I came up with this in grade school; while I certainly wasn't the first to ponder number systems not based on ten, I think I did a fair job of explaining the concept to myself. I share this hopefully simple explanation for those who, like myself, are not natural math pros and who do not think in binary.

Thanks to N-Wing for pointing out that there's a convention to use regular digits for number systems base ten or lower...but I had no idea then, and this is something like what I came up with.

The Humbabboid Base Nine Number System:
ð = 1
¬ = 2
¥ = 3
¤ = 4
µ = 5
§ = 6
Þ = 7
€ = 8
ß = 0

I reckon it's because humans generally have ten fingers (digits) that we have ten digits in our modern number system: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. The first nine digits always struck me as arbitrary--no mathematical function except to symbolize values that must simply be learned by rote. (Which is a good reason to keep their number small--imagine we had 255 digits to memorize in addition to the zero.)

The zero, however, has a mathematical function. It is used to denote a higher order, or multiple of the base (ten), in counting. When you have one ten, you represent it by the first digit multiplied by the base, shorthanded to 10. When you have ten tens, you again multiply by the base: 100. Two tens and change might be represented at 24, where the four in the zero slot indicates not only a multiple of the base but the addition of one of the arbitrary integers--24 means "two tens plus four." (Of course zero is used for lower orders as well in decimal values below 1, among other things, but the idea's basically the same.)

There's nothing wrong with base ten. I'm a base ten fan. Except, except.... When first learning to convert fractions to decimal values, I was dismayed to find that one-third could not be represented in base ten. I mean, 0.333333... didn't cut it for me. What do you mean it goes on like that forever? I knew from experience that you could have exactly one third of a cake, for example. It wasn't some never-ending Xeno's paradox--cut a thing into three equal pieces (assuming no crumbs--a Platonic form of the crumbless cake), and you've got perfect thirds that my handy solar-powered calculator was incapable of understanding:

1 ÷ 3 = 0.33333333, says Texas Instruments.
0.33333333 x 3 = 0.99999999, insists Texas Instruments.

It lost the crumbs. It couldn't remake the cake under perfect laboratory conditions, as I wanted. So I ignored the rest of the math lesson, and opened my spiral notebook to a blank page and started fiddling with numbers. Pretty soon, I began to suspect that a base nine system would have perfect thirds--to avoid muddling my brain further, I concocted the above set of arbitrary symbols rather than use the Arabic ones I associated so closely with base ten.

Here's how it works--the higher orders are groups of nines, rather than tens. So if we attach the Humbabboid digits for 1 (ð) and zero (ß), we get 1 x 9. ðß = 9, despite the fact that we're displaying it with the symbolic equivalent of 10. To represent ten in Humbabboid, you need one times the base plus one, or ðð. Other fun conversions:

42 (four times base ten plus two) = ¤§ (four times base nine plus six)
256 (twenty-five times the base plus six) = ¥ð¤ (twenty-eight times the base plus four)
1999 (199 times the base plus nine) = ¬§§ð (222 times the base plus one--that one hurt my brain, I'll stop here)

Where this complete waste of time becomes really useful is in the representation of the aforementioned thirds. (Humbabboid uses the decimal point, but refers to it as the "zwark horizon.")

One third, exactly = ß.¥
ß.¥ x ¥ = ð, exactly, with no damn lost crumbs.

(NB: ß.¥ does not equal 0.3, it equals that infuriatingly endless 0.333... deal.)

Now, we do lose a bit of base ten's neatness with halves:
0.5 = ß.µµ

That's misleading, though--for while ß.µµ does equal base ten's zero-point-five, zero-point-five does not represent one-half in Humbabboid. It's slightly more than half. In a base nine system, 0.45, or ß.µ, is half of the base.

I could have it all wrong. Again, I'm no math genius. But it was enough to impress my teacher at the time, and earned me a gold star. I even, briefly, started a campaign to get the base nine system accepted on all math homework at the school.

Then I discovered girls....

Update: A few math-inclined noders have pointed out that I did, indeed, get it all wrong where the halves were concerned. If this is so, I am bummed out all over again, but invite the active-synapsed to write up their proofs and corrections here.

Every base other than binary has some fractions that expand 'nicely', and some that repeat infinitely. It depends on the factors of your base.

When you're expanding a fraction, you're trying to match up each factor of the denominator with a factor of the base. Let me show you how it works in base 10.

10 factored is 2 x 5.

2 factored is 2.
So 1/2, in base 10, will be one digit long, because the 2s match up.

4 factored is 2 x 2.
So 1/4 won't be only one digit long, because the first 10 can only 'use up' one of the 4's 2s.
But if we take a second 10, we have another 2 (and another 5) to work with, so we can 'use up' the 4's other 2.
1/4 in base 10 is two digits long (0.25)

6 factored is 2 x 3.
We can use up the 2 on our first go-round, but then we're stuck. No matter how many 10s we take, we'll never get a 3.
1/6 in base 10 repeats infinitely (0.16666....)

8 factored is 2 x 2 x 2
1/8 in base 10 is three digits long, cause we need three tens to use up those 2s. (0.125)

This isn't limited to numbers lower than your base:

50 factored is 5 x 5 x 2
Our first 10 uses up the two and one of the fives; our second 10 takes care of the other five.
1/50 in base 10 is two digits long (0.02)

Clear as mud?

So base 9 doesn't solve the problem, it just moves it to different numbers. Anything with factors other than 3 will repeat infinitely in base 9. Sorry, but that does include 1/2.

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