“Now and then we pluck numbers from the blur…numbers which have no names except the ones we might now give them…souvenirs from alien, unknowable worlds.”
-Really Big Numbers by Richard Evan Schwartz
Really Big Numbers by Richard Schwartz, a mathematician at Brown University, is the first children's book published by the American Mathematical Society. (Disclosure: I am an editor of the American Mathematical Society Blog on Math Blogs.) But this is not Schwartz's first foray into talking about math with kids. He has two daughters, and in 2010, he wrote You Can Count on Monsters, which uses illustrations of strange and spiky beasts to explore the properties of numbers from 1 to 100.
Really Big Numbers is more. In it, Schwartz moves from counting by ones to counting by tens and thousands, to ever more abstract ways of representing quantities too huge for human comprehension. The book was inspired by the fact that when children find out that Schwartz is a mathematician, they often ask him about big numbers. It would be easy for a professional mathematician to dismiss questions like that; after all, what is more mechanical than big numbers? Name one, and I’ll tell you a bigger one. But Schwartz does not see the request as trivial. Sometimes the sheer magnitude of a number is unfathomable, and Schwartz embraces the challenge of expressing these magnitudes and understanding them once we express them. I recently spent some time grappling with impossibly huge magnitudes in my post on Graham’s number, and I appreciated the playful way Schwartz approached “plex” and other efficient ways to write about very large numbers. He invites the reader to jump in and play with them too.
The book is cheerfully illustrated and includes interesting factoids to help us get an intuitive feel for large quantities, from how many grains of sand you could fit into a basketball (a lot) to how many ways there are to visit the capitals of the lower 48 states (a lot more). Although it is written for children, I found it enjoyable, and I think many other adults will too. I hope others are just as stumped as I was on some of the questions in the text. (This one got me: is 10×9×8...×2×1 or 2^{22} larger?)
Schwartz recognizes the fact that not everyone will be able to understand his book the first time through. He writes,
“Before we get to the numbers, I want to tell you something about this book: It is like the game of bucking bronco I used to play with my daughters. The ride starts out slow and gets faster. The game is to stay on for as long as you can. If you fall off without finishing the whole book, don’t worry. This book isn’t something that you have to read all at once, or even all in one year. Just read as far as it makes sense and then save the parts you don’t understand for later.”
Many math writers, including me, work hard to make math accessible. But the approach Schwartz takes is refreshing. He works hard to make the ideas accessible, of course, but he doesn't want to limit the book to concepts that we can all grasp immediately. The things we don’t understand now do not represent failure but an opportunity for growth.