Is this number rational or irrational?

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?

• – lab bhattacharjee Jan 31 '14 at 8:50
• $\sum\limits_{n=1}^\infty F_n z^n = \frac{z}{1-z-z^2}$, so your sum is $100/9899$, a rational number. – achille hui Jan 31 '14 at 8:54
• – Hagen von Eitzen Jan 31 '14 at 9:40
• @HagenvonEitzen Thanks. This question was actually inspired by this topic on Hacker News. – Chris Taylor Jan 31 '14 at 9:46
• Generalization: a $k$-places shift. – Martín-Blas Pérez Pinilla Jan 31 '14 at 10:08

This is the series $$\sum_{n=1}^{\infty}{F_n\over100^n},$$ easily summable with the explicit formula for $F_n$.
• In case OP is interested, the sum is $100/9899$. – user98602 Jan 31 '14 at 8:53