Since studying mathematics at school, I've wondered what exactly complex numbers are useful for. There were plenty of lectures at university about complex number analysis, but those courses lost me after only one or two weeks. From school and university, I'd understood that complex numbers were a complete number system, able to provide roots for any number, even where real numbers were insufficient. But what I couldn't grasp was the usefulness of roots that consisted of that most mysterious of components: the imaginary number.
Natural numbers (from one to infinity) are easy to understand for modern folk. And most people understand that real numbers can indicate a fraction of something, and that a negative number just means to do something the other way round (changing from positive to negative velocity just means going into reverse, for example). But once imaginary numbers expand the numeric space into two dimensions, I had no idea how that could be interpreted in the real world.
So I recently bought An Imaginary Tale by Paul J Nahin, hoping that he could shed light on what complex numbers meant, and what they were useful for. And I think he succeeded.
Nahin offers a history of the creation and development of the complex number, from early mathematicians who determinedly side-stepped the issue of rooting negative numbers, through to the men who devoted time to making sense of them, and on to the genius of those that applied complex numbers to produce theorems and identities that are still used today. Nahin has a particular interest because, as an electrical engineer, he has frequently relied upon these famous results.
The book starts with a "note to the reader" which states that, while large chunks of the book will be understood by high school seniors, it will be most accessible to people who have taken freshman calculus at university. It's been a long time since I studied calculus at university, and I have to agree that large chunks of An Imaginary Tale left me behind. I enjoyed the early chapters about the problems that led to the formulation of complex numbers, and I enjoyed the early derivations of trigonometric identities. But as soon as the calculus arrived, I was barely able to even grasp the basic aims. And this book contains a large amount of calculus.
However, I think the book has answered my question: how do you interpet an imaginary number? As an example, Nahin produces a formula for the time it will take a man running at fixed speed to catch up with a bus that is slowly accelerating away from him. When the solution is a real number (with no imaginary component), it represents the number of seconds it will take the man to catch the bus. When the solution is a complex number with an imaginary component, it represents that the man will never catch the bus. But the size of the imaginary component is a measure of how close the man will get to catching the bus, so the imaginary component does have a direct physical interpretation, albeit from a different perspective.
The real value of complex numbers is that they can be used as a tool for deriving results where the use of real numbers alone is either impossible, or would take much more effort. Nahin demonstrates this by deriving identities from trigonometry, first using the old geometric methods, and then far more easily using the simple rules of complex number addition and multiplication. In further derivations, Nahin makes it clear that complex numbers allow a sort of shortcut from one real result to another, passing through the "imaginary" on the way.
Even though it's certainly true that this book is better suited to well-rounded students of mathematics, being particularly demanding in calculus, I'm glad I read this book. Nahin is knowledgable and enthusiastic about the subject of complex numbers, in their history and in their use.