Let us consider two transcendental numbers whose decimal representation is $$ 0.a_1a_2a_3a_4... $$ $$ 0.b_1b_2b_3b_4... $$ Is $$0.a_1b_1a_2b_2a_3b_3a_4b_4...$$ also a transcendental number?
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7$\begingroup$ My guess would be: this is unknown. I would say "interlace" the decimal representations, not "concatenate". $\endgroup$– Gerald EdgarOct 25, 2022 at 1:25
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$\begingroup$ @GeraldEdgar Thanks! $\endgroup$– KeiOct 25, 2022 at 2:29
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9$\begingroup$ Most likely no. If we de-interlace $\sqrt{2}$ to get two numbers, it's highly unlikely that either one might turn out to be algebraic! And question: if one were indeed algebraic, would the second one have to be too? $\endgroup$– Yaakov BaruchOct 25, 2022 at 3:40
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$\begingroup$ Maybe normality in base $10$ of both numbers is needed. $\endgroup$– Sylvain JULIENOct 25, 2022 at 17:54
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