From Graham’s number to the Googlelplex, the largest known numbers ever “invented”

Which is the largest number in the world? As we know, this is a question that actually doesn’t make much sense, given that if there were a number larger than all of them, we could always overcome it by adding 1. However, there are some large numbers that compete for the title of the largest number ever “invented”, that is, for which we invented a name. These are huge numbers like the googleThe googolplexThe Graham numberThe Rayo’s numberand the number TREE(3). Some of these numbers are so large that all the space in the universe would not be enough to write them down. Let’s see how big they are, and which of them is the biggest of all.

The Google

A googol is equal to 10¹⁰⁰that is, 1 followed by 100 zeros and owes its name to the mathematician Edward Kasner who, around 1930, called it this at the suggestion of his 9-year-old nephew. We can immediately notice that googlel looks a lot like Googleand it’s no coincidence! The inventors of the famous search engine, they were inspired by Googleto give an idea of ​​the immense amount of information that Google would bring to anyone’s fingertips.

Apparently a googol is not huge, and in fact it is the smallest of the numbers we will see in this article, but if we try to compare it with things we know we realize how huge it is for our “human perception” anyway! Consider that it is estimated that there are about 10 in the Universe⁸⁰ particles. This means that the Google number is 10²⁰ times, that is, 100 billion billion times, all the particles in our Universe.

The googolplex

The googolplex it was initially defined, again by Kasner, as

1 followed by as many zeros as we can write before we get tired.

This definition gives a good idea of ​​a huge number, too large to be written in full, but it is a bit vague and has been improved, again by Kasner, who defined it:

1 followed by a googol of zeros, that is 10^google

This number, in 1940, was recorded in the Guinness Book of Records as “the largest number that had been named” (today this is no longer the case), and it is so large that, according to Kasner himself and his colleague James Newman, the distance between us and the most distant star, there and back, would not be enough to write all the zeros in this number.

Graham number

The googol and googolplex are great, but they are nothing compared to the Graham number: this number is so large that throughout the Universe known there is not enough space to write all its digits.

It is a number born as a solution to a problem in the Ramsey theory, a particular branch of mathematics, and in 1980 it entered the Guinness Book of Records as “the largest number that has been used in a mathematical proof”. His greatness therefore also lies in the active role he played in mathematics.

Graham’s number, unlike the other numbers we have seen, it is not fully known to us. We know all the digits of googol and googolplex – also because they are simply 1 and 0 – but we don’t know all the digits of Graham’s number. However, we know that it is a multiple of 3 and that there is an algorithm that allows us to calculate the last digits, for example the last 500, which are very few in the immense totality of its digits, are those represented in the figure below.

Rayo’s numbers

As we have already mentioned, a number greater than all cannot exist, but there are numbers, i Rayo numberswhich are so large that it is impossible to describe them The mathematical symbols we have at our disposal are not enough for us. But in what sense? We have seen that the googolplex is too large to be written in full, but we can represent it as a power, i.e. 10^google. Similarly, there are notations, that is, ways of writing numbers, which allow the immense Graham number to be written using a relatively small amount of symbols. We can then ask ourselves, simplifying a bit, whether

Is there a number so large that it cannot be described with any mathematical writing of a given length?

This is the idea with which the Mexican philosopher Agustin Rayo won a competition to invent the highest number in 2007 against his colleague Adam Elga. A Rayo number, intuitively, is

the smallest number that cannot be described with a given amount of mathematical symbols.

Once we have fixed a number of symbols that we want to use, there exists a Rayo number that is greater than all the numbers expressible with that number of symbols. In practice these numbers are larger than the googol, the googolplex and even the Graham number, but, of course, we can always add 1 to these numbers too and exceed them.

The number TREE(3)

We conclude with the TREE(3)a seemingly innocuous but truly enormous number. Everything comes from the game, Game of Trees in which, following rules, stylized trees are drawn by marking the base, leaves and forks of the trees with colored dots. TREE(3) corresponds to the number of different trees that can be created by following the rules of the game and using up to 3 colors for the dots. In the image below you can see examples of trees built following the rules of the game.

The surprising thing is that, although you only need a few colored markers to play Games of Treesthe number TREE(3) it is so big that, in a certain sense, we can’t even describe how big it is. We know, in fact, that it is a finite number and that it is possible to demonstrate that it is finite, but in fact we cannot prove it because to write such a demonstration would require such a large quantity of symbols that the entire life of the universe, past and foreseen future, would not be enough to write them all. An enormous number, therefore, but which can be written with a finite number of symbols and therefore there are Rayo numbers that exceed it.