What Bad Bases Are Good For

Base two numerals can be awkward to compute with, and Roman numerals are even more awkward. But awkward is not the same as bad. Today I’ll tell you about a truly awful numeral system—possibly the very worst ever proposed as a way of doing arithmetic—but I claim that even such an abominable system is useful for something. EQUAL TIME FOR THE FAR LEFT First let’s talk about a funny way of extending decimal notation that lets you write negative numbers without using a minus sign. In this system −1 is written as …999, which should remind you of the infamous decimal 0.999…, except that where the dots in 0.999… mean “forever rightward” the dots in …999 mean “forever leftward”. Here’s how negative integers work in this system: −1 = …999,−2 = …998,−3 = …997,…−9 = …991,−10 = …990,−11 = …989,…−99 = …901,−100 = …900,−101 = …899,… But what does this mean? Since the ordinary decimal 99 means 90 + 1 or equivalently 1 + 90, and 999 means 900 + 90 + 1 or equivalently 1 + 90 + 900, and so on, it’s…