the clog

numbers

hello there.

I've had this idea for some time now, and I took advantage of it being my birthday to justify playing with it. the idea is that if you have sets of natural numbers, you can encode them in a fractional (real) binary number. the nth digit is 1 if the number n is in the set, and 0 if it isn't.

if you do this with the prime numbers, you get (approximately) 0.41468250985111166... after looking around a bit, I found that there's this smart man Simon Plouffe, who made a website Plouffe's inverter, where he has a database of real numbers. I also wanted to see what numbers I got if I looked at pi/4, and I found this page with a binary expansion of pi/4.

it's a bit sad when you get an idea and you see someone else had it before, but that's the way it is. anyway, here are some more of these numbers: 0.6349187201195740649... (lucky numbers; didn't even know until today that there was such a thing as lucky numbers), 0.7656250596... (factorials), 0.9102787972... (Fibonacci).

from the way they're constructed, it's easy to realize if these numbers are rational or not. I wonder what can be said about whether they're transcendental or algebraic. I'm also curious if the distribution of digits in binary notation is related to the distribution of digits in decimal notation.

one thing to note is that this kind of correspondence (real betwen 0 and 1) <-> (set of natural numbers) is not a bijection. because you can write the same real number in two ways (0.1 = 0.0111111... in binary). however, there are at most two sets of natural numbers that correspond to a rational number, and that's it.

another interesting thing is that this can be used to define a distance between functions defined on the natural numbers. if v(A) is the number corresponding to the set A of natural numbers, then v({n|f_1(n) not equal to f_2(n)}) is a distance between the functions f_1 and f_2. It has all the properties of a distance, even though it might be a bit weird that d(n, n+1)>d(n, n^2).

anyway, end of transmission.