Fibonacci (the nice guy's kid) has credit for introducing Hindarabic numeration into Europe and the usual calculation algorithms along with them (they beat the abacus or trying to calculate in Roman numerals). In the process he mentions a strange little sequence of numbers <1, 1, 2, 3, ...> where each number is the sum of the two before (after the first pair, obviously). The sequence turns out to be one of those things that turn up all over the place in nature (the pattern of hairs on your head or bumps on pineapples fit into it and many more as well).
One idea for a Fibonacci number system is to take the Fibonacci numbers (those in the sequence) in order as the successive bases for the successive places. So, the rightmost place is base 2 (base 1 is a real pain to deal with, though it could be done, I suppose), the next base 3, the next 5, and so on. It pretty quickly starts needing either new symbols or the old Babylonian trick of lumping a string decimal -- or whatever -- symbols together in one "place" -- as in officially base 60 minutes and seconds.
I can't think of any practical advantage to using this sytem rather than a regular powers system, but it is cute. >|8}
An alternative idea is to use a system where each position represents a Fibonacci number. So the last few places would be worth ...34, 21, 13, 8, 5, 3, 2, 1. In this Fibonacci number system, the number 19 could be decomposed as 13 + 5 + 1 and could be written as 101001. Note, however, that the representation of a number is not unique, unless a further rule is added.