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The square root of minus one, the building block of imaginary numbers, seems to be a contradiction in terms. Some say that admitting the possibility of such a number is what separates the mathematicians from the rest. A creative leap is required to gain access to this bit of the mathematical world. At first sight it looks as if it has nothing to do with the physical world. The physical world seems to be built on numbers whose square is always a positive number. Imaginary numbers, however, are more than just an abstract game. They hold the key to the twentieth-century world of subatomic particles. On a larger scale, aeroplanes would not have taken to the skies without engineers taking a journey through the world of imaginary numbers. This new world provides a flexibility denied to those who stick to ordinary numbers.

The story of how these new numbers were discovered begins with the need to solve simple equations. As the ancient Babylonians and Egyptians recognised, if seven fish were to be divided between three people, for example, fractional numbers – , and so on – would have to come into the equation. By the sixth century BC, the Greeks had discovered while exploring the geometry of triangles that these fractions were sometimes incapable of expressing the lengths of the sides of a triangle. Pythagoras’ theorem forced them to invent new numbers that couldn’t be written as simple fractions. For example, Pythagoras could take a right-angled triangle whose two shortest sides are one unit long. His famous theorem then told him that the longest side had length x, where x is a solution of the equation x² = 1² + 1² = 2. In other words, the length is the square root of 2.

Fractions are the numbers whose decimal expansions have a repeating pattern. For example, In contrast, the Greeks could prove that the square root of 2 is not equal to a fraction. However far you calculate the decimal expansion of the square root of 2, it will never settle down into such a repeating pattern. The square root of 2 starts off 1.414213562 … Riemann used to idle away the hours calculating more and more of these decimal places during his years in Göttingen. His record was thirty-eight places, no mean feat without a computer but perhaps more a reflection on the dull Göttingen nightlife and Riemann’s shy persona that this was his evening entertainment. Nonetheless, however far Riemann calculated, he knew that he could never write down the complete number or discover a repeating pattern.

To capture the impossibility of expressing such numbers in any way other than as solutions to equations such as x² = 2, mathematicians called them irrational numbers. The name reflected mathematicians’ sense of unease at their inability to write down precisely what these numbers were. Nevertheless, there was still a sense of the reality of these numbers since they could be seen as points marked on a ruler, or on what mathematicians call the number line. The square root of 2, for example, is a point somewhere between 1.4 and 1.5. If one could make a perfect Pythagorean right-angled triangle with the two short sides one unit long, then the location of this irrational number could be determined by laying the long side against the ruler and marking off the length.

The negative numbers were discovered similarly out of attempts to solve simple equations such as x + 3 = 1. Hindu mathematicians proposed these new numbers in the seventh century AD. Negative numbers were created in response to the growing world of finance, as they were useful for describing debt. It took European mathematicians another millennium before they were happy to admit the existence of such ‘fictitious numbers’, as they were called. Negative numbers took their place on the number line stretching out to the left of zero.

The real numbers – every fraction, negative number or irrational number is represented by a point on the number line.

Irrational numbers and negative numbers allow us to solve many different equations. Fermat’s equation x³ + y³ = z³ has interesting solutions if you don’t insist, as Fermat had, that x, y and z should be whole numbers. For example, we can choose x = 1 and y = 1, and put z equal to the cube root of 2 – and the equation is solved. But there were still other equations which couldn’t be solved in terms of any of the numbers on the number line.

There seemed to be no number which was a solution to the equation x² = −1. After all, if you square a number, positive or negative, the answer is always positive. So any number that satisfies this equation is not going to be an ordinary number. But if the Greeks could imagine a number like the square root of 2, without being able to write it down as a fraction, mathematicians began to see that they could make a similar imaginative leap and create a new number to solve the equation x² = −1. Such a creative jump marks one of the conceptual challenges for anyone learning mathematics. This new number, the square root of minus one, was called an imaginary number and given the symbol i. In contrast, mathematicians began to refer to the numbers that could be found on the number line as real numbers.

To create an answer to this equation, seemingly out of thin air, seems like cheating. Why not accept that the equation has no solution? That is one way forward, but mathematicians like to be more optimistic. Once we accept the idea that there is a new number that does solve this equation, the advantages of this creative step far outweigh any initial unease. Once named, its existence seems inevitable. It no longer feels like an artificially created number but a number that had been there all along, unobserved until we’d asked the right question. Eighteenth-century mathematicians had been loath to admit there could be any such numbers. Nineteenth-century mathematicians were brave enough to believe in new modes of thought which challenged the accepted ideas of what constituted the mathematical canon.

Frankly, the square root of −1 is as abstract a concept as the square root of 2. Both are defined as solutions to equations. But would mathematicians have to start creating new numbers for every new equation that came along? What if we want solutions to an equation like x⁴ = −1? Are we going to have to use more and more letters in our attempts to name all these new solutions? It was with some relief that Gauss finally proved, in his doctoral thesis of 1799, that no new numbers were needed. Using this new number i, mathematicians could finally solve any equation they might come across. Every equation had a solution that consisted of some combination of ordinary real numbers (the fractions and irrational numbers) and this new number, i.

The key to Gauss’s proof was to extend the picture we already had of ordinary numbers as lying on a number line: a line running east-west on which each point represents a number. These were the real numbers familiar to mathematicians since the Greeks. But there was no room on the line for this new imaginary number, the square root of −1. So Gauss wondered what would happen if you created a new direction. What if one unit north of the number line were used to represent i? All the new numbers that were needed to solve equations were combinations of i and ordinary numbers, for example 1 + 2i. Gauss realised that on this two-dimensional map there was a point corresponding to every possible number. The imaginary numbers then simply became coordinates on a map. The number 1 + 2i was represented by the point reached by travelling one unit east and two units north.

Gauss would interpret these numbers as sets of directions in his map of the imaginary world. To add two imaginary numbers A + Bi and C + Di just meant following two sets of directions, one after the other. For example, adding together 6 + 3i and 1 + 2i gets you to the location 7 + 5i.

Following directions – how to add two imaginary numbers.

Although this was a very potent picture. Gauss was to keep his map of the imaginary world hidden from public view. Once he had built his proof, he removed the graphic scaffolding so that no trace of his vision remained. He was aware that pictures in mathematics were regarded with some suspicion during this period. The dominance of the French mathematical tradition during Gauss’s youth meant that the preferred pathway to the mathematical world was the language of formulas and equations, a language that went hand in hand with the utilitarian approach to the subject. There were also other reasons for this aversion to images.

For several hundred years, mathematicians had believed that pictures had the power to mislead. After all, the language of mathematics had been introduced to tame the physical world. In the seventeenth century, Descartes had attempted to turn the study of geometry into pure statements about numbers and equations. ‘Sense perceptions are sense deceptions’ was his motto. Riemann had come to dislike this denial of the physical picture when he’d been reading Descartes in the comfort of Schmalfuss’s library.

Mathematicians around the turn of the nineteenth century had been burnt by an erroneous pictorial proof of a formula describing the relationship between the number of corners, edges and faces of geometric solids. Euler had conjectured that if a solid has C corners, E edges and F faces, then the numbers C, E and F must satisfy the relationship C − E + F = 2. For example, a cube has 8 corners, 12 edges and 6 faces. The young Cauchy had himself constructed a ‘proof’ in 1811 based on pictorial intuition, but was rather shocked to be shown a solid which didn’t satisfy this formula – a cube with a hole at its centre.

The ‘proof’ had missed the possibility that a solid might contain such a hole. It was necessary to introduce an extra ingredient into the formula which kept track of the number of holes in the solid. Having been tricked by the power of pictures to hide perspectives that weren’t initially apparent, Cauchy sought refuge in the security that formulas seemed to provide. One of the revolutions he effected was to create a new mathematical language which allowed mathematicians to discuss the concept of symmetry in a rigorous way without the need for pictures.

Gauss knew that his secret map of imaginary numbers would be anathema to mathematicians at the end of the eighteenth century, so he omitted it from his proof. Numbers were things you added and multiplied, not drew pictures of. Gauss eventually came clean some forty years later about the scaffolding he had used in his doctorate.