I remember once heard that a spy in some scifi story stored the entire library of data from the military with a needle. He marked a point on the needle, which divide the needle into 2 parts. The length of each part became the numerator and denominator of an fraction, and the binary expansion of the fraction is the compressed data of the library.

How innovative. Or is it? Fit a large amount of data by converting it into a fraction and store it in a needle. How convenient. Why isn't it happening in real life?

One must note that currently, we can only manipulate the world discretely. We can't continuously divide the needle into infinite small parts. Assuming the smallest object can be precisely altered is an atom.
Let's make some assumptions about an ordinary needle. Suppose it have 4 cm long, and the diameter of an atom is 0.1nm and all very tightly packed. Then there are 40000 0000 atoms, 4*10^8. Call this number k.

WLOG, the first part of the needle has x atoms, the 2nd part has y atoms. It's ok to assume x != y. Why put the data on a needle when the number "1" is all he have to remember.

The binary digit expansion of x/y is the compressed data. The expansion digit is almost always infinite, but only the digits before the first repeat is useful.

What is the largest possible binary expansion for this?
Less than k.

So, the entire library can be stored in less than 50MB of data? Please...
And even if that IS the case, what's the possibility of that? there are 2^k combination of k bits. k/2^k chance. It's more likely to win the lottery and get hit by lightning... 3 times... in 20 seconds... than that the library actually can be stored with such an needle.

I don't know what sci-fi thing it is from. I wish to see the author and tell him how this is unpossible(p(x) = impossible ->p(x) = ε. p(x) = unpossible -> p(x) = 0)