Duodecimal System Help?

Started by Stranger Come Knocking, November 25, 2012, 08:10:48 AM

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Stranger Come Knocking

No, definitely not nìNa'vi, but math-related anyway.  Trying to fill in this chart:



I think I know what I need; I think I need to drop one letter, but I can't quite split my mind to wrap it around the problem.  Can someone help me organize my thoughts? :) Much appreciated.

EDIT: NEW THOUGHT: After 10012, should the next base be 100,00012 instead of 1,00012? >.>
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

#1
1.728 = 1.00012 :-\
2.985.984 = 1.000.00012
5.159.780.352 = 1.000.000.00012

The root of 2.985.984 = 1.000.00012 is 1.728 = 1.00012
5.159.780.352 = 1.000.000.00012 = 2.985.984 = 1.000.00012 x 1.728 = 1.00012

My head burning... :-X

Quote from: Stranger Come Knocking on November 25, 2012, 08:10:48 AM
EDIT: NEW THOUGHT: After 10012, should the next base be 100,00012 instead of 1,00012? >.>
1.728 = 1.00012. The next step would be 20.736 = 10.00012

I don't understand, why it "jumps" to 2.985.984 = 1.000.00012. :-\

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Stranger Come Knocking

Quote from: Tìtstewan on November 25, 2012, 08:31:19 AM
1.728 = 1.00012 :-\
2.985.984 = 1.000.00012
5.159.780.352 = 1.000.000.00012

The root of 2.985.984 = 1.000.00012 is 1.728 = 1.00012
5.159.780.352 = 1.000.000.00012 = 2.985.984 = 1.000.00012 x 1.728 = 1.00012

My head burning... :-X
You and me both. @.@

Quote from: Tìtstewan on November 25, 2012, 08:31:19 AM
Quote from: Stranger Come Knocking on November 25, 2012, 08:10:48 AM
EDIT: NEW THOUGHT: After 10012, should the next base be 100,00012 instead of 1,00012? >.>
1.728 = 1.00012. The next step would be 20.736 = 10.00012
So instead of starting "eu" at 1,000, it would start at 10,000?  And then "ju" would be 1,000,000 and "ou" 100,000,000? ???

Quote from: Tìtstewan on November 25, 2012, 08:31:19 AM

I don't understand, why it "jumps" to 2.985.984 = 1.000.00012. :-\
Me either.  Let's ask someone. >_>
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

Quote from: Stranger Come Knocking on November 25, 2012, 01:37:14 PM
Quote from: Tìtstewan on November 25, 2012, 08:31:19 AM
Quote from: Stranger Come Knocking on November 25, 2012, 08:10:48 AM
EDIT: NEW THOUGHT: After 10012, should the next base be 100,00012 instead of 1,00012? >.>
1.728 = 1.00012. The next step would be 20.736 = 10.00012
So instead of starting "eu" at 1,000, it would start at 10,000?  And then "ju" would be 1,000,000 and "ou" 100,000,000? ???
In the table "eu" is 172810 => 1.00012.
But in the table "ju" 2.985.98410 = 1.000.00012, that is 100012 time of "eu" (1.72810 x 1.72810 = 2.985.98410)
The "ou" 5.159.780.35210 = 1.000.000.00012, that is 100012 time of "ju" (1.72810 x 2.985.98410 = 5.159.780.35210)

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Stranger Come Knocking

Quote from: Tìtstewan on November 25, 2012, 02:06:16 PM
In the table "eu" is 172810 => 1.00012.
But in the table "ju" 2.985.98410 = 1.000.00012, that is 100012 time of "eu" (1.72810 x 1.72810 = 2.985.98410)
The "ou" 5.159.780.35210 = 1.000.000.00012, that is 100012 time of "ju" (1.72810 x 2.985.98410 = 5.159.780.35210)
:/ Then what about 144/10012?  ɬəj = 10012.  [so on and so forth] wəj = 100012 -> 10x100



100
...
B00




1,000
...
B,000




10,000
...
B0,000




100,000
...
B00,000




1,000,000
...
B,000,000




10,000,000
...
B0,000,000




100,000,000
...
B00,000,000




1,000,000,000
...
B,000,000,000




10,000,000,000
...
B0,000,000,000




100,000,000,000
...
B00,000,000,000




1,000,000,000,000
...
B,000,000,000,000


Therefore:


ɬəjbflsçʒmntp'w
1001,00010,000100,0001,000,00010,000,000100,000,0001,000,000,00010,000,000,000100,000,000,0001,000,000,000,00010,000,000,000,000

And then:

ɬɛu = 100,000,000,000,000

And ɬju = ..................................@_@
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

Have you created the table at the first post? ???

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Tìtstewan

#6
So, my brain is not exploded yet... :o ;D
I tryed to create this table:


Edit:
Error fixed

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Stranger Come Knocking

I know it looks like that. ??? But the "w" is left open and that's not quite what is wanted.  But I actually figure out that you can move up two places from a starting point going by ones (0-100 by 1s) and fill every block, but not up one place (10-100) by changing the place of the first and still fill in all the blocks.  Yup, this is how it is. ^_^

Irayo, ma Tìtstewan. :D
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

I hope you understand this system...
I do not know if I have understood it correctly. :-[ :-X :o
I just calculated it...
*search for a brain cooling system* ;D

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Stranger Come Knocking

*gives you slushie* :P

Actually, the calculations were helpful. :)

Irayo again.

Topic closed. (Unless something monumental is contributed)
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

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?
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.
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.)
Ooh, care to share? *intrigued*
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

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. >:(
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*
*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
:o It's a work in progress. ::)

Quote from: Clarke on November 27, 2012, 07:58:10 PM
Quote
Ooh, care to share? *intrigued*
*cracks his knuckles, as if about to play piano* ( ;D)


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 ::)

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?
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

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 ::)

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?
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.



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Stranger Come Knocking

Quote from: Tìtstewan on November 28, 2012, 11:15:58 AM
Ma oeyä Eywa.... :o :o

I have reworked my table.


Look at the red and the blue field.
I don't think so.  "g" numbers are used to rep. "gajillions" or other impossible numbers.  But I think just having two words for "one thousand" and whatnot might be the best option here. :)


Quote from: Tìtstewan on November 28, 2012, 11:15:58 AM
I think you may will see, some thing what.
??? Eh?
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

Quote from: Stranger Come Knocking on November 28, 2012, 02:34:31 PM
Quote from: Tìtstewan on November 28, 2012, 11:15:58 AM
I think you may will see, some thing what.
??? Eh?
Just forgot this line...
I thought everything had to be started with the zero...
  0 --> 1012
  0 --> 10012
....
And between the zero and X the 12th parts...

My problem is, I do not understand how the numbers associated with the letters.
When I think more and more about it, my head will burning. :o :-X

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Stranger Come Knocking

Quote from: Tìtstewan on November 28, 2012, 02:59:26 PM
My problem is, I do not understand how the numbers associated with the letters.
When I think more and more about it, my head will burning. :o :-X
In duodeci, the numbers between nine and ten are A and B because there simply aren't any more digits.

Or if you are talking strictly about the graph,

ɬɑ = 0
bɑ = 1
fɑ = 2
lɑ = 3
sɑ = 4

and so on.  There are thirteen letters along the top.  Discounting the "g" it leaves 12.  Assuming the chart would keep going on and on and on forever, there would be no problem.  But the my friend who made this up made it so that there is an extra letter.

The 0-143 chart is fine; there is nothing wrong with it.  The problem comes in with the next few rows.  In base 10, it would look like this:

100 200 300 400 500 600 700 800 900 1,000

1,000 2,000 3,000 ...........................10,000

10,000............................................100,000

When it should be

100 200..........................................900

1,000 2,000.....................................9,000

10,000............................................90,000

There's just that extra infuriating letter.  That's why I think having two words for 1,000 (the first example) would be easier than trying to squish and rearrange everything to fit the second example.  That or go with my original theory in which ɬɛu = 100,000,000,000,000. :o

I've sent her the link to this thread, so I guess it's up to her.

(And if you're reading this, it was pointed out that ɛj is used twice.)
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.

Tìtstewan

Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
Quote from: Tìtstewan on November 28, 2012, 02:59:26 PM
My problem is, I do not understand how the numbers associated with the letters.
When I think more and more about it, my head will burning. :o :-X
In duodeci, the numbers between nine and ten are A and B because there simply aren't any more digits.
Thats no problem, I know it from the hexadecimal system.

Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
Or if you are talking strictly about the graph,

ɬɑ = 0
bɑ = 1
fɑ = 2
lɑ = 3
sɑ = 4

and so on.  There are thirteen letters along the top.  Discounting the "g" it leaves 12.  Assuming the chart would keep going on and on and on forever, there would be no problem.  But the my friend who made this up made it so that there is an extra letter.

The 0-143 chart is fine; there is nothing wrong with it.
Ah ok.
So
  120 = ɬji
  109 = boj
and so on - correct? :-\

Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
The problem comes in with the next few rows.  In base 10, it would look like this:

100 200 300 400 500 600 700 800 900 1,000

1,000 2,000 3,000 ...........................10,000

10,000............................................100,000

When it should be

100 200..........................................900

1,000 2,000.....................................9,000

10,000............................................90,000

There's just that extra infuriating letter.  That's why I think having two words for 1,000 (the first example) would be easier than trying to squish and rearrange everything to fit the second example.  That or go with my original theory in which ɬɛu = 100,000,000,000,000. :o

I've sent her the link to this thread, so I guess it's up to her.
A question: When 144 = әj, what would be 145?  Not ɬәj = 145? :-\

Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
(And if you're reading this, it was pointed out that ɛj is used twice.)
Oops! :-[ fail :-X
I will fix it in my document.

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Stranger Come Knocking

Quote from: Tìtstewan on November 28, 2012, 04:16:54 PM
Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
Or if you are talking strictly about the graph,

ɬɑ = 0
bɑ = 1
fɑ = 2
lɑ = 3
sɑ = 4

and so on.  There are thirteen letters along the top.  Discounting the "g" it leaves 12.  Assuming the chart would keep going on and on and on forever, there would be no problem.  But the my friend who made this up made it so that there is an extra letter.

The 0-143 chart is fine; there is nothing wrong with it.
Ah ok.
So
  120 = ɬji
  109 = boj
and so on - correct? :-\
Yup. :)

Quote from: Tìtstewan on November 28, 2012, 04:16:54 PM
A question: When 144 = әj, what would be 145?  Not ɬәj = 145? :-\
ɬәjbɑ 10012+1

Because ɬәj would be the "0" on the next 0-99 chart (if the chart was expanded to go up to 199).

Quote from: Tìtstewan on November 28, 2012, 04:16:54 PM
Quote from: Stranger Come Knocking on November 28, 2012, 03:57:50 PM
(And if you're reading this, it was pointed out that ɛj is used twice.)
Oops! :-[ fail :-X
I will fix it in my document.
I was addressing her and I think it was one of those drive-by errors. o_o

Which do you think would be better?  Right now, because of the extra letter, we have two words for 50012.  (And two for 5,000, and so on).  Do you think 500 would be better or 1,000?
I will not die for less
I dug my grave in this
Will I go before I fall
Or live to slight the odds?

These are my books.  You should check it out.  Speculative sci-fi murder mystery historical fiction.