General Discussion > Mathematics and Statistics

Duodecimal System Help?

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Clarke:
If you don't mind me asking, why are you drawing tables instead of working out which letter corresponds to which digit? Once you've established the correspondence between Arabic numerals and your new digits, then you can simply take established base-12 arithmetic and substitute the digits, e.g.

1
2
3
4
5
6
7
8
9
A
B
10
11
...
19
1A
1B
20
...

(If one cannot simply substitute the symbols, then it isn't strictly a base-12 system, but is instead something more complex.)

Stranger Come Knocking:

--- Quote from: Clarke on November 26, 2012, 06:47:55 pm ---If you don't mind me asking, why are you drawing tables instead of working out which letter corresponds to which digit?

--- End quote ---
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. :-\

--- Quote from: Clarke on November 26, 2012, 06:47:55 pm ---Once you've established the correspondence between Arabic numerals and your new digits, then you can simply take established base-12 arithmetic and substitute the digits, e.g.

--- End quote ---
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. >:(

--- Quote from: Clarke on November 26, 2012, 06:47:55 pm ---(If one cannot simply substitute the symbols, then it isn't strictly a base-12 system, but is instead something more complex.)

--- End quote ---
Ooh, care to share? *intrigued*

Clarke:

--- Quote from: Stranger Come Knocking on November 27, 2012, 07:38:47 am ---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. >:(
--- End quote ---
The same symbol is used multiple times in the left hand column, so go ask him for clarification.  :P

--- Quote ---Ooh, care to share? *intrigued*

--- End quote ---
*cracks his knuckles, as if about to play piano* ( ;D)

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=d1b0+d2b + d3b2+d4b3+...+ds+1bs
to represent every number by the digits ds+1ds...d3d2d1.
(The fact that this combination both exists and is unique can be proven fairly easily, but that seems unnecessary. :P)

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. :D) 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.  :P ::)

Stranger Come Knocking:

--- Quote from: Clarke on November 27, 2012, 07:58:10 pm ---The same symbol is used multiple times in the left hand column, so go ask him for clarification.  :P

--- End quote ---
:o It's a work in progress. ::)

--- Quote from: Clarke on November 27, 2012, 07:58:10 pm ---
--- Quote ---Ooh, care to share? *intrigued*

--- End quote ---
*cracks his knuckles, as if about to play piano* ( ;D)

--- End quote ---

--- Quote from: Clarke on November 27, 2012, 07:58:10 pm ---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=d1b0+d2b + d3b2+d4b3+...+ds+1bs
to represent every number by the digits ds+1ds...d3d2d1.
(The fact that this combination both exists and is unique can be proven fairly easily, but that seems unnecessary. :P)

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. :D) 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.  :P ::)

--- End quote ---

So how does this help fill in the rest of the chart (100s, 1000s, etc.) using all top letters (except g)?  As of right now, "w" is being left out in the cold.  And it is very cold out.

Or does the whole stinking thing have to be rewritten?

Tìtstewan:

--- Quote from: Stranger Come Knocking on November 28, 2012, 07:38:09 am ---
--- Quote from: Clarke on November 27, 2012, 07:58:10 pm ---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=d1b0+d2b + d3b2+d4b3+...+ds+1bs
to represent every number by the digits ds+1ds...d3d2d1.
(The fact that this combination both exists and is unique can be proven fairly easily, but that seems unnecessary. :P)

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. :D) 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.  :P ::)

--- End quote ---

So how does this help fill in the rest of the chart (100s, 1000s, etc.) using all top letters (except g)?  As of right now, "w" is being left out in the cold.  And it is very cold out.

Or does the whole stinking thing have to be rewritten?
--- End quote ---
Ma oeyä Eywa.... :o :o

I have reworked my table.

Look at the red and the blue field.
I think you may will see, some thing what.