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Let $x=0,a_1a_2a_3\cdots a_i\cdots$ be a number such that $a_1=0$, $a_i=1$ if $i\in\mathbb{N}$ is a prime number and $a_i=0$ otherwise. So $x=0,01101010001\cdots$.

Is $x$ a rational number? How can I show?

Thanks.

kurtzdoni
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1 Answers1

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It is not rational because the decimal expansion of a rational number is eventually periodic. However, there are arbitrarily long sequence of consecutive composite numbers (for example, by the Chinese remainder theorem), and of course infinitely many primes, so the decimal expansion for this number is not eventually periodic

user115940
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  • If $x$ is a rational number then, necessarily, the decimal expansion of $x$ is periodic? – kurtzdoni Jan 20 '15 at 11:39
  • Yes, a number is rational if and only if its decimal expansion is eventually periodic, and this is true in any base. See for example http://math.stackexchange.com/questions/198810/proof-that-every-repeating-decimal-is-rational/847020#847020 – user115940 Jan 20 '15 at 12:55