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Approaching Infinity

One of the cool (and terrifying) things about math is that it’s almost a trivial task to construct a number which is not only larger than any number a human being will ever be able to use, but is also larger than any number that occurs in the Universe, even if you measure its mass in electron masses or its volume in Planck volumes.

The average human’s mathematical circuits are not that hard to overload. If I give you a deck of one hundred photographs and give you one hour to memorize all of them, you might very well be able to do it, but odds are you’ll miss some details. If I ask you to remember a 500-digit number, unless you’re a savant (like Daniel Tammet, who once recited pi to over 22,000 digits from memory, and who allegedly has a distinct mental image for every integer from 1 to 10,000), you’ll need some sort of fancy technique to do it. When it comes to counting objects, human beings don’t need very many numbers. I am one person. You (the reader), and I are two people having a sort of conversation. When I’m talking to a friend and somebody annoying butts in, that’s three people. If I have three apples, and I can ford a river to get to a tree with four apples, at the cost of dropping the ones I already have, I’ll do it. Numbers like three, five, and seven show up in most of the world’s myths and superstitions. Occasionally, you’ll get to seven or nine or even eleven, but rarely much farther than that. On a basic hunter-gatherer level, one hundred is a bit excessive. It’s only science, mathematics, and economy that have made a hundred of anything meaningful.

Take the number one trillion. That’s 10^12, or 1,000,000,000,000. (According to the American number scale, anyway.) It’s a big number. Draw a square. Divide it with ten vertical and ten horizontal lines. Divide each of those boxes with ten pairs of lines. Do this eight more times, and you’ve got a trillion squares. I should, of course, point out that, if you’re working on regular letter-size paper, by the time you get to a trillion boxes, the lines will be so close together that a virus will take up more than one square. Even if you drew your grid in the heart of Asia, where there’s a nice big squarish landmass 3,780 kilometers on an edge (stretching from the coast of China to the Caspian Sea along the east-west axis and from Siberia to the Himalayas on the north-south axis), the squares would be the size of a small closet.

But I already talked about a trillion at length in a previous post. A trillion green peas would just about fit on a football field (for most reasonable definitions of “football”). It’s a lot, but it’s a sensible, comprehensible number.

And a trillion is the largest number you’ll see mentioned frequently in serious astronomy, although there’s also the pleasant-sounding number “ten sextillion.” It sounds like something Lewis Carroll would’ve come up with. Ten sextillion is 10,000,000,000,000,000,000,000. That’s how many stars there are in the visible universe, according to Carl Sagan’s estimation. If you took the heart of Asia from before and divided it into ten sextillion squares, the lines would be separated by less than a hair’s breadth: about 30 microns. Cramped lodgings even for an amoeba.

But with nothing but digits and a few symbols, we can effortlessly construct numbers so massive that there’s no sensible way to describe how massive they are. Consider one trillion again. One trillion is 10^12. That’s the number 10 to the 12th power: 10 multiplied by itself 12 times. Here’s a number that will hurt your head: 10^(10^12). Simple: just take 10, and multiply it by itself one trillion times. I thought I’d be able to actually copy-and-paste that number, but it turns out that, in a 12-point font, I’d need almost 218 million pages (printed both sides). That’s a whole library’s worth of dictionary-sized books, just to hold the digits of a number I described using ten characters a second ago. If you divided the diameter of the observable universe into 10^(10^12) pieces, the distance between them would be 999,999,999,938 orders of magnitude smaller than the Planck length, which is just about the smallest length that makes sense, according to our current physics.

It’s easy as pie to create a scarier number. I’ll do it right now! (10^12)^(10^12). That is, (1,000,000,000,000)^(1,000,000,000,000). Multiply a trillion by itself a trillion times. This is where things not only get horrifying and migraine-inducing, but where they start to get strange: (10^12)^(10^12) isn’t really all that much larger than 10^(10^12). A trillion to the trillionth power is only 10^(12,000,000,000,000), or ten to the twelve trillionth power. That’s because of the way exponents work: the twelve in the first exponent gets multiplied by the trillion in the second exponent. Simple.

Don’t worry, though. With hardly any work, we can construct a function which will generate numbers as scary as you like with almost no effort.

Let’s say that the function M[1](a,b) is just a * b. Simple multiplication (which is really just adding a to itself b times, or vice versa). Let’s extend the concept by saying that M[2](a,b) is a^b, or a multiplied by itself b times. There’s no reason we can’t define M[3](a,b). It would just be a nested series of M[2] being applied over and over, exactly b times. For example, M[3](2,8) is 2^(2^(2^(2^(2^(2^(2^2)))))). You know you’ve wandered into the weird part of mathematics when you get a headache just from dealing with the damn parentheses…

There is no way to write M[3](2,8) out. As a matter of fact, there’s no way to write out its number of digits (which, after all, is only ten times smaller than the number itself). Here’s the closest I can get to writing M[3](2,8). Prepare for an absolutely horrific number-salad. I PROMISE this is the only time in this article that I’m going to do this:

[HUGE NUMBER GOES HERE]

If you’re mad at me right now, I understand. But if it makes you feel better, trying to work out that formatting has both given me a headache and made me physically nauseous. And I still screwed it up.

Nope. Couldn’t do it. It was just too damn hard to look at. Suffice to say, most of the article would have been digits, if I’d pasted that ugly bastard in.

But even the M[3](2,8) thing is unwieldy. We need better notation. Thankfully, Donald Knuth (who, at the age of nineteen, created an entire system of measurement based partly on the thickness of Mad magaizne, issue 26) provided a more elegant solution.

(I should, at this point, mention that the enormous number I copy-and-pasted above was so big that it was making the WordPress text editor lag, so I had to copy it into a Notepad file so that I can continue writing. You’ll never see it, but up above, I’ve written “[HUGE NUMBER GOES HERE]”. I have a headache.)

Knuth’s up-arrow notation is just like my M-notation, but it’s (slightly) easier on the eyes. No easier on the brain, though. In Knuth notation, a ^ b is replaced by a↑b, that is, a multiplied by itself b times. For example: 2↑8 is 256.

Things get scary very, very fast. a↑↑b is defined as a↑(a↑(…↑a)), where a is repeated a total of b times, with all the associated symbols. That’s too damn abstract for me, so let’s compute 2↑↑3. That’s 2↑2↑2, or 2^(2^(2)), which is only 16. 2↑↑4 is 2^(2^(2^(2))), or 65,536.

We can go further, though, although my headache is telling me to stop. a↑↑↑b is just a↑↑a↑↑a…↑↑a, with a repeated b times. 3↑↑↑2 is 3↑↑3 or 3^(3^(3)), which is about seven and a half trillion. 3↑↑↑3, on the other hand, is a number so large that I can’t express it in decimal notation. Hell, I can’t even express it using exponentials or up-arrows. It’s equal to 3↑↑3↑↑3, which is equal to 3↑↑(3^(3^3)), which is equal to 3↑3↑…↑3, where there are over seven trillion threes. That means a tower of exponents seven trillion threes tall. My word processor tells me that a superscript is 0.58 times the size of a regular letter, and by the time we get to the 7.6 trillionth three, it’ll be infinitely smaller than a proton.

That’s the level we’re at. Even trying to describe the typography of this ridiculous number is impossible.

What about a↑↑↑↑b? Well, 3↑↑↑↑2 is 3↑↑↑3, which we just saw was the most horrible thing in the world. 3↑↑↑↑3 is 3↑↑↑(3↑↑↑3). That is to say, 3↑↑↑3↑↑↑…↑↑↑3, with 3↑↑↑3 threes.

But I’m not letting you get off that easy. Let’s say that a↑[c]b means a↑…↑b, with c arrows in total. So a^b would be a↑[1]b. 3↑↑↑3 would be 3↑[3]3.

You know what I’m going to do. I can’t stop myself. If I knew any Medusas, I’d be a statue by now, because I wouldn’t be able to resist sneaking a peek.

There’s no turning back. It’s too late for you now. Too late for me.

Consider the number 3[3↑↑↑3]3. That’s 3↑…↑3, with seven trillion arrows. Think of the endless eternities of parentheses and arrows and evaluations, and that wouldn’t even get you close to the number of digits in this horror. Let’s call this horror X.

Now consider 3↑[X]↑3. Call it Y.

I imagine that my punishment in Number Hell will be evaluating 3↑…↑3, with Y arrows. And that’s infinitely smaller than 3↑[3↑…↑3 with Y arrows]↑3.

I’m not exaggerating for dramatic effect: I am genuinely smelling rotten eggs right now. I think I might have given myself a stroke. But before the aphasia sets in, let me introduce you to the Devil Incarnate: the Ackermann Function.

The Ackermann Function is the kind of thing they must’ve tortured Winston Smith with in 1984. It’s the reason some mathematicians walk around with that horrified thousand-yard stare. It’s an honest-to-goodness nightmare.

The Ackerman function is dead-simple. You write it A(a,b), for positive integers a and b. Here’s how you evaluate it.

If = 0, then A(a,b) = b+1

If a > 0 and b = 0, then A(a,b) = A(a-1,1)

If > 0 and b > 0, then A(a,b) = A(a-1,A(a,b-1)).

Simple rules. Not simple to apply. For instance, A(2,2) = A(1,A(2,1)) = A(1,A(1,A(2,0))) = … a horrifying mess of parentheses that ultimately gets you to 7. At least it’s a sensible number. So is A(3,2). It’s 29. A(4,2), on the other hand, is over 19 thousand digits long. When I typed Ackermann(4,4) into WolframAlpha, it actually told me “(too large to represent).” It’s always nice when a computation engine built by one of the masters of symbolic computation says “Hell with this. I give up.”

You know how evil I am. You know what I’m going to do. You know how psychotic and depraved I’ve become after looking at unfathomable numbers for an hour.

The Number of the Devil isn’t 666. It’s Ackermann(666↑↑↑↑↑↑666,666↑↑↑↑↑↑666).

Sleep well. I know I won’t.

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