An interesting family of functions
You probably already know that we represent numbers using base 10 (or decimal) notation. You may also know that this choice is, ultimately, arbitrary. In principle any integer (greater than 1) can be used as a base. The representation of a number in a given base b need not be unique though. For example, in base 10 we have 1 = 0.999… In general, if a number x > 0 has a terminating base b expansion, then we can come up with a second expansion by subtracting one from the last non-zero digit after the decimal and replacing all the zeroes after this digit by the digit for b - 1. This follows from the identity It can be shown that if b is an integer, this is the only way for a number to have more than 1 base b representations. For non-integer bases this is false. For instance, since φ + 1 = φ² (where φ is the golden ratio), the base φ expansion of φ² can be 100 or 11. This is one of the reasons why non-integer bases are a pain, so we will not use them. For any integer b...