# An Imaginary Tale: The Story of √-1

This is a reprint in the *New Princeton Science Library* of a classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by E. Maor, J. Havil, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another classic, reprinted at about the same time is Maor's *e: the story of a number* which follows a similar idea that may have inspired the current author.

The current reprint is of the first paperback edition of 2007 which is an updated version of the original from 1998. Paul Nahin is an electrical engineer who wrote several successful popular science books. His first one was a biography of Heaviside, and this book about complex numbers (it contains even an introduction to complex functions) was his second. Several other were to follow, some of which have been reviewed in this EMS database: *Chases and Escapes. The Mathematics of Pursuit and Evasion* (2007), *Digital Dice. Computational Solutions to Practical Probability Problems* (2008), *Number Crunching* (2011), *The Logician and the Engineer. How George Boole and Claude Shannon Created the Information Age* (2012), *Holy Sci-Fi! Where Science Fiction and Religion Intersect* (2014), Of course complex numbers and functions are important tools in electrical engineering. The book has a strong historical component of course, but, unlike the book by Maor about the history of the number e, this book has much more mathematics in it. Hence it requires some mathematical affinity to understand much of what is presented here. It requires the knowledge of advanced secondary school or even freshman's university level, in particular when it turns into an introductory course on complex functions in the trailing chapter. Nevertheless, Nahin avoids a textbook structure of definitions, theorems and proofs, but keeps the level of a casual account, cheering up the reader with some witty remarks now and then.

The historical background starts with the solution of the cubic equation and the search for a formula that gives its roots, which was a hot topic in the 16th century. Knowing such a formula was a strong weapon for `mathematicians' that made a living as (human) computers, so it was important not to share it with competitors. Obviously this includes the well known story of del Ferro who knew how to solve the cubics and who told it to Antonio Fior before he died. Niccolo Fontana, better known as Tartaglia, also knew how to solve them. It came to a public duel between the latter two to solve the most equations in a given time. Tartaglia won much to Fior's surprise. Cardano who was a well respected mathematician in those days, stalked Tartaglia to tell him the magic formula, and after much pressure, Tartaglia eventually told Cardano, but made him promise to keep it a secret. However Cardano could consult the letters by del Ferro and considered his promise to Tartaglia not valid anymore and published the result anyway, which resulted in a vigorous priority fight.

Solving cubic equations is important for the history of complex numbers because the square roots in the relatively complex formulas could give complex conjugate solutions. However, even though the square roots of negative numbers made no sense to them, it turned out that when computations were performed as if these were genuine numbers, this could lead to valid results, which was most puzzling at first. The formulas are known as Cardano's but historically this is clearly a mistake. They were re-discovered a couple of times by others (e.g. Leibniz).

Since in antiquity (think of the Greek) many computations were done by geometric constructions. Descartes, Wallis, Newton, and others thought about a construction of the square root with compass and ruler. In geometric constructions it is difficult to give a meaning to a negative number when it concerns the length of a line segment or the area of a polygon. It was not before Bombelli had the idea of drawing a line with marks for the numbers (which we now call the real line) that negative numbers finally made sense. It was even more staggering in those days to make geometric sense of an imaginary number. Some possible interpretation was that a line intersected another one outside an certain interval, which made the intersection `imaginary'.

Of course, as we know, the proper interpretation of a complex number should be a point in the complex plane. That idea and the modulus-argument representation of the complex numbers with a complex exponential was first given by the Norwegian Caspar Wessel in 1797. He was not even a professional mathematician and succeeded where many great minds had failed before him. His finding went unnoticed though, until much later. Argand and Buée came to the same solution about a decade later and published their findings almost simultaneously which started another row between them. Again these names do not sound very familiar to us. Argand was a French amateur mathematician and Buée was a French priest who published a confusing, almost mystical paper on the subject. Just like Wessel's, their results faded away and were only re-discovered much later. Of course once the complex plane is accepted, one gets the goniometric representation, the formulas of De Moivre, the multiplication with i corresponds to a rotation over 90 degrees, etc. This is close to the vector interpretation by Hamilton, who finally gave a formal definition of the complex number field where a complex number was a couple of reals with vector addition and scaling and with a particular way of multiplying the vectors.

Once the complex playground has been fixed in the first three chapters, the next three chapters deal with applications of complex numbers. Since the complex numbers are like vectors in a plane, some geometric problems can be easily solved with complex arithmetic. A less known theorem of Cotes and a puzzle problem from Gamow's book *One, Two, Three... Infinity* are given as examples. Other applications discussed include the `imaginary' time axis in the space-time indefinite metric of relativity theory, the maximal distance of a random walk with decreasing step sizes, Kepler's laws, and electrical circuits. More mathematics is found in the chapter on Euler and the famous Euler formula (exp(ix) + 1 = 0) but also infinite series, infinite products, the calculation of i to the power i, and even the gamma and zeta functions. The final chapter is an introduction to complex functions, derivatives, contour integration, Cauchy integral theorem, Green's theorem, analytic and harmonic functions.

Concerning the structure of the text, I can mention that it is occasionally interrupted by `boxes' that discuss some topic closely related to the surrounding text, but that is not essential and could be skipped without a problem. On the other hand, some technical material is moved to appendices. The reprint is the unaltered version of 2007. That means that no additions or corrections are added since and the little defects that remained are still there. For example, the strange looking capital gamma symbol on page 177 (it is at least a different font from the surrounding pages) is still there. The preface of 2007 does explain what corrections and additions were made on that occasion. This book is not recommended if you are allergic to formulas, but if you want to peek behind the formulas and theorems in a textbook (a textbook is more `to the point' and hence necessarily `duller'), this is a the book that I recommend to read. You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies.

**Submitted by Adhemar Bultheel |

**27 / Apr / 2016