When Ed Scheinerman’s seven-year-old granddaughter asked the question that every mathematician secretly loves—“What is the number before infinity?”—he knew he was onto something. “You’re not going to like the answer,” he recalls telling her, “but it’s a marvelous question,” says Scheinerman, professor of Applied Mathematics and Statistics and vice dean for special projects at the Whiting School of Engineering. There isn’t a number before infinity, but that impossibility sparked an idea: what if he could take readers on a journey through the many faces of infinity?
This curiosity led to his recently published book, A Guide to Infinity: Ten Mathematical Journeys (Yale University Press 2026), which draws on decades of mathematical thinking to present many concepts of infinity in accessible ways. Unlike traditional math books aimed at specialists, Scheinerman’s work is dedicated to his six grandchildren, serving as a playful reminder that curiosity, and sometimes the best questions, starts young. “The audience is anyone who likes math,” he says. “You don’t need calculus, just a memory of high school math. If you remember that, the book should be accessible.”
When asked why he frames infinity as a series of journeys rather than a single concept, Scheinerman explains, “There are many ways to think about infinity. Each chapter explores a different concept. It’s a guide that takes you on a tour of infinities.” The ten journeys mentioned in the book range from familiar ideas to more surprising ones, all unified by the principle that infinity is not a number to fear, but a concept to explore.
Even after decades of research in discrete mathematics—where he developed random dot product graphs used to model social and biological networks—Scheinerman was surprised by what he uncovered while writing the book. Concepts like hyperreal numbers, which extend the idea of numbers in new ways, and tropical arithmetic, where all algebraic curves are straight lines thanks to a playful nod to Brazil, revealed that even a veteran mathematician can stumble upon new ideas when examining old questions from fresh perspectives.
Part of the book’s charm lies in its approach to a concept that can feel intimidating. “Most people use numbers as adjectives,” Scheinerman says. “Mathematicians think of numbers as nouns. Infinity is just another number, like negative seven. It’s a thing you can manipulate.”
His goal is not just to explain, but to normalize. Just as zero and negative numbers once surprised the human imagination, infinity can be mastered with perspective and a little practice.
One chapter explores projective geometry, the concept that parallel lines meet at infinity. Here, the abstract becomes practical: a line on the horizon is just another line and manipulating it mathematically creates the illusion of depth in a painting or a digital scene. “That’s infinity as a place, not just a number,” Scheinerman says.
While not written as a textbook, the book reflects Scheinerman’s teaching philosophy. “I hope students, especially non-math majors, get a sense of how mathematicians think,” he says. “It’s different from how an engineer or economist might think about a problem. Infinity shows there are many ways to approach a concept, and not all of them are compatible—but that’s okay.”
Scheinerman hopes readers will walk away both unafraid of infinity and inspired by the creativity inherent in mathematics. “Mathematics isn’t just about solving equations,” he explains. “It’s about inventing ideas, exploring them, and seeing how they connect to the world.”
After several years of writing and research, the book has already sparked interest far beyond the university. Upcoming interviews include the Museum of Mathematics and a feature interview with Rorotoko, a platform known for in-depth discussions with authors across disciplines. But the moment that remains most meaningful to him? Delivering a copy to the granddaughter, now 11, whose question started it all. He says with a smile, “She was the spark for this whole thing.”