[Coda]

44

A base is basically how many digits you have. Our normal math uses base 10 ("decimal"), because we have ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. You'll hear about binary, which is base 2, because it has two digits: 0 and 1. Base 13 is a number system that has, then, 13 digits; you might choose to label the extra ones A, B, and C.

When you work with a number in a given base (say, base B), then each place value is B times more than the place on its right. In base 10, each digit is worth 10 times more than if it were in the place to its right -- 200 is 10x more than 20, and 30 is 10x more than 3.

So in base 13, you might have the number 2C, which is equal to the number 38 in decimal: 13 * 2 + C (which is 12) = 26 + 12 = 38.

All of the other math you're familiar with works the same way in other bases. You've got to memorize different times tables (3 * 4 = C, for example) and addition wraps around in a different place (9 + 1 = A instead of wrapping around to two digits) but the rules are all the same.

In this case, 6 * 9 can be calculated as adding 9 to itself 6 times, just like in decimal.

9 + 9 = 15 (which is 18 in decimal)
15 + 9 = 21 (which is 27 in decimal)
21 + 9 = 2A (which is 36 in decimal)
2A + 9 = 36 (which is 45 in decimal)
36 + 9 = 42 (which is 54 in decimal)

So, in base 13, 6 * 9 = 42, but 42 is actually 54.

(EDIT: I'm actually not very good with math in base 13; I actually did all of that in decimal and then converted the answers to base 13 afterward. :P Base 2, I'm good with. Base 16, I can add fairly quickly but I can't multiply very well, because it's used a lot in computing.)