by Jonathan Kujawa
To say math is about numbers is like saying writing is about words. You can use words well or badly, but in the end it is the things and ideas they represent which are important. Just so with numbers.
I have a clear memory of learning in middle school that the plots of Shakespeare's plays were nothing but retreads of older tales. With the certainty of youth I wrote off Shakespeare as nothing but an over-glorified plagiarist. It took a few years to come around to the realization you don't read Shakespeare after all these years for the plots, but for his deep study of human nature and unmatched skill with words. Will could put the right words in the right order and really zing: “How well he's read, to reason against reading!” or “Things won are done, joy's soul lies in the doing”.
But, as every logophile knows, words as words are fun, too. Every language has a rich vocabulary. It can be great fun to flip through a dictionary and find words which are unexpected, funny sounding, intriguing, insightful, etc. I am terrible at foreign languages. Not only am I tone deaf to pronunciation, but my memory always locks onto the words which are interesting, obscure, and utterly useless in everyday communication. Twenty years after studying Hungarian I remember hepehupás is the word for bumpy. Why? For the singularly silly reason that saying it makes you sound like you're bumping along a rutted country road.
In just the same way there are numberphiles who enjoy curious numbers. They have raging internet arguments over the great Pi vs. Tau controversy. They know why 1729 is called a “taxi cab” number. Or why Google is called Google.
I myself have a fondness for what I call Brobdingnagian Numbers. Brobdingnag is the land of giants in Gulliver's Travels. When doing math you find yourself using such staggeringly large numbers that you become numb to how big they really are. But it's worth taking note of them and it can be quite fun to collect the ridiculously large numbers you come across in your travels.
My snarky middle school self wouldn't have cared about such things since you can always add one to get a bigger number. But we've already established that he was an idiot who didn't know how to appreciate the finer things in life and we choose to ignore him.
Small Numbers
How big does a number have to be to be Brobdingnagian? Let me give you an example or two of small numbers. The estimated US federal deficit for 2015 is $564,000,000,000 [1]. The US federal debt is around $18 trillion; that is, $18,000,000,000,000 [2].
Remember, these are small numbers. If we keep writing numbers out like this, things are going to completely get out of hand. Rather than waste precious internet electrons on vast numbers of zeros and commas, let's just keep track of how many zeros there are in total. Instead of 564,000,000,000, I'm just going to round this up to 600,000,000,000 and count this as 11, because it's some number followed by eleven zeros. Using this rule, the federal debt counts as a 13 [3].
You are probably objecting that these are outrageously bad estimates. It's true we're wildly rounding by billions and trillions, but where we're going we won't care about such trivial amounts. Bill Gates doesn't bother to pick up pennies from the sidewalk and we won't bother with a few trillion here or there.
Medium Numbers
If these are small numbers, what are some medium sized numbers? A great mathematical game is Sudoku. Probably all of us have played this game. You have to fill in a nine by nine grid of empty squares in such a way as to have every row, every column, and each three by three grid filled with the numbers one through nine. An interesting question is exactly how many legal Sudoko puzzles are there? For a very rough estimate, we can say that each of the 81 boxes has to be filled with one of nine numbers, so at the very most there is
981=196627050475552913618075908526912116283103450944214766927315415537966391196809
of them. That is, in our shorthand of only counting zeros after rounding, there is definitely no more than 77 of them.
But this count overestimated the number of legal puzzles. After all, if you put a one in the upper left corner box, the rules say you can't have another one in the top row or leftmost column and still have a legal Sudoko. You can count more carefully and determine exactly the number of Sudoko puzzles. In our shorthand there are 21 of them.
Even with medium sized numbers it's easy to forget how jaw-droppingly large they are. A very rough estimate is that there are, in our shorthand, 18 grains of sand in all the beaches of the world. But remember, our shorthand is only counting the number of zeros. So this means if you are the meticulous sort of person who might like to write down every Sudoko puzzle on a grain of sand, you will need no less than the beaches from 1,000 Earths to do the job. If you are undaunted, I should also mention that the Universe is approximately 13.75 billion years old. If you wrote one puzzle per second, you'd need over 15,000 Universe Lifetimes to record them all!
We know math isn't about numbers, so why did I say Sudoku is a mathematical puzzle? What makes it mathematical is not the numbers in the boxes, but rather what you do when you fill those boxes: you make deductions, you eliminate possibilities, you consider scenarios and follow them until they lead you to an impossibility or a new insight. When it's all said and done you have the satisfaction of seeing all the pieces fit together into a beautiful pattern. A pattern which was hidden when you started. Given how many puzzles there are, if you start with a randomly created Sudoku the odds are staggeringly high you are the first person to ever solve that particular puzzle.
The puzzle isn't about the numbers in the boxes, it's about the relationships between them. What's allowed, what's required, and what's impossible. This is what mathematics is all about and what mathematicians do every day. After all, as Ron Brown shows, you can just as well play Sudoku with something other than numbers:
What are some other medium-sized numbers? Last year at 3QD we discovered that there are, in our shorthand, 63 ways to arrange a deck of fifty-two playing cards. You are more likely to win the lottery five times in a row than to find a well shuffled deck of cards in an arrangement which has ever before occurred in the history of card playing. The number of atoms in the observable Universe is estimated to be around 80. A googol is a 100 and is Google's namesake. The number of possible chess games is approximately 120 and the number of go games is approximately 360. Remember, since we're only counting zeros here this means if you had a variant of chess for each atom in the universe, the number of possible games in all those variants would still be far, far less than the number of possible games of go. No wonder computers can beat the very best chess players but still struggle at go!
Large Numbers
If the number of atoms in the universe is at best a medium-sized number, then what the heck is a large number? For this, we'll need Donald Knuth's “up arrow” notation. It starts with good ol' exponents.
As we know, we write 103 as a shorthand for 10 x (10 x 10) = 1,000. The exponent three means we multiply ten by itself three times. For compactness, Knuth wrote the exponent as a single arrow:
So a single arrow followed by a three means we should multiply ten by itself three times. Knuth then tells us that we should understand a double arrow followed by a three to mean doing a single arrow of ten with itself three times:
Holy Moses! This number has 10,000,000,000 zeros and so counts as ten billion in our shorthand. Our medium-sized numbers seem tiny in comparison! But of course there is nothing to stop us at two arrows. If we do three, then that should mean we double arrow ten with itself three times:
This in turn means we should up arrow ten with itself 1010,000,000,000 times! If we write it with exponents, that's the number you get by doing
where there is a tower of tens in the exponents is 10 billion tens high. This is why Knuth introduced the up arrow. In the end you're just doing exponents, but the up arrow is a vastly more compact way to write them.
This number is already so vastly large that our shorthand no longer saves us. In our shorthand this number would be whatever you get if you do the same tower but with a mere 9,999,999,999 tens in the tower. I shudder and weep at the thought of
But surely Knuth's up arrows are a monstrosity used only to give nightmares to first year graduate students. Right?
In fact it comes up in serious math research. More than a year ago here at 3QD we talked about Ramsey Theory. It is the area of math which says that in a sufficiently large collection you cannot help but have order and structure. Depending on which structure you would like to have, the collection may have to be very large indeed! Ramsey Theory is shot through with gargantuan numbers.
In the 1970s Ron Graham solved a problem in Ramsey Theory. He wanted to know if you were to color the edges connecting the corners of a hypercube (i.e.. a cube in higher dimensional space) with red and blue, how high of dimension would you have to be to ensure you could find a certain specified structure? Graham showed that the dimension you needed was somewhere between six and what's now called Graham's Number. Graham's Number, G, is the very real and very large number
This means Graham's Number is three up arrow three, where the number of up arrows between the threes is given by another up arrow calculation. And the number of up arrows in that calculation is in turn given by an up arrow calculation. And so on for 64 layers! The number at the very bottom layer is already Brobdingnagian but all it is giving you is the number of arrows in the second layer. By you get to the top and finally compute G you are beyond what we can even imagine. Even so, this is an honest to goodness actual finite number. It appeared in the Guinness book of world records as the largest number used in a serious math problem [5].
Some progress has been made on Graham's original problem. We now know that the number of dimensions required is somewhere between 13 and 
Amazingly, even though there isn't enough room in the universe to write down Graham's Number, we do know that if you were to write it down you would end with …262464195387. It turns out that computing the last few digits isn't so difficult.
Numberphile has a pair of fantastic videos about Graham's Number. The first one gives a brief explanation of the Ramsey theory problem Graham was working on when he used Graham's Number, and the second one explains the up arrow notation and gives you some idea of how monstrous Graham's Number really is [6].
Lastly, I wanted to be sure to mention Richard Schwartz's book, Really Big Numbers. Close readers of 3QD will recall that we came across Dr. Schwartz last year in his work on the mathematics of billiards. Not only does Dr. Schwartz do cool math, he writes children's books about math. They are fantastically illustrated in Dr. Schwartz's singular style and are a big hit with math-minded kids everywhere. His most recent is on Brobdingnagian Numbers and is the perfect gift for the budding numberphile.
[1] That is, the amount they will borrow in 2015.
[2] That is, the total amount owed. To be more precise, as of this writing it was $18,346,890,978,321 (see the US debt clock for the latest figures).
[3] Those who know will recognize that our shorthand is just noting the exponent in scientific notation and disregarding the rest.
[4] From wikipedia.
[5] Amazingly, even larger numbers have since been used in the solution of similar sorts of Ramsey Theory type math problems.
[6] Note that there is an error in the Numberphile video (which they themselves readily point out). Their number is staggeringly large, but Graham's Number is even larger!