The study of really, really big numbers


'With names like "Bowers' Exploding Array Function", how could you not like this stuff?'
— someone, probably.


Think of a number. Go on, think of the biggest number you can, as big as you can imagine. Think of a number bigger than a billion, or a trillion. Bigger than a billion trillion. Most people struggle, as I did when I played this game as a child. We ended up with invented numbers like a bajillion, simply because in the everyday world we don't deal in numbers that big.

In the field of cryptography there are some pretty big numbers, gained by multiplying huge prime numbers together, but we rarely see them, leaving then to our computers to calculate and handle them. These numbers have lengths we can count, being a few thousands of characters. But there are people who deal in improbably large numbers which fall way outside the size of these numbers. These are numbers you can't possibly ever write down without using mathematical tricks. These are numbers that would literally blow your mind. The field is not a formal one, but it exists. This is the world of Googology.


As I grew out of the childhood game, I discovered that really big numbers often don't even have names, rather they are expressed as powers of other numbers. Some do, and the first big named number I came across was the googol, 10100. This is one followed by a hundred zeroes: 10,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000,​000. This number is mostly famous for inspiring the name of the search engine Google; it has no real significance in the field of mathematics, it exists only for the purpose of being a big number. But the googleplex is even bigger. It is 10googol, or 1010100.

To put these numbers into perspective, the number of atoms in the observable universe is ~1080. To publish all the digits of a googolplex would take 1094 400-page books, with a total mass of ~1093 kilograms, far exceeding the mass of all matter in the known universe (~2 × 1052kg).

These numbers are clearly big, really big. But to get really stupidly huge numbers we need more than exponentiation, and we need to move into really abstruse areas of mathematics. Now I'm no mathematician, but numbers do fascinate me. A few years ago I had an introduction to the concept of stupidly large numbers through a Numberphile video on Graham's Number. (I've linked that video and others below; they are informative and entertaining to boot.)

To solve a puzzle in graph theory, Ronald Graham came up with a solution whose upper bound was vast. After becoming well known in Martin Gardner's column Mathematical Games it became known as Graham's number, and was so stupidly large that even a power tower of exponentiation could not adequately express it. Instead he had to use a system devised by Donald Knuth, known as arrow notation. This uses up arrows (↑) to denote levels of exponents. The more arrows, the higher the tower and the more ridiculous the number. Graham's number is so much larger than even a googolplex that it boggles the mind even to think of it. For the longest time, this was the largest number to have been published in a mathematical proof, and was in the Guinness Book of Records as such.

Even as my head was spinning from this, I discovered Tree(3), an even bigger number. Occasionally I stumble across others; Rayo's number being the latest. Mexican professor Agustin Rayo devised the number to win a "big number duel" against Adam Elga at MIT on 26 January 2007.

The best way to explore them if you're a non-mathematician is to watch various Numberphile videos, some of which I have linked below. It's beyond me to explain them, I can only look on in fascinated wonder.

There are others, equally mind-buggeringly large and impossibly absurd. Of course, there's a website that collects, catalogues, celebrates and explains them. These people are nuts, and of course the whole field is suitably spoofed at the Fictional Googology Wiki. Of course you can count on the internet to take it to even more absurd levels.




Googology wiki
¹https://en.wikipedia.org/wiki/Googolplex#Size

Numberphile video on Graham's number
Numberphile video on TREE(3)
Numberphile video on Rayo's number

⍼ xclip -o | wc
     23     697    4249

C-Dawg says re Googology: An interesting thing about stupidly large numbers: because we can't write them out, sometimes we can't even say which of two of them, described in different systems, is larger!

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