Cont'd Decimal Expansion, rational or not?

Question

This is a follow up from this question.

Since it's proven by Calvin Lin that $0.11235813213455...$ (Fibonacci Sequence), I'm not wondering if the sequence $$0.123456789101112131415...$$

(which is just natural numbers) is rational or not. Thanks for the help in advanced!

The decimal expansion does not terminate or repeat, so the number is not rational. — Michael Albanese, Sep 26 '13 at 07:23
This is the Champernowne constant. It is known to be transcendental. — André Nicolas, Sep 26 '13 at 07:26

score 3 · Accepted Answer · answered Sep 26 '13 at 07:23

3

A number is rational if and only if its decimal expansion terminates or eventually begins repeating. This number does neither.

answered Sep 26 '13 at 07:23

Thanks again for the again! – Commander Shepard Sep 26 '13 at 07:24
You can actually leave off the "terminates" option since it is included in the "eventually begins repeating" option. For example, $\frac14=0.250000\cdots$. Repeating doesn't preclude that the repeating part is the digit zero. So: A number is rational if and only if its decimal expansion eventually repeats. (And there's nothing special about "decimal" here. Any integral base $>1$ will do.) Also, if you don't like repeating zeroes, you can always rewrite such a case with repeating nines: $0.250000\cdots \equiv 0.249999\cdots$. +1 for you. – MPW Feb 23 '16 at 19:51

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