There’s an odd branch of mathematics called googology, in which people identify extremely large numbers and give them names. You might, for example, claim that a one followed by three billion and fourteen zeroes is called a skoodyhoodenfroody, and then someone else comes along and asserts that if you multiply that by seventeen thousand four hundred and twenty it becomes a skoodyhoodenfroodyplex, and so on.
All very entertaining, of course, and people enjoy it. However, staggeringly gigantic numbers also appear in more serious mathematical studies. One of the most famous, and one of the most difficult to contemplate, is Graham’s number.
It’s named after Ronald Graham, who proposed it as the largest possible answer to a certain problem in multi-dimensional geometry. The actual answer still isn’t known, but it’s now believed to smaller than this (though still enormous) and certainly higher than twelve, which leaves a lot of room for further research.
In principle, but definitely not in fact, Graham’s number is easy to calculate. It’s all about threes. You start off by writing a number three, and there you are – that’s the first line done. In every subsequent line, you multiply three by itself the number of times shown in the previous line, so line two simply shows three multiplied by itself three times, which equals 27.
Following the same process, the number you reach on the third line is 7,625,597,484,987. If you want to tell a friend about Graham’s number, you can become a little vague at this point and say the answer is “about 7.6 trillion”. An inaccuracy of over 25 billion might appear concerning, but it’s going to become very trivial very quickly.
Now, 7,625,597,484,987 is a number with thirteen digits. For obvious reasons, nobody knows exactly how many atoms there are in the observable universe, but it’s estimated to be a number with around eighty digits. There’s an awful lot of empty space in the observable universe, and if you jammed that space full of atoms, assuming you could find enough, the number would now have around a hundred and twenty digits. (This might not seem like much of an increase, but the second number would in fact be ten thousand billion billion billion billion times larger than the first, which is actually quite an increase after all.)
The reason for that brief diversion is to illustrate how quickly the calculation of Graham’s number develops from now on. On the fourth line, we multiply three by itself 7,625,597,484,987 times and arrive at a number not with eighty digits, not even with a hundred and twenty digits, but with 3.6 trillion digits. There is no adjective adequate to describe this. Large, gigantic, astronomical and even universal don’t come close. There aren’t that many things in existence, because there isn’t room for them.
You might have noticed that not only are the figures expanding vastly, the rate at which they are expanding is doing the same. In comparison with 7,625,597,484,987 on the third line, 27 on the second line was as close to zero as makes no difference, and is itself effectively zero compared with the monster on the fourth line, which is in turn basically zero relative to the majestic horror on the fifth line.
That horror is called G, which stands for Graham’s number. So there we are. Except no, we’re not, because it isn’t really called G at all. It’s called G(1). We have to keep going, calculating yet more numbers, every one of them so colossal it makes the immediately preceding number seem invisibly small, until we reach G(64). That is Graham’s number.
There is not the slightest hope of writing it all out. The universe isn’t big enough for that, and according to some hypotheses it won’t last long enough. But it has been possible to calculate what the last five hundred digits are, which is something. (The last is 7.) Considered separately, those five hundred digits make up a number which is useless for counting because there isn’t that much stuff to count, but removing them would make about as much difference to the scale of Graham’s number as the difference in the risk of Mount Kilimanjaro crumbling under the weight of a mosquito, depending on whether or not the mosquito blew its nose before it landed.
So Graham’s number must be the largest there is, right? Oh, my sweet summer child. No. No, it isn’t. There’s a series of numbers called TREE, in which TREE(1) equals one and TREE(2) equals three. After that, it gets complicated. The value of TREE(3) can’t even be guessed at, but it has been demonstrated that it must be vastly, incomprehensibly, terrifyingly larger than Graham’s number. Imagine what TREE(4) must be like, if you dare.
Even TREE(3) is tiny compared with Rayo’s number, which is said to be the largest ever named. It is, in practical terms, absolutely impossible to calculate, but there’s a set of instructions on how to do this, so whatever it is, it definitely exists.
It’s tempting to say that Rayo’s number must be closer to infinity than TREE(3), which must be closer to infinity than Graham’s number. That makes no more sense than comparing their relative closeness to Thursday, or to Rio de Janeiro, because Thursday and Rio de Janeiro are not numbers, and neither is infinity. And since this article is only about numbers, we can safely avoid that question and wander off to do something else instead.
Ronald Graham (1935-2020)
Photo copyright Cheryl Graham
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