The number of primes less than a given integer \(n\).

The value of \(n^2\).

The number of graphs with \(n\) vertices.

The number of random walks in \(k\)-dimensional space with \(n\) steps.

The value of the integral \( \int_1^n \frac{1}{x} dx \).

The number of steps it takes to multiply two matrices with \(n\) rows and \(n\) columns.

As \(n\) gets large, each of these values goes to infinity, but which ones go to infinity faster, and which slower? These functions come from different branches of mathematics, but one can use the language of asymptotics to think about what happens to each function for large values of \(n\). And this is precisely what Joel Spencer does in the book *Asymptopia*. This is not a textbook in any one area of mathematics, but instead it considers how a certain family of techniques and way of thinking can be applied to a number of different parts of mathematics.

After a few introductory examples, as well as some anecdotes about Paul Erdős, whom Spencer credits with introducing him to this way of thinking, the book introduces the reader to the language of asymptotics. In particular, he discusses big-oh and little-oh notation and what we mean when we say that certain functions are asymptotic to each other or negligible compared to each other. Several theorems are proved, and several others are stated without proof, but Spencer quickly moves on to his real interest, which is applying this language and theory throughout mathematics.

The chapters that follow look at examples related to integration, binomial coefficients, graph theory, probability theory, prime numbers, triangle geometry, algorithms, and more. A final chapter, entitled “Really Big Numbers” (not to be confused with the recent children’s book of the same name) discusses the Ackermann function, the WOW function, and other ways of creating really big numbers.

I found the idea of this book to be very intriguing and would like to see more books that show how a single technique can be used in a wide variety of ways. But in the case of *Asymptopia* the execution does not live up to the idea. While Spencer does a good job of not assuming too many prerequisites beyond Calculus, this means that many of the topics are not very well motivated, so that at times it was not clear why I would care about estimating the value of a certain function, even if Spencer’s methods would help me to do so. More fundamentally, I never felt that the general philosophy of the book, which the author frames as “living in Asymptopia,” was as cohesive to me as it was to him, and many of the chapters felt disconnected from each other. I kept waiting for a clearer statement of what it meant to “think asymptotically,” and it never came.

Despite my reservations, there is quite a bit of interesting mathematics in *Asymptopia*, and at times Spencer is a good tour guide, although I felt the quality of the exposition varied from one chapter to the next. By limiting the prerequisites and focussing only on questions that can be analyzed using his asymptotic methods, Spencer is able to touch on some truly deep mathematics in a way that a beginning college student could understand. The book works better as a collection of independent articles than it did as a cohesive book, and I can imagine referring a student to a specific chapter as their studies demanded, even if I would be reluctant to recommend the book as a whole.

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Darren Glass is an Associate Professor of Mathematics at Gettysburg College. His research interests include number theory, cryptography, and algebraic geometry. He can be reached at [email protected].