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How is it possible to prove a number is irrational?

First part of that question: How it possible to know that a number will go on infinitely?

Second part: How is it possible to know that no repetition will occur during the infinite sequence of digits?

Any examples of proofs of irrationality?

  • 1
    See [this question](http://math.stackexchange.com/q/4467/11619) and its answers for an example.2011-09-08
  • 4
    It can be hard in general. For example, nobody's been able to show that the Euler-Mascheroni constant is irrational.2011-09-08
  • 2
    ...and it took quite a while before $\zeta(3)=\sum\limits_{k=1}^\infty \frac1{k^3}$ was proven irrational.2011-09-08
  • 1
    As an example, http://en.wikipedia.org/wiki/Proof_that_e_is_irrational2011-09-08
  • 13
    You don't prove "goes on indefinitely" (since that is not the definition of irrational); instead you prove "not a quotient of integers".2011-09-08
  • 0
    @GEdgar, the point is not whether "goes on indefinitely" is *the* definition of irrational (indeed, concepts do not have unique definitions, since every characterization could, in principle, be used in place of the usual definition), the point is whether "goes on indefinitely" is *equivalent* to the usual definition of irrational. Of course, as you rightly point out, it is not.2014-05-07

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