Models for Rational Number Bases

We think it’s important that you have some background on how this article came to be written. One of the authors sometimes plays a game he invented, called “hats,” in which he puts on a funny hat and converts base-ten numerals into another base representation. For example, he puts on a beanie and calls for anyone to say a favorite number. Someone says “six.” He goes to the chalkboard and writes “110,” The object of the game is for the students to figure out what he is doing, Someone who figures it out says, “Gotcha!” And then that person gets to wear the hat until someone else says, “Gotcha!” (In case you haven’t “got it” yet, the beanie hat converts numbers to the binary representation.) Several hats are used in the course of play, and each represents a different number base, but it’s always a natural number base greater than one. Recently we began to wonder if perhaps you can have a negative number base system in which the base is not an integer. And if you can, what will the numerals look like?