An Imaginary Tale: The story of √-1               

Mathematicians are always complaining that there is more to their subject than numbers. Rightly so: geometry and topology draw intuition from visual, haptic and kinaesthetic sources rather than the discrete, static regularities of numbers; and even numerically oriented calculus and analysis and the many types of algebra that generalize number long ago developed their own patterns and narratives quite separate from the purely numerical. The constant interchange and cross-fertilization between geometry and algebra makes it possible to talk about equations or curves or such hybrids as topological groups and the geometry of numbers.

Nevertheless, numbers form the conceptual as well as the historical backbone of the subject. The story of numbers, from its beginnings in the familiar whole numbers, notated by the literate cultures of China, Egypt and Babylon several millennia ago, to its present sophistication, is one of the grand narratives of Western science. The story starts with Pythagoreans, for whom the universe was constructed out of whole numbers, and their loss of innocence at the discovery of a length (the hypoteneuse of the right-angled triangle whose other sides are units) that could not be expressed as the ratio of such numbers. This so-called crisis of incommensurables introduced irrationals, i.e., non-rationals such as 2 into the concept of number.

Next came the introduction, in the Renaissance, of the number zero along with the negative numbers. Resisted as meaningless as long as the presiding metaphor for number was quantity of physical objects, they were accepted when the metaphor changed to position on a line: negative numbers being the reflection in the point zero of the positive ones; or, somewhat differently, when the metaphor became the less concrete one of quantity of money: negative numbers being interpretable as quantities of debt.

Next came the introduction of imaginary numbers. If an irrational number solves the equation x2 = 2 and a negative number solves x + 1 = 0, then could there be a number solving x2 = -1? The idea that there could, prompted by sixteenth-century investigation of the solutions to cubic equations, was met with extreme suspicion by the mathematical community; some, finding the notion of multiplying a “number” by itself to get a negative quantity scandalous and contrary to all logic, rejected it out of hand, others were merely uncomfortable and ignored it. Vestiges of this unease continued until the mid-nineteenth century, and even today many on their first encounter with -1, forewarned no doubt by its infamous history, while no longer finding the idea absurd or paradoxical, are fascinated and not a little mystical about it.

Such seems to be the starting point of Paul Nahin, a professor of electrical engineering who has written books on the science of radio, and who confesses to a lifelong enthrallment by the complex numbers that -1 gave rise to. An Imaginary Tale: The story of -1 is his attempt to share that fascination. To this end, we get a history of complex numbers – that is, numbers of the form x + -1 y where x and y are ordinary (real) numbers – from their early appearance in Cardano’s attempts to solve the cubic equation to their modern-day role as a fundamental and unproblematic tool for physics, especially quantum physics, and electrical engineering. Along the way are many anecdotes and apercus – Leibniz, for example, thought them elegant and wonderful, “an unnatural birth in the realm of thought, almost an amphibium between being and non-being”; and many mathematical derivations wherein the power and surprise of complex numbers are demonstrated.

As with negative numbers, the crucial step in the general acceptance of complex numbers was finding an appropriate metaphor. This turned out to be pictorial: if one interprets -1 as an operator that effects a right-angle turn, then complex numbers became identifiable as points or vectors in the plane, and everything about them takes on a geometrical meaning. Though due to others – Wessel, deMoivre, Argand – before him, it was the use of this geometrical interpretation by Gauss that won the day. Gauss counselled thinking of -1 not as “imaginary”, which only increased its obscurity, but of calling +1, -1, and -1 “direct, inverse and lateral units”, and he incorporated the idea in his proof of the Fundamental Theorem of Algebra (which asserts every equation of degree n has n roots). Thus, notwithstanding that Euler had already opened up the subject of complex numbers (it was he who named ∫-1 as i) a generation before him, it was Gauss’s use of their geometrical properties that removed any serious opposition to complex numbers within the mathematical community.

Conversely, the identification of complex numbers with points in the plane allowed new mathematical fields, such as complex differentiation and contour integration, to be developed; and it made complex numbers an indispensable tool for the study of electricity. Nahin takes time to explain the latter, but it is the former that sets him on fire; and his determination that we share his enthusiasm leads him far beyond the suitably cryptic and much cited equation eⁱπ= -1 that appears in every popular exposition of complex numbers to explaining the heavy machinery behind calculus as it applies to functions of complex variables. This entails many definitions and pages of partial differentiation and contour integration, all necessary to sketch the Cauchy-Riemann equations and be able to receive Nahin’s contention that Cauchy’s second integral theorem is “one of the most beautiful, profound, indeed mysterious results in all of mathematics”. An opinion that Nahin’s treatment of working through the steps – however mathematically correct – is unlikely to justify for the general reader not privy to the differences between “superficial” and “profound” or “ugly” and “beautiful” or “obvious” and “mysterious” that are routine for those with the relevant mathematical expertise; contrary to Nahim’s evident expectations, much of his book will bypass those without such expertise. But that is perhaps less important that the singular achievement involved in writing a book-length hymn of praise to the square root of minus one.

TLS June 11, 1999

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