Really Big Numbers

As any parent knows, there are inevitably some things that your child will learn on the playground from classmates at school that you would rather have them learn from you. I know parents who have had to deal with their kids learning about the birds and the bees, or about drugs, or the Israeli-Palestine conflict, or even the fact that Darth Vader is Luke Skywalker’s father before the parents felt their child was ready for the information. Even though my son is only six years old, I have already had to face this stark reality, as he came home from school one day last spring with the story that his friend Paul had told him about googolplex and how it is the biggest number in the world. I was pretty outraged by this, but luckily I was able to keep my cool, and I sat down next to him and calmly tried to talk to him about the true nature of large numbers while correcting the misinformation his friend had shared with him.

As a parent, I felt alone in this experience, but if the conversation took place now then I would feel like I had some help. I would just pull out my copy of Really Big Numbers by Richard Evan Schwartz and read it to him so that we could learn together. This is the first children’s book published by the American Mathematical Society, but readers may recall Schwartz’s earlier book You Can Count On Monsters which explained prime numbers and factoring at an elementary school level using lots of colorful drawings. As its title suggests, Schwartz’s new book deals with large numbers. REALLY large numbers. Sure, he starts by representing the number one with a single dot, and then represents two with two dots, but within a few pages there are drawings with 500 dots, or 1000 cheese cubes, or 5000 dots. And it just gets bigger from there. As Schwartz himself writes, “the ride starts out slow and gets faster. The game is to stay on for as long as you can.”

Obviously, before too long the conceit of representing numbers by pictures is inadequate, so he starts to illustrate numbers in other ways, using his own art and various factoids to help the reader wrap their mind around big numbers. There are roughly 10,000 minutes in a week, or 100,000 hairs on the head of a person. A person who lives to be 114 years old has lived a million hours, and 20 billion grains of sand would fill a basketball. Jupiter has four octillion pounds of material, and there are 100 billion times as many ways to travel between the 48 state capitals as there are atoms in the Earth. Pretty soon, Schwartz is introducing exponential notation and recursion to the reader, as well as giving the names of some of the large numbers. Did you know that 10¹⁰⁸ is also called a quinquatrigintillion? Now you do.

About halfway through Really Big Numbers, Schwartz has introduced a googol, and is ready to talk about what it means to “plex” a number so that he can introduce a googolplex. He does this in a very interesting way, using pictures and notation to explain ways to make bigger and bigger numbers. He tries to give a good way of thinking about a googolplex (“If you took a 10⁹⁸-dimensional cheese cube and sliced it…”) but quickly admits that “if you want to think about REALLY big numbers, you have to give up the idea of picturing them.” That said, he goes on to write about new ways of defining and notating extremely large numbers with names like “Fred” and “Big Jim,” and discusses how mathematicians might think about them and how big finite numbers can really be, let alone infinitely big numbers. Schwartz does a very nice job of mixing very elementary ideas with mathematical content that will be new even to mathematicians, making for a fun and enlightening reading experience.

I should say that while writing about a picture book may not quite be like dancing about architecture, it is certainly true that a written review such as this cannot capture the spirit of Schwartz’s book, every page of which is illustrated with bright colors and beautiful drawings that are appealing to look at even beyond the mathematics that they convey. To get a sense of his work, you should check out the sample pages at the AMS website or this promotional video. I have read the book to my son a half dozen times, and while he does not request it at bedtime quite as often as books about Spiderman or sharks, there have been many times that we refer back to the ideas it contains. I asked my son if he recommended the book and he gave two very enthusiastic thumbs up. When I asked what he liked about the book he was quick to say that he liked “the pictures and the big numbers,” as well as the fact that he now knew a bigger number than his friend Paul.

Sorry, Paul’s dad.

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Darren Glass is an Associate Professor of Mathematics at Gettysburg College. He has many mathematical interests including number theory, Galois theory, and cryptography. He can be reached at dglass@gettysburg.edu

Elliott Glass is a first grader at Vida Charter School. He was an active participant at the MOVES conference at the Museum of Mathematics, and reports that he likes math, but he likes basketball and Star Wars more.