MAA Reviews

A Guide to Infinity: Ten Mathematical Journeys

Book cover with a dark, gradient background shifting from deep green and black to hints of red and blue. Fine, curved white lines form a geometric, wave-like pattern sweeping diagonally across the cover, creating a sense of motion and depth. Centered in large white uppercase text is the title, “A GUIDE TO INFINITY,” with the subtitle “Ten Mathematical Journeys” in smaller text below. At the bottom, the author’s name, “Edward R. Scheinerman,” appears in white uppercase letters.
  • Author: Edward R. Scheinerman
  • Publisher: Yale University Press
  • Publication Date: 01/06/2026
  • Number of Pages: 192
  • Format: Hardcover
  • Price: $28.00
  • ISBN: 9780300284799
  • Category: gen

[Reviewed by Geoffrey Dietz, Ph.D. , on 05/15/2026]

Edward Scheinerman possesses great talent in both mathematics and writing. His skills allow him to reach out to audiences over a wide spectrum of mathematical talents and backgrounds. I have previously read and reviewed Scheinerman’s book The Mathematics Lover’s Companion: Masterpieces for Everyone. That book made me a fan of his writing and so I was eager to read his 2026 book A Guide to Infinity Ten Mathematical Journeys.

Whereas The Mathematics Lover’s Companion was pitched to a very wide potential audience, I believe A Guide to Infinity is aimed at a slightly more mathematically mature and engaged audience. For example, the former book could easily appeal to someone with a high school background in mathematics that does not necessarily include calculus, but I believe A Guide to Infinity is best put in the hands of someone who has at least started the pursuit of an undergraduate degree in mathematics based on the use of complex numbers, functions, limits, axioms, and mathematical notation. In other words, A Guide to Infinity is the more advanced book with deeper topics. The two books share some overlapping themes, such as repeating decimals, transfinite numbers, fractals, and hyperbolic geometry. Both books are excellently written.

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$\textit{A Guide to Infinity}$ covers ten "mathematical journeys" that all involve the infinite in one manner or another. The opening chapter talks about extending the real numbers to include both $-\infty$ and $+\infty$ as new elements. The benefits and costs of treating $\pm \infty$ like a number are then seen in ways that should be greatly appreciated by anyone who has ever learned L'Hospital's Rule or dealt with limits involving infinity. Chapters on tropical arithmetic and hyperreals are likely to be new even to readers with a typical mathematics education. Each topic introduces ways to expand the real numbers but in different ways. In tropical arithmetic the set of $\mathbb{R}$ along with both $\pm\infty$ is given new operations (minimum and addition) that satisfy some ring-like axioms leading to a discussion of B$\acute{e}$zout’s Theorem from algebraic geometry. The hyperreals on the other hand, add "infinitely small" numbers into the reals and form the foundation of nonstandard analysis. I found this chapter on hyperreals teasingly short and wished it had continued on longer.

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The middle chapters focus on geometry with great treatments of the projective plane and the hyperbolic plane. The chapter on the projective plane starts with a discussion of the line at infinity where parallel lines meet and then develops homogeneous coordinates and the natural duality between projective points and lines. Desargues's Theorem is shown and Bézout's Theorem returns. Even the Fano plane makes an appearance. This chapter is longer and went much deeper into its topics than any previous chapter. It personally reminded me of learning about projective geometry in the context of algebraic geometry at an REU in 1998, which really helped convince me that I should pursue a Ph.D. in mathematics. The hyperbolic chapter concentrates mostly on the Poincaré disk model and will be familiar to anyone who has studied geometry at the undergraduate level.

The closing section of the book has chapters on transfinite cardinals and ordinals as well as a discussion on fractals. I really liked the chapters on transfinite numbers as they go well beyond the basic Cantor diagnonalization arguments that undergraduates see that distinguish countable and uncountable infinities. These chapters wade into the deeper waters found in an introductory Set Theory course (which is sadly lacking for many mathematics students). The Continuum Hypothesis eventually makes an appearance along with the analogy to the parallel postulate of classical geometry as both statements are logically independent of their axiomatic compatriots. They can either be accepted or rejected and still lead to consistent theories of sets or geometry.

As with Scheinerman’s The Mathematics Lover’s Companion, his Guide to Infinity is an excellently written expository work that presents some fascinating, engaging, and beautiful mathematics more like “bedtime reading” and less like a textbook.

Geoffrey Dietz is a Professor of Mathematics at Gannon University in Erie, PA. He is married and has seven children.