Consider the non-negative natural numbers:
$0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$
Encode the primes as $1$, the rest as $0$.
$E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$
The number $E_0$ is
$001.10101000101000101\dots$ (i.e. the binary pattern of the primes)
$E_0$ is irrational in every base due to a simple proof which shows that the binary pattern of the primes, $E$, is not periodic.
Since the location of the (binary) point is arbitrary, if we locate the point one place to the right, we obtain $E_1 (=0011.0101\dots)$.
So $\dfrac{E_1}{E_0}$ is the integer $2$.
Since $E_0$ is irrational in every number base, we are led to the interesting aphorism:
"Every integer greater than one is the ratio of two irrational numbers. Each of these irrational numbers expresses the infinite pattern of the prime numbers".
Now, prove that $E_0$ is transcendental.
Note that $C_0 = 110.01010111010111010\dots$ (i.e. the binary pattern of the composites)
and $E_0+C_0 = 111.11111111111111111\dots$
All contributions acknowledged.