Re: Math trivia
One can find 2aleph0 by considering binary representations of numbers between 0 and 1. Their cardinality turns out to be C. That is likewise true for any other number base b:
baleph0 = C
The openended real line segment (0,1) has the same cardinality as the closedended one [0,1] by a pushingintheends argument. The real numbers are easily mapped onto (0,1) and vice versa:
x <> (1 + x/sqrt(1+x2))/2
So both (0,1) and [0,1] in the reals have cardinality C.
One can find C*C by considering an ordered pair of numbers from [0,1] and interleaving the digits. One gets another real number, making C*C = C. That is also true of any finitelength ordered ntuples, by the same argument.
If one finds all (real number, integer), one also gets C, because 1 < aleph0 < C.
The number of infinite series of rational numbers, (aleph0)aleph0 is also C.
The number of permutations of positive integers is the number of their selfbijections: (aleph0)! = C.
Interestingly, Caleph0 = C. Thus, the total number of continuous functions from real numbers to real numbers is C, and that is also true of all the infinite sequences of real numbers.
