Big Numbers: Part 2
Read Part 1 here.
This is it.
This is the biggest, most insanely huge, mind-blowing, utterly incomprehensible number you have ever heard about, or ever will hear about.
You will, naively, optimistically, arrogantly, and ultimately, hopelessly, try to wrap your mind around it, only to discover that before we’re even ready to consider the magnitude of the beginning of Graham’s number, everything you ever thought you knew about big numbers was wrong, and that you are absolutely, totally, completely unable to grasp the very first term.
It’s impossible to overstate the sheer heaviness of this, and frankly, I find that I’m lacking the vocabulary necessary to impress upon you how profoundly immense this number is. You, my friends, are about to enter some very, very deep water. Your mind will twist and warp; it may even hurt, and you’ll be somewhat frustrated that you are having so much trouble coming to terms with the sheer massiveness of the very first layer of this number; and then you’ll sit there, mouth agape, utterly speechless, as you consider that there are 64 layers that make up Graham’s number, each progressively, horrifyingly, inconceivably larger than the one before it.
Grahams number is, in fact, so unfathomably huge that it cannot even be written using any sort of normal notation, such as exponents, or even exponents piled upon exponents. So, in order to talk about it, in order to even start talking about it, I’m afraid you’re going to have to suffer through a brief (I promise) course on Knuth’s Up-Arrow Notation. Don’t feel bad; I’d never heard of it before a week or so ago either.
Abandon All Hope, All Ye Who Continue Reading… I’m Just Sayin…
Preliminary Step 1 of 3: Tetration = Repeated Exponentiation
Just as exponentiation is the process of repeated multiplication (23 = 2 x 2 x 2), Tetration is the repeated process of exponentiation. It can be represented as an ever increasing staircase of exponents, but when the numbers get too big, as they are wont to do with this type of procedure, they become too unwieldy, and so we need to start using up-arrow notation to keep things clear. Tetration is represented by two up-arrows together like so ↑↑, which then represent a “power tower” of exponents piled upon exponents, so that a ↑↑ b = a power-tower of a’s, b-levels high. Thus, 3 ↑↑ 2 = A power-tower of 3′s, 2 levels high:
3 ↑↑ 2 = 33
And so on, so that:
3 ↑↑ 3 = 3^3^3 = 3^(33) = 327 = 7,625,597,484,987
3 ↑↑ 4 = 3^3^3^3 = 3^327 = 37,625,597,484,987
Note: It’s important to start from the top right exponent, and work your way down. 3^3^3 = 3^(3^3), which equals a little over 7.6 trillion, NOT (3^3)^3, which equals a little under 20,000. Big Difference.
Anyway, pretty simple, don’t you think? With ↑↑ notation, whatever the last number is behind the 2nd arrow is how many levels of exponents there are. Got it? Good. Now hang on, ’cause it only gets worse…
Preliminary Step 2 of 3: Pentation = Repeated Tetration
Continuing with the up-arrow notation, we can add yet another up-arrow to form a ↑↑↑ b, which can be understood as a stack of such tetration towers, each describing the size of the one above it:
For example, 3 ↑↑↑ 3 = 3 ↑↑ 3 ↑↑ 3 = 3 ↑↑ (3 ↑↑ 3) = 3 ↑↑ (7,625,597,484,987) = a power-tower of 3′s, 7,625,597,484,987 levels high, which is, as I think you’ll agree, is completely beyond comprehension.
Starting at the very top of that tower, start working your way down:
|7,625,597,484,987th term (top of the tower)||3|
|7,625,597,484,986th term||33 = 27|
|7,625,597,484,985th term||327 = 7,625,597,484,987|
|continue this process another 7,625,597,484,983 times||[insert big number here]|
That, my friends, is 3 ↑↑↑ 3
OK. Take a breath. Run that around in your mind a bit. Let it sink in. Still with me? Of course not, nobody is; nobody could be.
Sure, you can still understand the process by which this number was created, but the scope of the number itself escaped your imagination about the 3rd or 4th terms down from the top of that tower, to say nothing of the 7.6 trillion steps, each getting progressively larger, below.
Hang on; we’re not quite ready to get started yet. We still have yet another step ahead of us.
Preliminary Step 3 of 3: Hexatation = Repeated Pentation
Furthermore, a ↑↑↑↑ b might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:
So for Grahams Number, it looks like this: 3 ↑↑↑↑ 3 = 3 ↑↑↑ 3 (3 ↑↑↑ 3) = 3 ↑↑↑ 3 (a power-tower of 3′s, 7,625,597,484,987 levels high)
Which is3 ↑↑ 3 (3 ↑↑↑ 3) times. 3 ↑↑↑ 3 is, as you will recall, the horrifying number resulting from the power-tower of 7-trillion 3′s)
3 ↑↑↑↑ 3 = Really Big Number
Let’s pretend for a second that we have some sort of grasp on the number resulting from 3 ↑↑↑↑ 3. Consider for a moment how few up-arrows we’ve used:
3 ↑ 3 = 27
3 ↑↑ 3 = 7,625,597,484,987
3 ↑↑↑ 3 = the number resulting from a power-tower of 3′s, 7,625,597,484,987 levels high
3 ↑↑↑↑ 3 = You’re high if you think you can conceptualize this, or even think about it. In fact, stop looking directly at it; show your respect, bow your head, lower your gaze, and prostrate yourself before the sheer awesomeness of it.
4 up-arrows. Four. That’s all we’ve used, and if we’re being honest, we left reality with 3 up-arrows.
Got it? Still with me?
Good, because 3 ↑↑↑↑ 3 is where Graham’s number starts! That’s right; it’s where it starts!
3 ↑↑↑↑ 3 is the very first layer of Graham’s Number, known as G1
G1 = 3 ↑↑↑↑ 3 (4 up arrows)
G2 = 3 ↑↑↑↑…G1 up arrows…↑↑↑↑ 3
Stop right there. Before we continue any further, I want you to think about what just happened. In step #1, we constructed a number which is probably bigger than anything you’d ever considered before. Step #1 (G1) was/is actually beyond the point where I thought infinity hung out. And we did that with only four up-arrows. Now, in Step #2 (G2), we’re supposed to use G1 up-arrows! Whatever number G1 is, it’s huge. There literally aren’t words to describe it, and that’s how many up-arrows we’re supposed to use. And then we continue this same process 62 more times so that…
G3 = 3 ↑↑↑↑…G2 up arrows…↑↑↑↑ 3
G4 = 3 ↑↑↑↑…G3 up arrows…↑↑↑↑ 3
Etc, etc, until we reach G64
G64 = 3 ↑↑↑↑…G63 up arrows…↑↑↑↑ 3
I don’t know about you, but I’m speechless right now. Wordless. My brain is incapable of thinking about the enormity of this number on any meaningful level. And it’s not just me (and you), it confounds smart dudes like Eliezer Yudkowsky:
Graham’s number is far beyond my ability to grasp. I can describe it, but I cannot properly appreciate it. (Perhaps Graham can appreciate it, having written a mathematical proof that uses it.) This number is far larger than most people’s conception of infinity. I know that it was larger than mine. My sense of awe when I first encountered this number was beyond words. It was the sense of looking upon something so much larger than the world inside my head that my conception of the Universe was shattered and rebuilt to fit. All theologians should face a number like that, so they can properly appreciate what they invoke by talking about the “infinite” intelligence of God.
The number above was forged of the human mind. It is nothing but a finite positive integer, though a large one. It is composite and odd, rather than prime or even; it is perfectly divisible by three. Encoded in the decimal digits of that number, by almost any encoding scheme one cares to name, are all the works ever written by the human hand, and all the works that could have been written, at a hundred thousand words per minute, over the age of the Universe raised to its own power a thousand times. And yet, if we add up all the base-ten digits the result will be divisible by nine. The number is still a finite positive integer. It may contain Universes unimaginably larger than this one, but it is still only a number. It is a number so small that the algorithm to produce it can be held in a single human mind.
I know. It’s a bit much. This may help:
OK folks, Graham’s Number is a hard act to follow, so I’m not even going to try. I’ll go ahead and pass on a few links I stumbled across while researching this number and hopefully, that’ll save you some time if you’re interested in looking into it a bit further.
Well, maybe just one more thing: Imagine Graham’s number as a grain of sand. Then imagine an entire beach of these “Graham Granules”. No, that’s not enough; imagine an entire planet (like Arrakis) of sand, where each grain contains Graham’s Number.
Infinity is still bigger.
Wanna hear something else that’ll blow your mind? Some infinities are bigger than others, and there’s an infinite staircase of ever-increasing infinite sets.
Numbers: They’re a trip.
Values of 2 ↑m n
|Tetration||2||2||4||16||65536||2 ↑↑ n|
|Pentation||3||2||4||65536||A power tower of 2′s, 65,536 levels tall
6.0 x 1019,728
|2 ↑↑↑ n|
|Hexation||4||2||4||A power tower of 2′s, 65,536 levels tall
6.0 x 1019,728
|2 ↑↑↑↑ n|