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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.

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    $\begingroup$ 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). $\endgroup$
    – lulu
    Mar 30, 2019 at 10:11
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    $\begingroup$ Well the first looks like Champernowne's constant: en.m.wikipedia.org/wiki/Champernowne_constant. Better than merely transcendental, it is normal. $\endgroup$
    – badjohn
    Mar 30, 2019 at 10:31
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    $\begingroup$ @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. $\endgroup$
    – Wojowu
    Mar 30, 2019 at 10:43
  • $\begingroup$ @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. $\endgroup$
    – badjohn
    Mar 30, 2019 at 10:55
  • $\begingroup$ Thanks, badjohn & Wojowu, for identifying the constants as Champernowne and Copeland-Erdõs. $\endgroup$ Mar 30, 2019 at 11:57

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$n_1$ 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 $10$.

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  • $\begingroup$ But a normal number can be either algebraic or transcendental (conjecturally). And the same is true for a non-normal number (proved). In fact the poster asked about transcendence, so you answered something else. $\endgroup$ Jun 6, 2022 at 9:33

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