The Four Rules in dozenal

Addition

In base ten we arrange our numbers in groups of ten and powers of ten; in base twelve we use groups of twelve and powers of twelve.

Counting from 0 is adding a unit at a time:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E, 10, 11...
and so on.

In dozenal we need a symbol for ten ( we use "T" here) and one for eleven ("E") as our base is now twelve and not ten, and the symbols "10" mean "one dozen and no units".

Compare these additions in decimal and dozenal:

14 + 40 = 54
*12 + 34 = 46
The first tells us that fourteen plus forty is fifty-four, and the second (the star * means we are in dozenal) that one dozen and two plus three dozen and four is four dozen and six

When we used shillings and pence this second expression could be shown as 1s2d + 3s4d = 4s6d, and we wouldn't have thought twice about the calculation. We can also write this second expression as if it were an addition sum in feet and inches (as we haven't any shillings left...) as 1′2″ + 3′4″ = 4′6″.

Here are some more sums in feet and inches, using the current notation and the dozenal notation:

To continue the parallel we need to imagine a twelve-foot unit (a "rod" perhaps) so that we can continue into three columns:
e.g. 27′ = *23′ (two rods, three feet) = *230″: