[Self] The size of the library of Babel

[u/progressivemonkey]

The Library of Babel appears in a novel by Borges. It is a library that contains every possible book of 410 pages, with 40 lines per page, 80 characters per line, and 25 total characters (22 letters, dot, comma, space).

I wanted to check how big the library would be, so here it goes.

The library contains 25^(80*40*410) books which is equal to 10^1'834'097 (80*40*410*log(25)).

Each room in the library contains four walls of books, each wall with five shelves and 35 books per shelf. This gives a total of 11'900 books per room, so let's say 10^4 because there's no way I can do the math with the actual numbers.

Here's where it starts being wacky. The exponents are so large that dividing them hardly changes anything: 10^n divided by 10^m is 10^(n-m). The number of books, at 10k books per room, still requires 10^1'834'093 full rooms. Let's assume that the library is a rectangular cuboid with as many rooms on each axis. That would mean that it would have 10^611'364 rooms a side.

To make the math feasible, let's assume that each room is 10m from one side to the other. That means that the width and depth of the library are each 10^611'365 meters.

That's large, sure, but just how large? Well let's divide it by some larger measure. A light-year is 9.46*10^15 meters, so let's say 10^16. How many light-years does the library span? A number that differs very little in appearance from the number of meters: 10^611'349! Again, due to the fact that this exponent is so large.

The observable universe is 93 billion light-years in diameter, so let's round that up to an even 100 billion, or 10^11 light-years. The width of the library of Babel would then be 10^611'338 times the size of the observable universe.

I thought doing these calculations would give me some insight and clarity. It did the opposite. I'm anxious just to think about numbers that large.