Five Numbers [Simon Singh's Numbers]

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Infinity20020315Given the old maxim about an infinite number of monkeys and typewriters, one can assume that said simian digits will type up the following line from Hamlet an infinite number of times:

'I could confine myself to a nutshell and declare myself king of infinity.'

This quote could almost be an epithet for the mathematician Georg Cantor, one of the fathers of modern mathematics. Born in 1845, Cantor obtained his doctorate from Berlin University at the precocious age of 22. His subsequent appointment to the University of Halle in 1867 led him to the evolution of Set Theory and his involvement with the until-then taboo subject of infinity.

Within Set Theory he defined infinity as the size of the never-ending list of counting numbers (1, 2, 3,

The Imaginary Number20020314The imaginary number takes mathematics to another dimension. It was discovered in sixteenth century Italy at a time when being a mathematician was akin to being a modern day rock star, when there was 'nuff respect' to be had from solving a particularly 'wicked' equation. And the wicked equation of the day went like this: 'If the square root of +1 is both +1 and -1, then what is the square root of -1?

Previously, mathematicians had rolled their eyes skyward and prayed for divine intervention. But where others failed, the creative Italian Rafaello Bombelli triumphed with his invention of the imaginary number. The imaginary number is the square root of -1 and is known as 'i'.

Imaginary numbers are real numbers multiplied by i. If, like many, you find yourself saying 'but what's the point?' then think on this. Imagine a world without electric circuits. No circuits, so no computers. No computers, so you wouldn't be reading this now. And while engineers need the imaginary number to analyse electrical waves, physicists need it to calculate the fundamental forces that govern our Universe via quantum mechanics.

And, most bizarrely of all, time might have an imaginary dimension, which might explain how time started. The Big Bang gave birth to our Universe some 15 billion years ago. It was every mother's dream birth, taking less than a second for all time, matter and energy to be born, with no need for an epidural. But if the Big Bang marks the beginning of time and the beginning of matter, what was there before it? How could time just start?

Einstein's classical general theory of relativity linked together time with the three spatial dimensions we are all familiar with (up - down, left - right, and in - out), creating a four dimensional 'spacetime' in which time can move forwards only. A brilliant theory, but one that comes a cropper when applied to the creation of the Universe.

But invoke quantum theory and a dollop of imaginary time and things begin to make sense ... at least to cosmologists. Imaginary time is measured in imaginary numbers and unlike real time, it can move backwards and forwards like an extra spatial dimension. Apparently, this gives the Big Bang something to start from.

Simon Singh takes a quirky look at some of the most important numbers in mathematics.