Irrationality of 0.123456789101112 ... and similar numbers

Consider four numbers in $$(0,1)$$: $$n_1$$ in base $$10$$ is formed by listing the decimal digits $$1,2,3,4,\ldots$$; $$b_1$$ in binary is formed by $$0$$ and $$1$$ for each even and odd digit of $$n_1$$: $$n_1 = 0.123456789101112131415161718192021 \ldots$$ $$b_1 = 0.101010101101110111011101110110001 \ldots$$ $$n_2$$ and $$b_2$$ are formed similarly, but listing the primes $$2,3,5,7,\ldots$$: $$n_2 = 0.23571113171923293137414347535961 \ldots$$ $$b_2 = 0.01111111111101011111010101111101 \ldots$$ Which of these numbers is known to be {rational, irrational, algebraic, transcendental}? I presume that all four are irrational.

• Irrationality isn't hard, for any of them. There are natural numbers with arbitrarily long strings of $1's$ or strings of $0's$, for example. And, similarly, there are primes with such strings (there are infinitely many primes that start with any fixed sequence).
– lulu
Commented Mar 30, 2019 at 10:11
• Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal. Commented Mar 30, 2019 at 10:31
• @badjohn It's only known to be normal in base 10, and actually the same is known about $n_2$, which is the Copeland-Erdos constant. Also, normality is not "better than merely transcendental", all algebraic irrationals are expected to be normal. Commented Mar 30, 2019 at 10:43
• @Wojowu I considered mentioning that but it seems more than necessary for a comment. The "better than" was intended to be light hearted. Plenty of numbers are expected to be normal but relatively few are proven to be. Commented Mar 30, 2019 at 10:55
• Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs. Commented Mar 30, 2019 at 11:57

$$n_1$$ is the Champernowne constant, which is known to be normal. $$n_2$$ is the Copeland–Erdős constant, which is also known to be normal in base $$10$$.