8
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Start writing down the Fibonacci numbers, using two digits for each one

01 01 02 03 05 08 13 21 34 55 ...

Eventually you will reach three digit numbers. When that happens, any digits apart from the last two "overflow", and are carried back through the sequence like this

  ... 21 34 55 89
          +     1 44
          +        2 33
     -----------------
= ... 21 34 55 90 46 3? ...

This defines a real number, by concatenating the digits of the sequence

$$\phi = 0.01\,01\,02\,03\,05\,08\,13\,21\,34\,55\,90\,46\,3\dots$$

Is $\phi$ rational or irrational?

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6
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This is the series $$\sum_{n=1}^{\infty}{F_n\over100^n},$$ easily summable with the explicit formula for $F_n$.

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  • 2
    $\begingroup$ In case OP is interested, the sum is $100/9899$. $\endgroup$ – user98602 Jan 31 '14 at 8:53
  • $\begingroup$ @Martín-BlasPérezPinilla How long did it take you to come up with this argument? Did you see it instantly? $\endgroup$ – Chris Taylor Jan 31 '14 at 8:56
  • 1
    $\begingroup$ Seconds. The hard problem is without the carry digits. Someone overlooked this and answered it in a removed answer! $\endgroup$ – Martín-Blas Pérez Pinilla Jan 31 '14 at 8:59

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