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We write all postive whole integers after the comma, how do we prove that this is an irrational number? ($0.1234567891011121314...$)

Ant
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user176791
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    Does it ever end or repeat? – user28375028 Sep 17 '14 at 18:52
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    The given number is known better as Champernowne's Constant: http://en.wikipedia.org/wiki/Champernowne_constant – Xoque55 Sep 17 '14 at 19:29
  • Count of this sentence from wikipedia: "Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.". So prove 'no repeating decimals'... – Remigijus Pankevičius Sep 17 '14 at 22:06
  • @r.pankevicius: that's not the usual definition though, so in practice you may or may not need more justification of that statement than "it's on Wikipedia" ;-) The proof that rational numbers have repeating or terminating decimal form isn't hard, and bears doing once. And you can consider terminating to be a special case of repeating: $0$-recurring. – Steve Jessop Sep 18 '14 at 00:44

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The decimal expansion we were given is not ultimately periodic, for it has arbitrarily long sequences of $5$'s (or any other digit).

But any rational number has ultimately periodic decimal expansion.

André Nicolas
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    You're right of course, but I note that $5/9$ also has "arbitrarily long sequences of 5s", so the "(or any other digit)" is critical, or some other qualification what we mean as a sufficient condition to prove non-recurrence ;-) – Steve Jessop Sep 17 '14 at 19:32