“Really Big Numbers”

A few years back I was disconcerted to hear a kid at my son’s preschool mention googol and googolplex. But what really shocked me was when, a year later, one of his kindergarten classmates dropped Graham’s number on me. Can you believe it? Back when I was a boy, a million really meant something. Googol and googolplex were numbers that middle-school and high-school kids would try to awe each other with — not preschoolers! And Graham’s number? Ha! Kids didn’t learn about “the G-bomb” until much, much later, if at all. I know I sound like a prude, but what will today’s kids have to look forward to if they’re exposed to such ginormous numbers at such a tender age? I ask you!

All kidding aside: Graham’s number was a glamorous by-product of joint work of mathematicians Ronald Graham and Bruce Rothschild in the area of mathematics called Ramsey theory (see chapter 17 of Gardner’s book “Penrose Tiles to Trapdoor Ciphers and the Return of Dr. Matrix”). In the intervening years, mathematicians have gone on to study problems that force them to consider even bigger numbers (see the Endnotes), but back in 1977, when Gardner’s essay was published in Scientific American, Graham’s number was the largest number that had ever been introduced in a serious mathematical context. (As opposed to a “Hey, I know bigger numbers than you do” context.) I don’t want to digress to define Graham’s number here, but see Gardner’s article or the Wikipedia page on Graham’s number if you’re curious. Unlike googol (ten to the hundredth power, or 10¹⁰⁰) and googolplex (ten to the googoleth power, or 10¹⁰¹⁰⁰), Graham’s number can’t be written using standard mathematical notations; you need to introduce new ways of representing numbers. And once you’ve got new ways of writing big numbers, why not push them as far as they can go, and then some? This is the spirit behind the book “Really Big Numbers“, written and illustrated by Richard Schwartz.

“Really Big Numbers”, by Richard Evan Schwartz. All images from the book are included with the author’s permission.

Schwartz is a mathematician, and this fact shows itself in lots of ways — above all, in the patient, persistently cumulative, and sometimes helical approach to learning mathematics that he advocates. 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.” This is good advice for learning math at all levels, from kindergarten to graduate school.

The first two thirds of the book treats “merely astronomical” numbers (the “-illions“) and numbers slightly beyond the needs of astronomy and physics (like a googol); it introduces exponential notation where this is helpful, and focuses on trying to build readers’ intuition for big numbers by relating them to imagined experiences. For instance, if the human population level were equal to Avogadro’s number, our species would fill about 15 hollowed-out copies of our planet.

HOW GARGANTUAN IS A GOOGOL?

How big is 10¹⁰⁰ ? It’s bigger than the number of atoms that could in theory be packed into a volume the size of the observable universe, but not by much (that is to say, not by more than, oh, a dozen orders of magnitude — which is small potatoes when you’re talking about a number like a googol, which exceeds the number 1 by a hundred orders of magnitude).

A painter selecting from a googol artistic options.

Schwartz points out that a googol is exactly equal to the number of ways to make a painting consisting of 100 colored squares, using a palette of 10 colors.  That brings the number down to a humanly comprehensible scale. But does it really make a googol comprehensible, or does it merely seem to? I’m reminded of Jorge Luis Borges’ short story The Library of Babel which Schwartz alludes to glancingly in his book. The scholars who inhabit Borges’ fictional library believe that the library contains all 25^1,312,000 possible books consisting of 1,312,000 symbols from a 25-symbol alphabet. You can imagine what a typical book from this library is like (or, if you can’t, a Borges-fan has created software that will generate one for you); but that’s not the same as imagining the extent of the library itself, which would have to be larger than our observable universe.

Schwartz’s “googol monster”.

Schwartz also introduces a charming if daunting creature that I’ll call the googol monster. When she is fully using her powers, her eyeless head has ten necks leading to ten one-eyed heads, each of which has ten necks leading to ten two-eyed heads, each of which has ten necks leading to ten three-eyed heads, and so on, leading finally to a whole lot of hundred-eyed heads. The beast has 10¹ (ten) one-eyed heads, 10²  (one hundred) two-eyed heads, 10³ (one thousand) three-eyed heads, …, and 10¹⁰⁰  (a googol) hundred-eyed heads. The googol monster doesn’t appear in the Vedic literature, but it should; the Indians who composed the Vedas loved large numbers, and in one sutra the Buddha, squaring off against a mortal mathematician, introduces numbers even larger than a googol.

GOOGOLPLEX AND FURTHER

10¹⁰⁰ is 1 followed by 100 zeroes, or
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
That might at first look big, but there are numbers so big we can’t even write them down this way at all, such as 1 followed by a googol zeroes, which is the decimal representation of the number called googolplex.

In advancing to googolplex and beyond, Schwartz’s book distinguishes itself from the growing collection of books in the how-big-is-a-billion genre. In the final third of the book the physical universe, having served its purpose, is cast aside entirely, along with any pretense that the numbers we are talking about are humanly comprehensible. Schwartz invites us to take leave of our senses and to put our trust in language instead, letting it take us to places that our senses cannot. As Schwartz puts it, “Mathematics gives us a language to name all kinds of things, but we can’t relate to everything we can name. If you want to think about REALLY big numbers, you have to give up the idea of picturing them.”

Schwartz’s implacable guard, barring our intuitions from apprehending really large numbers.

Schwartz makes several attempts at conveying the bigness of googolplex in concrete terms, but concedes that the task is beyond him and may be beyond the human mind. “In my experience, it is impossible to picture googolplex in concrete terms. Any attempt will scramble your brain. An implacable guard blocks the door to that kind of intuition.” Instead, we must take symbolic constructs like exponential notation that can be applied to imagineable numbers and then, with a faith born of experience, trust those constructs to give us truths when applied to unimagineable numbers. For instance, we can’t imagine 10¹⁰¹⁰⁰ , but using the laws of exponents we can predict that when we raise it to the tenth power, we get the equally unimagineable number 10¹⁰¹⁰¹ . (If you’re wondering why, see the Endnotes.)

SMALL PICTURES OF BIG NUMBERS

When discussing really big numbers like googolplex and its larger kin, Schwartz stops trying to give intuition for their bigness, but he doesn’t stop using pictures; now the pictures he gives us are whimsical ways of representing big numbers (embodiments of computer scientist Don Knuth‘s arrow notation and other similar schemes). The whimsicality eases the transition from the sensory-thought-experiment approach (appropriate for merely big numbers) to the purely formal approach (appropriate for really big numbers), and gives us a way to think about really big numbers (or rather, to fool ourselves into thinking that we’re thinking about them). For instance, the symbols 1, 2, 3, etc. stand for the numbers 10, 10¹⁰, 10¹⁰¹⁰, etc. That is, N stands for the value of an exponential tower consisting of N stacked 10’s (see the Endnotes for a guide to interpreting such towers properly).

A deceptively compact way of writing very big numbers.

These numbers may seem to grow really quickly, and so you might think that googolplex is about as far as we can go, but we’re just getting started here. Consider that googolplex is equal to 10¹⁰¹⁰⁰, which is smaller than 10¹⁰¹⁰¹⁰ (since 100 is smaller than 10¹⁰), and 10¹⁰¹⁰¹⁰ is the number we’re calling 4. So a googolplex is smaller than 4, from which it follows that googolplex is smaller than 4 (a box containing a box containing a 4). An even bigger number than that is 10 — a box containing a box containing a 10. And an even bigger number than that is 10— a box containing a box containing a box containing a 10. And so on.

Schwartz has a different way of writing this last number: it’s the numeral 3 inside a pentagonal box. More generally, an N inside a pentagon represents the value of a nested collection of N square boxes with the numeral 10 inside the smallest of the boxes. But why stop with 5-sided boxes? Schwartz defines an N inside a hexagon as the value of a nested collection of N pentagonal boxes with the number 10 inside the smallest of them, and defines an N inside a heptagon as the value of a nested collection of N hexagonal boxes with the number 10 inside the smallest of them. Already 7 inside a heptagon is much, much bigger than Graham’s number.

If these numbers aren’t boggling your mind, then you’re probably not letting them infect your brain. And that might bode well for your ability to function in the real world. Millions of years ago, arboreal primates prone to moments of radical amazement probably smacked into tree-branches more often than their less flappable kin. So there could be evolutionary advantages to being the sort of person whose reaction to the above is “Uh huh, sure, that makes sense” — as opposed to the sort of person whose reaction is “Wait … but … that’s … how is that possible … ? This is making my head hurt!” People of the latter sort tend to be the ones who go on to take honors math classes, and attend math grad school, and invent new pure mathematics. Mental quickness is not as essential as a capacity for wonder.

FASTER, FASTER!

7-in-a-hexagon is huge, but we’re still just getting started. More notations giving vastly bigger numbers can be defined, exploiting the power of recursion. “Each new addition to the language is a chariot moving so quickly it makes all the previous ones seem to stand still,” writes Schwartz. “Unhindered by any ties to experience, giddy with language, we race ever faster through the number system. Now and then we pluck numbers from the blur; numbers which have no name except the ones we might now give them, souvenirs from alien, unknowable worlds.” Even a comparatively small really-big-number like googolplex can pose problems for the philosopher: in what sense can we be sure that the googolplex-eth digit of pi exists, when it is likely that no physical process of computation in our universe would ever enable us to determine it? And what’s true of googolplex holds all the more for the much, much, much bigger numbers Schwartz goes on to describe. I’m reminded of what Dean Fogg, the head of Brakebills Academy for Magic in Lev Grossman’s Magicians trilogy, tells his students on the topic of other worlds: “I have never been to these worlds, and you will never go there.”

And yet, even though our bodies and intuitions can’t visit these worlds, we can still at least point at them with our symbols, coming up with ever-more twistily recursive schemes for pointing at places further up the number hierarchy, in a never-ending climax. Schwartz’s book comes to an end, but the ideas it imparts can be extended beyond, and beyond, and beyond. I am reminded of the last paragraph of C.S. Lewis’ The Last Battle: “But for them it was only the beginning of the real story. All their life in this world and all their adventures in Narnia had only been the cover and the title page: now at last they were beginning Chapter One of the Great Story which no one on earth has read: which goes on for ever: in which every chapter is better than the one before.”

But maybe that comparison isn’t so apt. The children in Lewis’ story, as the veils of reality part before them, feel an ever-increasing joy; we, in contemplating ever-larger numbers, are likely to experience something more akin to vertigo. To quote the last sentence of Schwartz’s book, “As you race through the number system, and find yourself in the middle of nowhere, with no end in sight, the main thing you will learn is that …” — but no, I’m not going to spoil it for you. Get a copy and find out for yourself.

THOUGHT-PROVOKING QUESTIONS

One of the approximately 262 sextillion spanning trees joining 20 points.

Another feature of Schwartz’s book is that, on the way to its mind-bogglingly big numbers, it innocently touches on some rather subtle math. For instance, near the middle of the book, Schwartz invites readers to count the number of ways to connect 20 dots without making loops. (Spoiler alert!) Putting it in more standard mathematical terminology, he’s asking us to count the spanning trees of a set of 20 vertices. In 1860, the mathematician Carl Wilhelm Borchardt showed that the number of spanning trees on n vertices is n^n-2 (a formula that is unfairly but universally called Cayley’s Formula); in the case n=20, this number turns out to be 20¹⁸, or 262,144,000,000,000,000,000,000 (not even close to what Schwartz would deem a “really big number”, but still respectable by ordinary standards). I’m not sure how many readers who didn’t already know Cayley’s formula will figure it out from the hint Schwartz gives (“I’m going to help you solve it by telling you the best trick I know for solving math problems. If you want to solve a problem involving big numbers, try it first with small numbers”). Fortunately, you can find a guide to Schwartz’s book on the web. This will be especially useful to home-schooling parents, or parents who want to supplement the math their kids are learning at school with math that the kids will definitely not learn at school in later grades.

Touring a set of 48 dots

A more manageable question, raised but not answered in the book, is whether every tour of a set of dots on a page can be drawn without crossings. You’re allowed to use curved paths, but you don’t want the tour to cross itself. When there are forty-eight dots to be joined up (the capitals of the continental states of the U.S., in Schwartz’s example), an attempt to meet the challenge with paper and pencil is likely to fail. Can you draw a tour that visits all 48 capitals in alphabetical order without crossings? Can every tour be drawn without crossing itself? Or do some tours inherently require crossings? See the Endnotes for the answer.

Spanning trees and tours can be tricky, but questions about them aren’t the hardest problem you’ll encounter in the book. Lurking just under the surface of Schwartz’s tour of numbers big and bigger is the task of deciding which of two given numbers is bigger. For the numbers that appear in the first two-thirds of the book, that isn’t a problem; decimal notation readily answers the question. (And if you know that 2¹⁰ = 1024, it’s easy to figure out whether 2²⁰ is bigger than 10⁶ or vice versa.) But for really big numbers defined by intricate recursive processes, it’s often unclear which of two numbers is larger. I suspect that ordering all the numbers that occur in the last part of the book would be a difficult task, and that a procedure for comparing any two numbers of the sort that Schwartz could have depicted using his pictorial notations might even be beyond the reach of existing mathematical techniques. I’m not saying it’s an important unsolved problem; but I do point to it as an example of the deep waters that Schwartz is leading his readers into.

I didn’t use this book as a springboard for much independent mathematical exploration with my two kids, but I expect that as they get older, I’ll find opportunities to re-visit it with them, and give them more of a feeling for mathematical research. Likewise for the book about Erdős that I reviewed last month. Not that I expect my kids to become mathematicians, but it’d be nice if they had a feeling for what I do for a living.

Speaking of my previous book review, remember my daughter? The one who asked “If I invent time travel, will I be as famous as Paul Erdős?”? I never told you what I said in reply, so I’ll tell you now. I told her the truth: I said that if she invents time travel, she will be much, much more famous than Paul Erdős. Because — let’s be honest — on the scale of world-wide celebrity, Erdős is no Galileo or Darwin or Einstein. Primes are very nice if you’re into that sort of thing, but time travel would be the biggest deal ever. Still, we live in a world in which some kids are beginning to hear at an improbably early age that there’s something out there called Graham’s number, so one can imagine a world similar to ours in which lots of kids actually know a little bit about how Graham’s number is defined, and know a little bit about Paul Erdős‘ work. These two children’s books bring our world a little bit closer to that one.

Next month: How to be wrong.

This article was written with help from John Baez, Ron Graham, Sandi Gubin, Henri Picciotto, Donna Propp, Tom Roby, Rich Schwartz, and Mike Stay.

ENDNOTES

I mentioned that there are books on “merely astronomical numbers”. To find them, search on titles like “Big numbers”, “How much is a million?”, “How big is a million?”, “On beyond a million”. And of course there’s nothing that beats videos like “Powers of Ten” for making numbers like 10⁶, 10⁹, 10¹², … more visceral. The idea of zooming in and zooming out is also helpful for solving certain sorts of geometric problems. For instance, consider the problem of drawing a noncrossing tour of 48 dots mentioned earlier. If you stay at the same scale throughout, it’s hard to see how to continue to add new segments to the growing tour. But if at each stage you imagine you and your pencil shrinking by a factor of ten, or equivalently the “map” getting larger by a factor of ten, you can see that there’s nothing that gets in your way when you try to connect the nth dot to the n+1st. Or rather, the things that get in your way (namely path-segments from earlier parts of the tour) can be skirted.

Once we start stacking exponential notation and talking about numbers like 10¹⁰¹⁰ , it becomes important to distinguish between (10¹⁰)¹⁰ and 10⁽¹⁰¹⁰⁾, which are very different. The laws of exponents tell us that (10¹⁰)¹⁰ equals 10^10×10, or 10¹⁰⁰, that is, a 1 followed by a hundred 0’s (also known as a googol). But 10⁽¹⁰¹⁰⁾ is much bigger: it’s 1 followed by ten billion 0’s. By mathematical convention, 10¹⁰¹⁰ (with no parentheses) denotes the latter, larger number.

I mentioned in the body of the essay that the tenth power of 10¹⁰¹⁰⁰  is 10¹⁰¹⁰¹ . That’s because (10¹⁰¹⁰⁰)¹⁰ = 10^{(10100 × 10)} = 10^(10100×101) = 10^10100+1 = 10¹⁰¹⁰¹. That is, if you take googolplex, also known as “100-plex-plex”, and raise it to the tenth power, you get “101-plex-plex”. (Yes, if you give Schwartz’s book to a child, be prepared to give some frank answers to questions about plex.)

Serious mathematical discussion of “really big numbers” involves going beyond mere plex-ing; you need things like Knuth’s arrow notation and/or the Ackerman function. To get a sense of what sorts of numbers count as “big” these days, check out John Baez’s blog post on enormous integers (but ignore the first half-dozen lines, which treat a different topic). Baez describes a combinatorial sequence studied by logician Harvey Friedman. The sequence innocuously begins 3,11,… but its third term is incomprehensibly huge.