What is the biggest number you can think of? Or better yet, what is the biggest number you can’t think of? Graham’s number is a quantity so mind-bogglingly large that if you tried to think of it, your head would quite literally turn into a black hole. The maximum amount of entropy you can store in your brain is related to a black hole with the same radius as your brain, and the entropy of this black hole carries less information than it would take to store Graham’s number in your head. The number is so large that the entire observable universe would not be able to store it, even if each digit was the size of a planck volume, the smallest measurable space. Graham’s number is a truly godly value, but where does it come from and why do we need to know about it? Come with me as we journey to the fringes of infinity as we explore one of the biggest number ever used constructively, Graham’s number.
Before we can consider Graham’s number, let us take a look at this math problem:
Let N be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n≥N, a complete graph K4 of one color with coplanar vertices will be forced.
If you are like most people who are not well versed in combinatorics, this question probably makes very little sense. Luckily, Hoffman proposed an equivalent analogy problem that is likely more accessible to the common person. The analogy problem is stated like this:
Consider every possible committee from some number of people n and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find N*, the smallest n that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees.
In a rather complex proof, Ronald Graham, an American mathematician, proved that the answer to this question is somewhere between 6 and Graham’s number.
To get an appreciation for how large Graham’s number is, we need to turn to “arrow notation”, proposed by the legendary computer scientist Don Knuth. First, let us begin with just one arrow:
So far, we are dealing with numbers we know and love. However, the numbers start to get really big, really fast. Let us explore two arrows now:
As you can see, adding just one arrow escalates things dramatically. However, 7.6 trillion is a number we can still fathom. It’s about equal to the number of bacteria on eight human bodies. When you add just one more arrow, the numbers become quite literally out of this world.
3↑↑↑3=3↑↑(3↑↑3)=33333333....333 where there are 7.6 trillion 3’s in the stack of 3’s
We aren’t even close to Graham’s number yet. However, we now have the tools to start making sense of Graham's number. Let us first define the first pivotal quantity, g1:
As you know by now, g 1 is absolutely gargantuan. We can now define g 2 :
g2 =3↑↑↑↑........↑↑↑↑3, where there are g 1 number of arrows
Naturally, g3 has g2 number of arrows, and so on and so forth. Onwards we go until we hit g64, which has g63 number of arrows. Finally, you’re done! Graham’s number is g64.
For a long time, Graham’s number was the largest number ever used in a mathematical proof. Nowadays, tree algorithms have produced bigger numbers, including the titanic TREE(3), but Graham’s number will always have a place in mathematical lore. For most of us, numbers this big will have no impact on our lives, but in our most philosophical moments, as we ponder the universe and what is beyond, we can remember that everything in existence cannot hold such a big value, and this colossal number is infinitely smaller than an infinite amount of numbers. Eternity is quite a lot bigger than you might think.
Gardner, Martin (November 1977). "Mathematical Games"
Padilla, Tony; Parker, Matt. "Graham's Number". Numberphile. Brady Haran.
Ron Graham. "What is Graham's Number? (feat Ron Graham)" Numberphile. Brady Haran