There are numbers that describe things like how many of atoms are there in the visible universe, (the estimate is 10^{80}) or the number of seconds in the last 13.7 billion years (which is on the order of 10^{26} seconds), and then there are numbers that are truly large. Graham’s Number was the first one I ran across. From 1,000,000 to Graham’s Number. (UPDATE: The formatting that WordPress is doing around the Power Towers is a bit wonky on the main page. Click “Continue reading” below, or the post title above.)

Huge numbers have always both tantalized me and given me nightmares, and until I learned about Graham’s Number, I thought the biggest numbers a human could ever conceive of were things like “A googolplex to the googolplexth power,” which would blow my mind when I thought about it. But when I learned about Graham’s Number, I realized that not only had I not scratched the surface of a truly huge number, I had been incapable of doing so—I didn’t have the tools. And now that I’ve gained those tools (and you will too today), a googolplex to the googolplexth power sounds like a kid saying “100 plus 100!” when asked to say the biggest number he could think of.

See the end of this post for a refresher on the definition of a googol and a googolplex. And contemplation on what is big…

So why do we care about Graham’s Number? Aside from the fact that it is big? There’s a problem which isn’t quite of the form, “What is the minimum number of dimensions you need to consider before hypercubes in that dimension MUST exhibit a certain property?” Graham was trying to answer that question, but he could only put bounds on it. Today we know the answer, or the minimum number of dimensions, is somewhere between 13 and Graham’s Number. It is from a branch of mathematics “that studies the conditions under which order must appear in relation to disorder.” Let’s build it.

If you’ve gotten this far, I am going to assume you know addition, and multiplication and exponentiation. We’ll start in on Knuth’s Up-Arrow notation.

Kunth’s Up-Arrow notation starts with exponentiation. a↑b is equivalent to a^{b} or a X a X a… X a where there are b occurrences of a. We need the up arrows, because it is going to get crazy.

Next up, is tetration, where a↑↑b will be defined in a minute. First let’s review something you know.

- a X b = a + a + a +… + a where there are b occurrences of a.
- a↑b is a X a X a… X a where there are b occurrences of a.

So a↑↑b = a↑ (a↑ (a ↑ (a…↑a))) where there are b occurrences of a.

Lets take a look at an example or two. 3↑3 = 27. Not too surprising. 3↑↑3 =3↑(3↑(3)) = 3↑27 = 7,625,597,484,987. See how things are getting big really fast? 3↑↑4 is equal to 3^{7,625,597,484,987} which is a 3.6 trillion digit number. (Type that into a calculator, or calculator app, and see what happens) Bigger than a googol, but less than a googolplex.

These are called “power towers” because if you write them the way you learned in middle school (or junior high, if you’re old enough) then they look like this.

3

^{333}is the same as saying 3 ↑ (3 ↑ (3 ↑ 3)). We bundle those 4 one-arrow 3s into 3 ↑↑ 4.

The tower power of (3 ↑↑ 7,625,597,484,987) would reach the sun.

And you can see how the superscript notation loses its utility after a tower of 3 high. But we haven’t even gotten started… Or as they say, Shit is about to get real. Real big.

Next is Pentation (↑↑↑) is defined as a↑↑↑b = a↑↑(a↑↑a(↑↑…a))) where there are b copies of a. Or b power towers of a.

Hexation is ↑↑↑↑ where a↑↑↑↑b is defined to be a↑↑↑(a↑↑↑(a↑↑↑(a… ↑↑↑a))) where there are b copies of a.

See the link for a longer-winded explanation of these 2 ideas. Can you guess where we go next? First, let us stop and take a look at where we are…

3 = 3

3^{3}= 27

3^{33}= 7,625,597,484,987

3^{333}= a 3.6 trillion-digit number, way bigger than a googol, that would wrap around the Earth a couple hundred times if you wrote it out

3^{3333}= a number with a 3.6 trillion-digitexponent, way way bigger than a googolplex and a number you couldn’t come close towritingin the observable universe, let alone multiplying out

So if we look at the general case of this notation, which is Knuth’s up-arrow notation if you’re interested, we get the following.

Now that we understand the notation, we can start on Graham’s Number. Fist a few preliminaries.

g_{1} is defined as g1 = 3 ↑↑↑↑ 3.

g_{n} is defined as 3↑^{gn-1}3. Or… (Where that fancy capital N is the set of Natural Numbers – or for our purposes the positive integers.)

Graham’s Number = g_{64}. So a pretty damn big number. Way bigger than things like a googolplexian, or the number of Planck distances in the visible universe.

And remember that original problem… The boundaries on the number of dimensions is 13 at the low end, and Graham’s Number at the high end.

There are numbers that are bigger than Graham’s Number. Tree(3), and Rayo’s Number come to mind. We may get to them eventually, though I don’t love them, the way I love Graham’s Number.

So here’s a 9 minute video on Graham’s number. Numberphile doesn’t usually let me down, but I don’t think they do a good job with Graham’s Number. They are mathematicians after all, and they aren’t always good at communicating with the man or woman in the street. Still here you are.

As for the review…. A googol is 10^{100}. So bigger than the number of atoms in the universe. A googolplex is 10^{googol}. The only thing is, while these are big, they aren’t really useful in anyway. See the text of the article linked at the top for the history of the googol, which is older than the history of the Google.

The post, Fun With 4 Dimensions is at this link. In case you need a refresher on the definition of a hypercube.

And remember, this number is so much bigger than space…

Power Towers should not be confused with the Tower of Power horn section… famous for accompanying Little Feat on the album

Waiting for ColumbusLikeLike

“Spanish Moon” by Little Feat from

Waiting for Columbusfeaturing the Tower of Power horn section.LikeLike