## Large numbers

Please read the short story “The Library of Babel” by J. L. Borges. In summary, it is the story of a large library containing books with all possible combinations of 25 symbols. Each line has 80 characters, there are 40 lines on each page and 410 pages on each book.

- How many words are, on average, on each book?
- How may books are there in The Library? It is ok to give an approximate number if you indicate the margin of error.
- How many rooms are in the library? Remember that each room has shelves in 4 walls and there are five shelves on each wall, with 32 books on each. (The rooms are hexagonal, so there are two other walls with doors connecting to other rooms.)
- What area is covered by such a library? Remember that the area of a hexagon of side \(s\) is \(3\sqrt{3} s^2/2\).
- If this library exists in a planet like Earth, how many floors does it need?
- Bonus: What is the weight of the library? How many trees are needed to make the paper for the books?

This is a question about *handling large numbers*.

## Interval arithmetic

What is the weight of this ferry? Estimate an interval for the weight, as narrow as possible.

## Drake’s equation

In 1961 the scientist Francis Drake proposed the following formula to estimate the number of alien civilizations that can contact us

\[N = R_* \cdot f_\mathrm{p} \cdot n_\mathrm{e} \cdot f_\mathrm{l} \cdot f_\mathrm{i} \cdot f_\mathrm{c} \cdot L\]

where

- \(N\) is the number of civilizations in our galaxy with which communication might be possible
- \(R_{∗}\) is the average rate of star formation in our Galaxy
- \(f_{p}\) is the fraction of those stars that have planets
- \(n_{e}\) is the average number of planets that can potentially support life per star that has planets
- \(f_{l}\) is the fraction of planets that could support life that actually develop life at some point
- \(f_{i}\) is the fraction of planets with life that actually go on to develop civilizations
- \(f_{c}\) is the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
- \(L\) is the length of time for which such civilizations release detectable signals into space

Drake made the following estimations

- \(R_{*}\) = 1 yr
^{-1}(1 star formed per year, on average) - \(f_{p}\) = 0.2 to 0.5 (1/5 to 1/2 of all stars will have planets)
- \(n_{e}\) = 1 to 5 (stars with planets will have between 1 and 5 planets capable of developing life)
- \(f_{l}\) = 1 (100% of these planets will develop life)
- \(f_{i}\) = 1 (100% of which will develop intelligent life)
- \(f_{c}\) = 0.1 to 0.2 (10–20% of which will be able to communicate)
- \(L\) = 1000 to 100,000,000 communicative civilizations (which will last somewhere between 1000 and 100,000,000 years)

Please use *interval arithmetic* to find the smallest and
largest value of \(N\) under this
model.`