Resonant Constellation

Every once in a while, I entertain myself by learning about random math stuff. A recent example is my foray into Fibonacci sequences which I mentioned in a previous post. This time, my friend Pete mentioned a peculiar number, called Graham’s number. As far as I can tell, this is the largest number to ever be used in a serious mathematical proof.

I know what you math-nerds out there are thinking: larger, even, than a Googol? (), or a Googolplex? (). Yes, my friends, Graham’s number is inconceivably big. It makes a Googolplex look like a mere handful. Interesting note: when I was little, no more than five, I remember writing out a Googol on an etch a sketch and trying to explain it to my grandparents. I was a weird kid.

Graham’s number is so absurdly large that there are not enough particles IN THE UNIVERSE to express it via any standard notation. Think about that-go outside to a high place and look around. Then think that everything you can see is made of inconceivably tiny particles which are so small, they cannot be seen by the human eye, nor any optical magnifier that has ever, or will ever, be made. Look at your hand-there must be millions, perhaps billions of particles in your hand alone. And yet including everything you can see, much less the entire friggin’ universe, there aren’t enough of these unfathomably tiny particles to write out this number, even using a series of exponents. Wow.

So how do you write it down? Well, mathematicians are relatively creative people (if not entirely practical), and they’ve come up with intriguing ways of expressing large numbers. One way is called “Up-arrow notation,” in which the number is expressed by a series of rows including numbers and arrows which signify computational steps to arrive at the number itself. Each higher row is predicated on how many arrows are in the last row. This is the only way to express Graham’s number. To show just how depressingly large this number is, you still can’t even express just the first row of up-arrow notation with all of the particles in the universe. There are 64 total rows.

What I can tell you about it is that it ends in the string “…262464195387″, where the … represents a whole lotta other numbers. So why would anybody in their right mind need such a comically large number? Were these mathematicians perhaps compensating for something (say, the budget differential between their department and a useful department like Astronomy)? From wikipedia:

Graham’s number is connected to the following problem in the branch of mathematics known as Ramsey theory:
Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices which lie in a plane?
Graham & Rothschild [1971] proved that this problem has a solution, N*, and gave as a bounding estimate 6 ≤ N* ≤ N, with N a particular, explicitly defined, very large number; however, Graham (in unpublished work) revised this upper bound to be a much larger number. Graham’s revised upper bound was later published — and dubbed “Graham’s number” — by Martin Gardner in [Scientific American, "Mathematical Games", November 1977].
The lower bound was later improved by Exoo[2003], who showed the solution to be at least 11, and provided experimental evidence suggesting that it is at least 12. Thus, the best known bounding estimate for the solution N* is 11 ≤ N* ≤ G, where G is Graham’s number.

Wow, that’s so useful (/sarcasm). It must have been a profoundly depressing result: “So, Ronald, how’s that proof you’re working on coming? Did you ever get a result?”
“Yeah, it’s somewhere between 11 and wharrgarbl.”

Even as an astronomer (a field which is known for large numbers, even coining the term ‘astronomical’), I’d probably just call it “effectively infinite for all foreseeable/sane purposes.” No wonder it was published in “Mathematical Games.”

In all honesty, stuff like this is probably good for the math departments-it will keep them at their desks during the inter-departmental war. I for one know that the physics department has long desired to vaporize the chemistry department with a large laser array. The mathematicians will likely be too busy coming up with crazy stuff like Graham’s number to be bothered by such events.

Last minute note:
I have just discovered that there is a larger named number, called TREE(3). It is part of a sequence of numbers: TREE(1)=1, TREE(2)=3, TREE(3)=Makes the word big seem hackneyed. Apparently, Graham’s number is “unnoticeable” next to a lower bound to TREE(3), which is itself unnoticeable next to TREE(3). I hear TREE(3), will anybody go to TREE(4)? Sold to the man in the straightjacket.

There are even bigger numbers yet, obviously, including “Totally Indescribable Cardinals,” (yes, that is the formal name), Transfinite numbers, and all sorts of made up names (Bajillion, Frumptillion, etcetera). For an incredibly humorous article on made up, unspecified numbers, look here.

Of course, infinity puts all of these so-called large numbers to shame. Compared to infinity, they might as well be 0. Maybe you should be careful next time you use the word “infinite” in casual conversation. You probably doesn’t mean that many.