The Fibonacci numbers hiding in 1/89

Question

Solve on paper if you must, but the punchline should land in your head. Show that 	frac{1}{89} has the Fibonacci numbers embedded in its decimal expansion, and read off its first several digits.

Accepted Answer

Stack the Fibonacci numbers, shifted by one place each. \begin{aligned} 0.0&1\ 0.00&1\ 0.000&2\ 0.0000&3\ 0.00000&5\ 0.000000&8\ 0.0000000&13\ 0.00000000&21\ &\vdots \end{aligned} Adding, the two-digit terms overlap and carry, but the running total is \frac{1}{89} = 0.011235955\ldots You can see 0, 1, 1, 2, 3, 5, \dots before 8 and 13 start overlapping their neighbors. Why it works. The Fibonacci generating function is \sum{n=1}^{\infty} Fn\,x^n = \frac{x}{1 - x - x^2}. Set x = 	frac{1}{10}. The denominator becomes 1 - 	frac{1}{10} - 	frac{1}{100} = 	frac{100 - 10 - 1}{100} = 	fr…

The Fibonacci numbers hiding in 1/89

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