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.`