What is the largest finite number that you can imagine? Is it all the grains of sand on all beaches in the world? Or, is it the total number of subatomic particles in the entire observable universe? Or, the famous googol number ($10 ^{100}$) that Google was inspired by? Or even, is it a googolplex ($10 ^{10^{100}}$)?
All these are nothing but absolute zero compared to Graham's Number, which is the largest finite number that human beings can ever imagine!
To be able to explain Graham's Number in detail, I will introduce Shannon's Number first, which is a comparatively small number. Then, using it as a base, we will calculate Graham's Number (at least up to some point) together.
1. Shannon's Number ($10 ^{120}$)
Shannon's Number, named after the famous American mathematician Claude Shannon, is the lower bound for all different possible chess games. To be more clear, I will give an example about chess:
The strongest chess engine that was ever created by now, Stockfish, is able to analyze 100 million different variants and play the best among them. Now, let's use this as the base point and dive as deep as possible:
$10^7$ variants -> 1 second
$10^9$ variants -> 100 seconds ( ~ 1.5 minutes )
$10^{11}$ variants -> 150 minutes ( 2.5 hours )
$10^{12}$ variants -> 25 hours ( ~ 1 day )
$10^{15}$ variants -> 1000 days (~ 3 years)
$10^{21}$ variants -> 3 million years
$10^{25}$ variants -> 30 billion years
Let's stop here. According to the calculations, the age of our universe is roughly 13 billion years. This means that if Stockfish had played 100 million different chess games every second twice the age of our universe, the total number of different chess games played would be $10^{25}$. But even this is roughly 1/googol of the lower bound of the total number of different chess games! In other terms,
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 of the total possible different chess games!!!
Now, let's use Shannon's Number as a base and try to understand what Graham's Number is.
2. Graham's Number (g(64))
Named after its founder Ron Graham, Graham's Number is the upper bound for the mathematical solutions of Ramsey problems. At the same time, it is the largest number that is actively being used to solve a practical mathematical problem.
Graham's Number is so incredibly massive that classical arithmetical operations are insufficient to express it. Therefore, we will use Knuth's up-arrow notation. It will help us to build a power tower. Let's start:
3↑3 -> $3^3$ = 27.
3↑↑3 -> 3↑3↑3 , which is $3^{27}$ = ~ 7 trillion.
3↑↑↑3 -> 3↑↑3↑↑3, which means there is a power tower that contains roughly 7 trillion 3's! This number is already impossible to be calculated using current technology, at least without the help of quantum computers. Let's try to visualize this number, instead of calculating it: If we want to write it by assuming the height of every 3 is one millimeter, we come up with a power tower that's height is roughly 76 million kilometers, which is almost 1/3 of the distance between the Earth and Mars!
So, how small is Shannon's Number compared to 3↑↑↑3 (or, how large)?
Shannon's number = $10^{120}$ = ~ $3^{240}$, which means the power tower contains more than two and less than three 3's. In other words, ${3^{3}}^{3}$ < $10^{120}$ < ${{3^{3}}^3}^{3}$. However; in 3↑↑↑3, the power tower contains roughly 7 trillion 3's! Seems like comparing 3 and 7.000.000.000.000 :)
Let's continue:
3↑↑↑↑3 -> 3↑↑↑3↑↑↑3 = g(1)
3↑...↑3 = g(2) (Number of ↑'s in between are g(1)!)
3↑...↑3 = g(3) (Number of ↑'s in between are g(2)!)
...
g(64) = Graham's Number.
You can check out my next post titled "Incredibly Large (But Finite!) Numbers - 2" to find out for which mathematical problem Graham's Number is used.
No comments:
Post a Comment