Because this isn't exactly my table; I'm trying to figure out how a friend of mine came up with this crazy idea of using a table.

I've figure out that the top letters are the ones and the left are the tens, so it's more of a one-plus-ten sort of system. But after 143, then suddenly the next proposed row is hundred, the next thousands, then millions, billions, etc. But that can't happen because then there is a double number somewhere.

The same symbol is used multiple times in the left hand column, so go ask him for clarification.

Ooh, care to share? *intrigued*

*cracks his knuckles, as if about to play piano* (

)

A positional base

*b* system uses an "alphabet" of

*b* symbols to represent numbers by using the fact that every number has a unique deconstruction,

x=d_{1}b^{0}+d_{2}b + d_{3}b^{2}+d_{4}b^{3}+...+d_{s+1}b^{s}

to represent every number by the digits d

_{s+1}d

_{s}...d

_{3}d

_{2}d

_{1}.

(The fact that this combination both exists and is unique can be proven fairly easily, but that seems unnecessary.

)

In the standard decimal system, the alphabet is the set {0,1,2,3,4,5,6,7,8,9}, and

*b* is the size of that set. (ten) In binary, the alphabet is {0,1} and b=2. In duodecimal, {0,1,2,3,4,5,6,7,8,9,A,B}, and b=12. However, in all systems that follow this format, the rules of arithmetic are the same: when adding two single digits,

*a* and

*b*, start from

*a* in the alphabet, step to the next letter

*b* times, and if you go off the end, start at the beginning again and add one (not "1", because that might not be your first digit.

) to the next column on your left. To add multi-digit numbers, just add the columns from right to left.

However, not all number systems work like this. The only restriction on how we write numbers is that our system has to have a unique correspondance between natural numbers and strings. For instance, the Roman system doesn't work like this at all, and is really closer to bizz buzz. In the Roman system, we write:

I

II

III

IIII

IIIIV

IIIIVI

IIIIVII

IIIIVIII

IIIIVIIII

IIIIVIIIIIVX

And for the next number, we write another "I", and then if it's a multiple of 5, another "V", and then if its a multiple of 10, another "X", and so on...

(This looks confusing, because Roman numerals aren't usually written this way, for a good reason: the Romans were just as lazy as the rest of us, so the IIIII leading up to the V were implicitly assumed, and not written down, as were the IIIIVIIIIIV before the X, and so on...)

Because this system produces exactly one string for every (positive, whole) number, and every string it produces corresponds to exactly one number, it still works as a counting system, even though it works nothing like a positional base system.

...And I think I've spouted enough abstract algebra for one post.