where indicates a prime and is the characteristic function of the primes.
The beginning of the decimal expansion of ρ is: (sequence A051006 in the OEIS)
The beginning of the binary expansion is: (sequence A010051 in the OEIS)
The number is easily shown to be irrational. To see why, suppose it were rational.
Denote the th digit of the binary expansion of by . Then, since is assumed rational, there must exist , positive integers such that
for all and all .
Since there are an infinite number of primes, we may choose a prime . By definition we see that . As noted, we have for all . Now consider the case . We have , since is composite because . Since we see that is irrational.