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.

Here I am almost seven hours later having completed a grand total of half of the work. I wasn’t the only one working on this, either, there was a rotating group of about 6 people working on the same problems. The best part of it all was when we thought we had one problem finished, so almost everybody left, and then the remaining two of use figured out that it was wrong and spent three hours getting it right, which we eventually did before going to fetch dinner. Of course, when getting dinner, we realized that it still wasn’t right, so we got back to work.

I wonder what the outside is like, it’s been so long. Please, help us.

3.14159265358979323846264338327950288419716939937510
5820974944592307816406286208998628034825342117067982
1480865132823066470938446095505822317253594081284811
1745028410270193852110555964462294895493038196442881
0975665933446128475648233786783165271201909145648566
9234603486104543266482133936072602491412737245870066
0631558817488152092096282925409171536436789259036001
1330530548820466521384146951941511609433057270365759
5919530921861173819326117931051185480744623799627495
6735188575272489122793818301194912983367336244065664
3086021394946395224737190702179860943702770539217176
2931767523846748184676694051320005681271452635608277
8577134275778960917363717872146844090122495343014654
9585371050792279689258923542019956112129021960864034
4181598136297747713099605187072114999999…and so on.

Ha ha! Pi is really irrational, it doesn’t continue with a string of nines forever (or does it? You’ll have to find out…). But anyway, pi day is a really special day for a lot of people. Today is Albert Einstien’s birthday for one, which is reason enough to celebrate, but on a more personal note, today is the anniversary of two very good friends of mine (one is a math teacher and the other is a physics student). For all these reasons, Pi day is one of my ever growing list of “Science Holidays,” (for example: Apple day or Gravmas, Issac Newton’s birthday. Which also happens to be Christmas.) I urge you to celebrate-bake a pie in the shape of pi, do some relativity, study brownian motion, there’s tons of things you could do!

Note: The point in pi at which I stopped (…999999…) is a very special sequence, known as the Feynman point. In a lecture once, Feynman said he wanted to memorize pi up until a point when he could say “Nine, nine, nine, nine, nine, nine, and so on,” implying that pi is rational and ends in a repeated sequence of nines. This is typical Feynman awesomeness. The fact that this sequence occurs so early in pi (starts at digit 762) is truly intriguing. Read more here.

I found this article on my favorite humorous news aggregator, FARK.com under the headline “100 – [10L - 7F + C(k - C) + T(m - T)]/(S – E) = OM NOM NOM.” The formula (for those of you too lazy to read the article) represents a scoring system (out of 100) for making the perfect pancake. This is an admirable goal, however the mathematician who concocted this formula (and pancakes) is in egregious error. Of course this just goes to show that mathematicians should stay out of experimentation and let those of us who actually know what we’re doing handle it. Here’s the formula again:

Those of you who are not used to looking at mathematical equations (or in my case reading them more than you read actual english) may be intimidated, but rest assured that this is a very easy to understand equation. The variables (letters) mean the following:

L is the number of lumps in the batter
F is the “Flipping score”
C is the consistency of the batter
k is the “ideal” consistency of the batter
T is the temperature of the pan
m is the “ideal” temperature of the pan
S is the length of time the batter stands before cooking
E is the length of time the pancake sits before being consumed

Then we can see what each of the terms mean here. Overall, since the score is out of 100 and you’re subtracting everything from 100, you want the combination of all the variables to be as close to zero as possible. It now becomes an optimization problem: how do we make things zero? This is easiest to do by looking at each of the terms individually and this is where the formula starts to break down.

The easiest thing to see is that the denominator of the fraction should be as large as possible. The larger the number you divide by, the smaller the result. That means that should be really big. You can do this two ways, make small and/or make big.
Sounds easy enough, right?

But what does it mean physically? Well, since S is the time the batter sits, and we want to make it arbitrarily large according to the formula, we should just let our batter sit out for years and our pancakes will be delicious. Also, since E is the time we let the finished pancakes sit after done and this should be minimized, the formula suggests that we should shove our faces straight into the sizzling pan without taking it off the burner for extreme heavenly pancake delight. Beginning to see the problems here?

Next, let’s look at the numerator. You’ll notice two similar terms: and . These terms should be as small as possible to make the numerator smaller. They also both represent an optimization of consistency and temperature, however, the tricky part in this is the so-called “ideal” parameters. These are completely arbitrary values, known to many as “Fudge factors” (tangent thought: chocolate chip pancakes=good idea) and really carry no experimental weight unless you actually assign a number to them. An interesting experiment would be to isolate the other variables and simply test one or the other and see what kind of relation the awesomeness of pancakes has to these variables. Sadly, this did not happen.

The other two variables are also fudge factors. Maybe somebody likes more lumps than somebody else and the so-called flipping score is not scientific at all. Rather it is a binary value: did you flip the pancakes correctly? YES, NO. Tsk, tsk.

I also would add some terms, for example: bubbles. Do you like your pancakes to have random air pockets in them (they form as you cook)? Or do you smash your pancakes down and tap the pan to develop wonderfully homogenous, purely awesome, irresistible airless pancakes? How many times should you hit the pancake? How hard? Clearly, this calls for experimentation. If you see me on the side of the road selling hundreds of pancakes, you’ll know why.

It’s thoughts like these that keep me out of normal company.