Boxing Pythagoras

On really, really, stupidly large numbers

When I was a kid and I got into an argument, inevitably it would devolve into a “Yuh-huh!” followed by a retort of “Nuh-uh!” After that, my brilliant counter argument would be “Yuh-huh, yuh-huh!” which was usually followed by “Nuh-uh, nuh-uh, nuh-uh!” It wouldn’t take us long to realize that repeating this, ad nauseum, would become irritating even to ourselves, so we soon came up with the idea of multiplying our answers: “Yuh-huh times ten!” would be followed by “Nuh-uh times a thousand!” But soon, we would reach the extent of big numbers that we could name. Most kids were familiar with “million,” “billion,” and even “trillion,” but numbers bigger than that often eluded us. Usually, after that, kids would either use nonsense words like “bajillion” or else they’d go back to repetition with “million million million.” However, occasionally, there were those few of us clever enough to learn about the bigger numbers. We’d learn “quadrillion” through “nonillion,” but the prefixes after that quickly became too confusing for little kid brains. Then we learned about a googol, , or (as we knew it) “a one followed by a hundred zeroes.” This number seemed insanely large, but remained easy (and fun!) for kids to say. Soon after learning about a googol, we would learn about the number googolplex, or “a one followed by a googol zeroes.” This number was so large, most of us couldn’t truly comprehend it, but since it was easy to say, we kept on using it.

I’m an adult, now, and even though my style of persuasive argument has become just a bit more sophisticated than it once was, I still find myself fascinated with really, really large numbers. Today, I want to talk about one of my favorites, called Graham’s number. It is so ridiculously, stupidly large, that a googolplex is only negligibly larger than 1, when compared to Graham’s number. The number was invented by Robert Graham in the late 70’s to represent the largest possible solution (or “upper bound,” in math speak) to a particular mathematics problem. I’d love to just tell you what Graham’s number is, but there’s a problem. You see, Graham’s number is so large that the usual mathematical operations with which people are familiar are entirely inadequate to describe it. A billion can be easily explained as “a thousand times a thousand times a thousand,” and a googolplex can be understood as “10 to the 10 to the 100th power;” but Graham’s number is so inordinately big that even nesting exponents is fairly useless in describing it. So, I’ll begin our journey, today, by talking about Knuth’s up-arrow notation.

This up-arrow notation was invented by Donald Knuth in 1976 for the express purpose of representing incredibly large integers. Knuth realized that our commonly utilized operations in mathematics just represent short-hand methods of repeating each other. For example, multiplication is just a short-hand way of doing repeated addition. So, if I want to know what is, I can just add three sets of 2 together; that is, . Similarly, exponents are short-hand for repeated multiplication, such that simply equals . Knuth decided to create a symbol (or “operator” in math-jargon) which could easily represent this sort of progressive repetition. He decided upon an up-arrow, or ↑ symbol, as his operator. A single up-arrow indicates exponentiation, so if we want to write 2 to the 3rd power, we would simply write .

Now, here’s where Knuth got really clever. Let’s say I wanted to represent repeated exponentiation (called “tetration”) in an equation. For example, let’s say I wanted to write out . Using Knuth’s up-arrow, this would be written as . Knuth wanted to describe this repetition, but he also wanted to be able to describe repeated tetration, and repeated repeated tetration, and any other arbitrarily large repetition of prior operators; but he didn’t want to have to add entirely new symbols for each of these operations– it would quickly become too convoluted to be useful. So, Knuth decided to represent further repetitions with additional arrows. So, if we wanted to write out 2 to the 2 to the 2nd power, we could write out , or we could simplify it as . Generalizing this, we would say that means to calculate  such that there are b copies of a. So, for example, would mean and would be , et cetera.

So, now, if we wanted to describe repeated tetration, we could just add another arrow. So, ,  which in turn evaluates as . Repeated repeated tetration would have four such arrows: , which equals . Each time you want to repeat a lesser operation, you simply add another arrow into the mix.

Now, I can finally start talking about Graham’s number. Graham’s number begins with the term . How big a number is ? Well, let’s try to evaluate it: . That, in turn, is equal to . Finally, we have some operators that we can start to evaluate using grade school math: . Plugging that back into our previous equations, we have . You can already see how insanely massive this number is starting to get. Even if we just isolate the term , remember that this means such that we have 7,625,597,484,987 copies of the number three. Already, we have a number so insanely large, we cannot actually calculate the whole thing. Then we would need to calculate so that there are copies of the number 3. Then, finally, we would have an answer to .

The number  is so large that it would be literally impossible to write the entire thing out, as the number contains more digits than there are atoms in the observable universe. But  is not Graham’s number. It’s just the first term in calculating Graham’s number.

Let’s say that is called . Now, let’s define  as , where the number of up-arrows equals . We could then find another term  which has  up-arrows,  which has  up-arrows, and generally which has up-arrows. By this process, Graham’s number would be . That is, we’d have to repeat this function 64 times in order to finally calculate Graham’s number.

This number is so stupidly large that, even if you were somehow able to write a digit so small that it only took up the smallest theoretically measurable volume of space, you would fill up the entire volume of the observable universe and you still would not even be close to writing down the whole number.

Now, here’s the real kicker. As insanely, ridiculously, ludicrously, stupidly large as Graham’s number is, we can always invent another number so large that Graham’s number is only negligibly bigger than 1, by comparison. Graham’s number is an infinitesimally small number if you are looking at the set of all numbers. The incredible nature of numbers is that no matter how large a number you describe, there are still an infinite amount of numbers which are even bigger.