Basic arithmetic in base twelve

It takes a while to get used to the idea that "15" in base twelve doesn't mean 
o o o o o
o o o o o
o o o o o
but that:
1512 =
1012 + 512 =
1210 + 510 =
1710.
1512
1 o o o o o o o o o o o o
5 o o o o o

Adding, and other sums

The method for adding numbers adding columns and carrying over and so forth is the same in base twelve as it is for base ten except each columns has twelve rather than ten digits. In fact, all the tradition school techniques are the same but take some getting used to: you may be surprised quite how wired in base ten is.

Convince yourself that the follow additions, subtractions, multiplication and divisions in the list below are correct. (I've omitted the subscript, these are all base twelve). Try to avoid thinking in base ten! Think in base twelve. Try to visualize the amounts as amounts rather than representing the number in your head in base ten. One idea that can help establish the relationship between the numbers is to use a clock face in the mind's eye: 3, 6, 9, 10. For example, breaking it into quarters so 3 is a quarter, 6 is a half and 9 is three quarters. 4 and 8 are at the third position.

Only convert to base ten if you absolutely can't see why. Don't be in a hurry, mull over each one and with any luck it'll start to feel more natural, or at least less unnatural!

Division

Division is where it gets interesting! Division is one of those subjects that used to give people nightmares back in school and probably would continue to do so if it weren't for calculators.

Why was it so hard?! Well, one of the reasons (and indeed one of the reason for this site) is that it is hard simply because of the numbering system we use. Here are some divisions in base twelve:

  • Division
    • 10 / 2 = 6
    • 10 / 3 = 4
    • 10 / 6 = 2
    • 20 / 6 = 4
    • 1 / 2 = 0.6 this is a half: think of the clock face!
    • 1 / 3 = 0.4 a third, again, the clock!
    • 1 / 4 = 0.3
    • 1 / 6 = 0.2
Already we've done a bunch of division and been able to express the results very easily: with a single digit after the decimal point. That's only possible with two numbers in base ten whereas with base twelve we're up to four. We're working up to a full side-by-side shoot-out between ten and twelve but first we need to explore what division really is and how divisors come to bear on the issue in deciding which is best...