Unary numbers.

February 13, 2018

A positional number system needs a base that is either greater than one, or smaller than minus one—yes, we can have a negative base for a number system. The system, however, seems to break down if the base we chose is base 1.

If the base is 1, then there are no permissible digits since the digits d, in a base b system, must be 0\leqslant{d}<b. But we can still represent numbers using just 1s. That's the unary numeral system, and numbers are just represented as repeated 1s. 15? Fifteen ones: 111111111111111. Operations? Not very complicated, just… laborious.

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The protoshadok number system

September 13, 2016

We’re so used to our positional notation system that we can’t really figure out how to write numbers in other systems. Most of the ancient systems are either tedious, complicated, or both. Zero, of course, plays a central role within that positional system. But is it indispensable?

shadok

In one of my classes, I discuss a lot of different numeration systems (like Egyptian, Babylonian, Roman and Greek) to explain why the positional system solves all, or at least most, of these systems’ problems. I even give the example of Shadok counting (in french) to show that the basis used isn’t that important (it still has to be greater than one, and, while not strictly necessary, preferably a positive integer). But can we write numbers in a positional system without zero?

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