens, though it took centuries to discover how much.

An imaginary number is the square root of a negative number. Such numbers have become essential tools in microchip design and in digital compression algorithms: your MP3 player relies on imaginary stuff. Even more fundamental than that, imaginary numbers underpin quantum mechanics, the theory that gave rise to the electronics revolution. Little modern technology would exist without complex numbers - numbers which have both a real and an imaginary component.

In the 16th century, when the Italian mathematician Gerolomo Cardano came up with imaginary numbers, even negative numbers were treated with deep suspicion. Though they were difficult beasts, Cardano pressed ahead. At one point, Cardano even wrote that they were “useless”, but it is clear that he found them intriguing as well as frustrating. “Cardano wrote a formal expression for complex numbers, he could add and multiply them, but he could not give them any practical or geometrical sense,” says Artur Ekert of the University of Oxford.

Rafael Bombelli built on Cardano’s work in the 1560s, but imaginary numbers were not taken seriously until mathematicians found links between them and constants such as π and e. In the 18th century, Leonhard Euler showed that e raised to the power i _ π equals -1 (where i is the square root of -1). Now imaginary numbers are indispensable.

It seems fitting that their role in quantum theory is to explain the most bizarre aspect of the

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theory: that quantum objects such as atoms and electrons can exist in two or more places at once. Physicists and philosophers still argue over what this means, but it is clear that the mathematics only works when it includes a complex number known as”probability amplitude”.

Without imaginary numbers, you won’t get an answer that reflects the reality of the physical world. And you won’t get an iPod either.

Source: New Scientist

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