One Over Eighty-Nine

The number 1/89 is a very special number, as we will see shortly. Expanded out into decimal form, it looks like this:

1/89 = 0.011235955056179...

Although it is a rational number, it's decimal form never terminates. It has an infinite number of decimal digits.

So what does this have to do with the Fibonacci Series? Have a look at the first few non-zero digits of the decimal expansion. Look familiar? Of course they do. The first five non-zero digits of the decimal expansion of 1/89 are the same as the first five terms of the Fibonacci series.

But that's not all. The fact that these numbers look so familiar clue us in that something else might be at work in this number. And, as a matter of fact, we would be correct in thinking that. Notice that the beginning of the decimal expansion of 1/89 is simply the terms of the Fibonacci series multiplied by increasing powers of 1/10.

Continuing in this fashion and adding, we find the true nature of the pattern:

(Note: the last few columns do not add up exactly because of carried numbers from previous additions. If you begin with the rightmost column and move left, you can verify the addition.)

So we know that this pattern holds for the first several digits of the decimal expansion of 1/89. But does it work for all decimal digits? That is, can we prove that the pattern always holds? In fact, we can. Click here to see how (recommended for those already familiar with summation notation).