It takes a while to get used to the idea that "15" in base
twelve doesn't mean
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!
- 4 + 5 = 9
- 4 + 6 = A
- 4 + 7 = B
- 4 + 8 = 10
- 15 + 7 = 20
- 10 - 3 = 9
- 12 - 3 = B
- 17 - 12 = 5
- 20 - 15 = 7
- 12 - 8 = 6
- 32 - 18 = 16
- 2 x 3 = 6
- 2 x 4 = 8
- 3 x 3 = 9
- 3 x 4 = 10
- 3 x 5 = 13
- 4 x 5 = 18
- 5 x 5 = 21 It's hard for those that aren't unusually gifted to visualize
multiplication, it's as much a rote learning exercise as base ten!
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
- 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
bear on the issue in deciding which is best...