Some numbers you can't count on

Imagining Numbers (Particularly the Square Root of Minus Fifteen)
by Barry Mazur
Allen Lane £9.99, pp288

Mathematics is the foundation of all science, and a pack of lies. Maths underpins physics, philosophy, economics and statistics. Without it we would be unable to build a democracy, which depends on the ability to count and the belief that counting matters. Democracy is defined, you might say, by the shared conviction that people count.

According to Barry Mazur, babies as young as five months are sensitive to the difference between 1+1 and 2-1. Numbers are with us throughout life. 'We sing numbers,' he rhapsodises; 'counting up the days of Christmas and counting down to the poignant monotheism of "One is one and all alone and ever more shall be so".'

But if a five-month-old baby is happy to count the fingers you're holding up, it may lose patience when you ask it to calculate the square root of minus 15. Even the square root of two might stump it. This is a number (1.41421356237...) which goes to infinite decimal places. It can be drawn by a child with a ruler - it's the length of a diagonal line across a 1cm square - but it can never be known in its entirety. The geometry is definitive, but the maths goes on for ever.

This is maths at its most lucid, and its most riddling. And, as Mazur shows in his quizzing, quizzical little book, this is only the beginning. He leads us into a world of mathematics so pure that sums become poetry. Divorced from the geometric world of shapes and their properties, maths gestures wildly towards a sphere of the unimaginable. As a focus, he takes the square root of minus 15, although - as any fool knows - negative numbers don't have square roots.

In pictorial terms, if a square has equal sides of 4 inches, then its area will be 16 inches. But in order to find the square root of a negative number, you have to imagine a negative area, with four sides of negative length. It's an impossibility, but, as Mazur argues, a mind-bendingly creative one: the calculation, he says, 'has more the ring of a Zen koan than of a question amenable to a quantitative answer'. But in the presence of this impossibility, new truths may emerge.

The square roots of negative numbers are not visible, but they can be used to calculate things which are. In Mazur's eloquent formulation, some calculations 'succeed in giving perfectly comprehensible answers to perfectly comprehensible questions, but only by dealing along the way with somewhat incomprehensible quantities'. This can be unsettling, he admits, 'rather like discovering that there is an efficacious way of getting from Brooklyn to Boston, but that somewhere in mid-journey one has to descend to the Underworld.'

He is paraphrasing GK Chesterton's old poem 'The Rolling English Road', which describes 'A merry road, a mazy road, and such as we did tread / The night we went to Birmingham by way of Beachy Head.' Mazur's book rolls us merrily along a mazy, mathematical road, almost to the edge of reason.

He reveals that whereas maths looks like the bedrock of rational thought, it is built on the slippery sand of imaginary numbers. The earliest mathematicians realised that alongside the positive integers which babies deal with are the negative numbers, known as fictae, or 'fictions'. Like a Borges short story, Mazur opens up these fictions to reveal further fictions within, and so on ad infinitum.

Mazur states that his aim in the book is not to give 'a historical account' of the search for negative square roots - which distances him from great popular maths writers such as Simon Singh and Robert Wilson - but to 're-create, in ourselves, the shift of mathematical thought that makes it possible to imagine these numbers'. He promises, rather disingenuously, that 'no particular mathematical knowledge is necessary', but that 'pencil and paper are good to have at hand, to make a few calculations'. I found that I needed more than pencil and paper to make these calculations. I needed a bigger brain. Yet, even without following all his workings-out, the window which Mazur cuts into the world of imaginary numbers is just as exciting, and almost as provocative, as anything in Philip Pullman.