The additive persistence of a number is related to iteratively casting out nines. Multiplicative persistence is analogous but more interesting.

Positional number systems typically have a integer base, but irrational and even complex bases are possible. The golden ratio was the first irrational base.

Powers of the golden ratio are nearly integers. This post explains why. Also, these integers are the sum of two Fibonacci numbers.

For any x, the behavior of multiples of x mod 1 is easy to classify. The powers of x mod 1 are more interesting. We give examples of different behavior.

How to calculate large powers of 3 + √2 numerically with bc and symbolically with Mathematica. Conjecture regarding the integer and irrational parts.