In Pursuit of the Unknown (16 page)

For centuries mathematicians engaged in a love–hate relationship with these ‘imaginary numbers', as they are still called today. The name betrays an ambivalent attitude: they're not
real
numbers, the usual numbers encountered in arithmetic, but in most respects they behave like them. The main difference is that when you square an imaginary number, the result is negative. But that ought not to be possible, because squares are always positive.

Only in the eighteenth century did mathematicians figure out what imaginary numbers were. Only in the nineteenth did they start to feel comfortable with them. But by the time the logical status of imaginary numbers was seen to be entirely comparable to that of the more traditional
real numbers, imaginaries had become indispensable throughout mathematics and science, and the question of their meaning hardly seemed interesting any more. In the late nineteenth and early twentieth centuries, revived interest in the foundations of mathematics led to a rethink of the concept of number, and traditional ‘real' numbers were seen to be no more real than imaginary ones. Logically, the two kinds of number were as alike as Tweedledum and Tweedledee. Both were constructs of the human mind, both represented – but were not synonymous with – aspects of nature. But they represented reality in different ways and in different contexts.

By the second half of the twentieth century, imaginary numbers were simply part and parcel of every mathematician's and every scientist's mental toolkit. They were built into quantum mechanics in such a fundamental way that you could no more do physics without them than you could scale the north face of the Eiger without ropes. Even so, imaginary numbers are seldom taught in schools. The sums are easy enough, but the mental sophistication needed to appreciate why imaginaries are worth studying is still too great for the vast majority of students. Very few adults, even educated ones, are aware of how deeply their society depends on numbers that do not represent quantities, lengths, areas, or amounts of money. Yet most modern technology, from electric lighting to digital cameras, could not have been invented without them.

 

Let me backtrack to a crucial question.
Why
are squares always positive?

In Renaissance times, where equations were generally rearranged to make every number in them positive, they wouldn't have phrased the question quite this way. They would have said that if you add a number to a square then you have to get a bigger number – you can't get zero. But even if you allow negative numbers, as we now do, squares still have to be positive. Here's why.

Real numbers can be positive or negative. However, the square of any real number, whatever its sign, is always positive, because the product of two negative numbers is positive. So both 3 × 3 and − 3 × − 3 yield the same result: 9. Therefore 9 has
two
square roots, 3 and −3.

What about −9? What are its square roots?

It doesn't have any.

It all seems terribly unfair: the positive numbers hog two square roots each, while the negative numbers go without. It is tempting to change the
rule for multiplying two negative numbers, so that, say, −3 × −3 = −9. Then positive and negative numbers each get one square root; moreover, this has the same sign as its square, which seems neat and tidy. But this seductive line of reasoning has an unintended downside: it wrecks the usual rules of arithmetic. The problem is that −9 already occurs as 3 × −3 itself a consequence of the usual rules of arithmetic, and a fact that almost everyone is happy to accept. If we insist that −3 × −3 is also −9, then −3 × −3 = 3 × −3. There are several ways to see that this causes problems; the simplest is to divide both sides by −3, to get 3 = −3.

Of course you can change the rules of arithmetic. But now it all gets complicated and messy. A more creative solution is to retain the rules of arithmetic, and to extend the system of real numbers by permitting imaginaries. Remarkably – and no one could have anticipated this, you just have to follow the logic through – this bold step leads to a beautiful, consistent system of numbers, with a myriad uses. Now all numbers except 0 have
two
square roots, one being minus the other. This is true even for the new kinds of number; one enlargement of the system suffices. It took a while for this to become clear, but in retrospect it has an air of inevitability. Imaginary numbers, impossible though they were, refused to go away. They seemed to make no sense, but they kept cropping up in calculations. Sometimes the use of imaginary numbers made the calculations simpler, and the result was more comprehensive and more satisfactory. Whenever an answer that had been obtained using imaginary numbers, but did not explicitly involve them, could be verified independently, it turned out to be right. But when the answer did involve explicit imaginary numbers it seemed to be meaningless, and often logically contradictory. The enigma simmered for two hundred years, and when it finally boiled over, the results were explosive.

 

Cardano is known as the gambling scholar because both activities played a prominent role in his life. He was both genius and rogue. His life consists of a bewildering series of very high highs and very low lows. His mother tried to abort him, his son was beheaded for killing his (the son's) wife, and he (Cardano) gambled away the family fortune. He was accused of heresy for casting the horoscope of Jesus. Yet in between he also became Rector of the University of Padua, was elected to the College of Physicians in Milan, gained 2000 gold crowns for curing the Archbishop of St Andrews' asthma, and received a pension from Pope Gregory XIII. He invented the combination lock and gimbals to hold a gyroscope, and he wrote a number
of books, including an extraordinary autobiography
De Vita Propria
(‘The Book of My Life'). The book that is relevant to our tale is the
Ars Magna
of 1545. The title means ‘great art', and refers to algebra. In it, Cardano assembled the most advanced algebraic ideas of his day, including new and dramatic methods for solving equations, some invented by a student of his, some obtained from others in controversial circumstances.

Algebra, in its familiar sense from school mathematics, is a system for representing numbers symbolically. Its roots go back to the Greek Diophantus around 250
AD
, whose
Arithmetica
employed symbols to describe ways to solve equations. Most of the work was verbal – ‘find two numbers whose sum is 10 and whose product is 24'. But Diophantus summarised the methods he used to find the solutions (here 4 and 6) symbolically. The symbols (see
Table 1
) were very different from those we use today, and most were abbreviations, but it was a start. Cardano mainly used words, with a few symbols for roots, and again the symbols scarcely resemble those in current use. Later authors homed in, rather haphazardly, on today's notation, most of which was standardised by Euler in his numerous textbooks. However, Gauss still used
xx
instead of
x
²
as late as 1800.