1/89
Type 1/89 into your calculator and you'll see a familiar sight:
0.011235955...
You'll see the start of the Fibonacci sequence. In fact, the whole sequence is buried there because
1/89 = 1/10² + 1/10³ + 2/10⁴ + 3/10⁵ + 5/10⁶ + 8/10⁷ + 13/10⁸ + ...
A surprising discovery, but why is it so, and why 1/89? And is it related to 89 being a number in the Fibonacci sequence? No, it's simply a result of our use of base 10 to write numbers and the relationship between 89 and 10² (=100).
The summation for 1/89 shown above can be derived by using the Taylor series (see here and here) of this:
1/(1-x) = 1 + x + x**2 + x**3 + x**4 + ... for |x| < 1
In order to get around the restriction on x, I'll use the function
F(x) = (1/100) x 1/(1-x)
For x=11/100, F(x)=1/89 and the series is:
1/100 (1 + 11/100 + (11/100)² + (11/100)³ + (11/100)⁴ + ...)
1/100 (1 + (1/10 + 1/100) + (100 + 20 + 1)/10⁴ + (1000 + 300 + 30 + 1)/10⁶ + ...
Regrouping and remembering that adding diagonals of Pascal's Triangle gives you Fibonacci numbers:
1/10² + 1/10³ + 2/10⁴ + 3/10⁵ + 5/10⁶ + 8/10⁷ + 13/10⁸ + ...
which is the series we started with: Fibonacci numbers divided by increasing powers of 10.
But why 89?
It's not because 89 is in the Fibonacci sequence, it's because
89 = 10² - 10 - 1 and we using base 10 to represent the numbers.
If we used other bases, the same pattern would result:
Base 2: 1/(2²-2-1) = 1 = 1/2² + 1/2³ + 2/2⁴ + 3/2⁵ + 5/2⁶ + 8/2⁷ + 13/2⁸ + ... (it converges very slowly)
Base 8: 1/(8²-8-1) = 1/55 = 1/8² + 1/8³ + 2/8⁴ + 3/8⁵ + 5/8⁶ + 8/8⁷ + 13/8⁸ + ...
I'm teasing you with choices that have fractions with Fibonacci numbers on the bottom. For base 11, the fraction to use is 1/109, for base 12 it is 1/131, etc. So for every integer (2 or more) that you use for a base, there is a fraction that can be written as a series where Fibonacci numbers are in the numerator and increasing powers of the base are in the denominator.
