The Mathematical Radio by Paul J. Nahin

The Mathematical Radio: Inside the Magic of AM, FM, and Single-Sideband

By Paul J. Nahin

Princeton University Press 2024

There are many, many impenetrable mysteries in The Mathematical Radio by University of New Hampshire emeritus professor of electrical engineering Paul Nahin.

That’s actually the entire review. This book is full of impenetrable mysteries. On literally every page, there’s at least one item — a contention, an allusion, a formula in what looks like High Klingon — that functions as a very thick unclimbable brick wall. You can’t scale it, you can’t skirt it, and you can’t blast it. You can either retreat or simply stand there staring at it (Note: your best route of retreat would be back to that freshman calculus class you slept through). 

But since The Mathematical Radio steadily, happily elaborates, a review of the book can hardly do less. 

Of course the first mystery of this or any book is why it was written in the first place, and here the answer is as obvious as the Fibonacci sequence on your face: Why, it’s the famous mathematician GH Hardy (1877-1947), of course! Hardy was a so-called “pure mathematician” who tended to heap scorn on people who employed mathematics for any grubby, practical purpose. As he put it in a 1940 essay titled “Mathematics in War-Time”: “I had better say at once that by ‘mathematics’ I mean real mathematics, the mathematics of Fermat and Euler and Gauss and Abel, and not the stuff which passes for mathematics in an engineering laboratory.” 

That essay in particular has stuck in Nahin’s craw since the moment he read it, about 750 years ago. He saw Hardy pizzling all over such applied sciences such as ballistics or aerodynamics, and it provoked his geeky heart to thoughts of dark vengeance. “Declaring ballistics and aerodynamics to be ‘sinister’ was pretty obnoxious (to me), but when I got to Hardy’s assertion that nothing ‘could be more soul-destroying than the numerical solution of differential equations,’ I had had enough,” he stirringly declares. “Although its author has now been dead for three-quarters of a century, Hardy’s essay clearly demands a response.”

Again, the certifiable insanity of this is simply an impenetrable mystery. You can keep reading, but not toward any kind of comprehension. 

Nahin organizes his book as a protracted response to Hardy’s contention that the only true mathematics is pure mathematics. His focus will be the electrical engineering behind FM and AM radio (the double-decker equations give way to triple-decker equations before the Preface is finished), but although he’s hoping on one level to convince Hardy that he’s wrong about applied mathematics even though he’s seventy years too late to do that, the real imagined reader is somebody else, somebody who’s also seventy years gone. “Like any other author, I hope this book appeals to a wide audience,” he writes hilariously, “but I have written with one particular person in mind: me, 67 years ago, when I was a 16-year-old kid full of excitement about electronics and math, but, alas, not so full a technical understanding.”

That 16-year-old read Popular Mechanics “cover to cover” every month all through high school, back when he wasn’t canvassing for Martin Van Buren, so all hope was lost fairly early. More broadly, Nahin mentions that is book is aimed specifically at mathematicians; he clarifies that this will not be a book about the historic or cultural dimensions of radio, and just to twist the knife, he mentions such books, like Empire of the Air by Tom Lewis and the wonderful Radio Voices by Michele Hilmes. Rather, this is a book about how radios work.

At least at the outset, Nahin attempts to extend a bit of empathy even to his long-dead nemesis, by reminding his readers that radio, bristling with resistors, vacuum tubes, transistors, transformers, capacitors, inductors, diodes, resonant tank circuits, oscillators, and bandpass filters, was mysterious and even ominous. “Interpreting the law on this subject is something like trying to interpret the law of the occult,” Chief Justice William Howard Taft wrote along those lines. “It seems like dealing with something supernatural.” 

Nahin realizes that he could take this kind of sociological approach to appeasing the shade of GH Hardy, and it might even be an intuitive approach, since Hardy was notoriously tech-averse (“If Hardy could have read this book,” Nahin writes right out of the starting gate, in the caption to the book’s first illustration, before he’s even started his Preface, “perhaps he would have learned how to solder two wires together without setting himself on fire”).  But instead, he picks a prickly and less friendly approach. “No, the only proper response to Hardy, one that would have a chance of success with him, would be a math response, with all (or as much as possible) of the electrical gadgetry and exotic tech talk replaced by the language he understood and loved – mathematics,” he writes. “And there’s plenty of math in radio, enough to satisfy even Hardy, I think.”

Which would be terrific news if Hardy were still alive, but as noted, he’s been dead for almost a century. For the living, Nahin has both lots and lots of math but also a foundational claim that’s startling at least in part for its inaccuracy:

Radio is perhaps the single most important electronic invention of all, surpassing even the computer in its societal impact (the telephone doesn’t depend on electronics for its operation, and television is the natural extension of radio). Even if we drop the ‘electronic’ qualifier, only the automobile can compete with radio in terms of its effects on changing the very structure of society.

This might be colorful, but, with no disrespect, it’s the kind of claim that’s going to come from somebody who was a teenager 70 years ago. This is reflected also in one of the book’s many entertaining footnotes:

There is no discussion of digital radio here and, as much as I might wish otherwise, I am unable to provide instruction on how to make a wonderful science fiction gadget called the Dirac radio (a radio that can receive signals from the future). I’ve also not said anything about the nature of Wi-Fi, a form of radio allowing digital devices to communicate with one another using microwave energy at frequencies far higher than those of AM and FM broadcast radio.

That mention of the Dirac radio seems like an odd kind of giveaway, one of the only genuinely crotchety notes in an otherwise exuberant book (digital radio and Wi Fi aren’t imaginary, after all, not “something supernatural,” as it were). As for the rest of the book’s impenetrable mysteries, well, do you even own a transistor radio? 

Steve Donoghue is a founding editor of Open Letters Monthly. His book criticism has appeared in The Washington Post, The American Conservative, The Spectator, The Wall Street Journal, The National, and the Daily Star. He has written regularly for The Boston Globe, the Vineyard Gazette, and the Christian Science Monitor and is the Books editor of Georgia’s Big Canoe News