Beyond base ten

As we saw in bases humans use base ten. Other bases, binary, octal, and hexadecimal, are commonly in use primarily thanks to digital computers.

Names for numbering systems
tendecimal, or denary
twelveduodenary, or duodecimal

Base twelve

Base twelve requires of course twelve digits for the amounts zero through eleven. We don't have any digits to represent ten and eleven so we'll do what the computer scientists (somewhat unimaginatively) did with hexadecimal and use letters instead, so A12 is ten and B12 is eleven.

But what's twelve in base twelve? 1012 of course! It follows exactly the same mechanism of returning the column to zero and incrementing the next one along. So 10012 is twelve times twelve (one hundred and forty four).

Recall the powers from bases:

By definition, this is true regardless of what base we're working in. "100" just means a different quantity in each case.

The important thing to remember is that these series of digits are just a way of writing down a particular quantity. The quantity "thirty six" is still the same number of things whether it's 3012 or 3610. The base of the numbering system is an agreed on standard and the lens and language with which we see and express numbers.

Let's look at some basic arithmetic in base twelve...