The Basics of Base Three

We give a little background to calculating in base three. It's actually very easy: it's the same as calculating in base ten (or base anything), but we just have to remember the columns stand for powers of three, rather than powers of ten.

For example, if were are working in base ten then 120 stands for "0 ones, plus 2 tens plus 1 hundred". In base three the same 120 stands for "0 ones, plus 2 threes, plus 1 nine". So 120 in base three amounts to fifteen (which we would normally write in base ten, as 15).

Performing arithmetic is very easy. For example the sum "two plus one equals three" would be written in base three as 2 + 1 = 10. Similarly, the multiplication "two times two equals four" becomes 2 x 2 = 11 in base three.

Bigger additions and multiplications are just as easy: we simply carry over just as we normally would. For example, 12 x 2 = 101. To see this, first note that 2 x 2 gives 1 and carry the 1; then 1 x 2 gives 2, and adding the carried 1 this gives 10. So, in total 12 x 2 = 101. In base ten this is just the multiplication 5 x 2 = 10. The point is, we don't have to convert to base ten to do the multiplication: instead, we directly apply the same rules in base three.

Now, what about squaring 120? We set this out exactly as if it were a base ten mutliplication. For the first line, put down the 0 and then multiply 120 x 2. And for the second line, put down 00 and multiply 120 x 1. Finally, add the two lines.

Notice that if we ignore the final zeroes, the result ends in a 1. And, it is easy to see that this will always be true. What goes into the rightmost place comes from a single multiplication, either 1 x 1 or 2 x 2, and both of these give a 1 in the rightmost place.