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Introducing Base Eight

So what if we had eight fingers, or for some other reason, we decided to start over every eighth number instead of every tenth? Then we would have "base eight" (also known as "octal") counting. In this system, there are eight symbols to work with:

0 1 2 3 4 5 6 7

We don't need an 8 or a 9 at all: out of just those eight symbols above, we are going to represent every possible number! So, we start by listing all the symbols after the "zero."


When we get to that point, we're out of symbols. So what do we do? We go all the way down to zero, and add a one to our left: we write "one-zero" (10). It means "the number that comes after seven," or what we normally call "eight." This is the key turning point in this paper, so make sure you're still with me: when I write "one-oh" (10) in base eight, I don't mean ten, I mean the number eight. The numbers in base eight look just like our normal numbers (except that they never use the symbols 8 or 9), but they don't mean the same things. You have to think in "eights" to understand them, just as you have to think in "tens" to understand our normal system.

Now we start counting on the right again: one-one, one-two, one-three, and so on. (11, 12, 13...) Soon we hit one-seven (17) and we run out of digits again, so we have to increment on the left: two-zero, or 20. Every eighth number, we start over again.

This system works great until we get to 77, and then we can't increment the left-hand digit any more. So we move to the left again and write one-zero-zero (100). It's important to remember again that this doesn't mean the same thing we normally call "one hundred" so it's best not to call it that: call it "one-zero-zero" and it will help keep things straight.

So, when I write the octal number 1235 what do I mean? I mean 5 "ones", 3 "eights", 2 "sixty-fours", and 1 "five-hundred-and-twelve." Each step to the left involves a multiplication by eightóbecause we are in base eight. So if there were another digit to the left, we know it would count as "four-thousand-and-ninety-sixes" since that is eight times "five-hundred-and-twelves," and so on.

If you've followed everything I've done so far, you have hopefully understood a few things. First, you understand that our normal system of counting is based on the principle that every tenth number, you start over and move to the leftóbecause there are only ten symbols to work with. Second, you understand that there's nothing sacred about the number ten: base eight is exactly the same, except it happens to start over every eighth number, because it has eight symbols to work with. Once you can do base ten counting, there's nothing fundamentally different about base eight.

But you may still be wondering, what does 1235 base eight really mean? Well, I said above that it is 5 "ones", 3 "eights", 2 "sixty-fours", and 1 "five-hundred-and-twelve." If you add all that up,

5*1 + 3*8 + 2*64 + 1*512 = 669

So we can say "1235 base eight equals 669 base ten." Or, to put it another way, "when you write 1235 in base eight, you really mean the number six-hundred and sixty-nine." That's a kind of base-ten-centric way to put it, but hey, we're base-ten-centric people. And in any case, the point I'm making is that it's easy to convert base eight numbers into their base ten equivalents, once you understand the system.