Dialogue on n-color Numbers (2)

DIALOGUE ON MANY-COLOR NUMBERS

Part Two: The Fascinating 3-Color Numbers

In the last dialogue Ki Algo sees the 2-color number system as a representation of polynomial number system. But we can also see the polynomial numbers or polyplex number system as the representation of colored number system. In the following dialogue we will listen how  Ki Algo teach Si Emo about 3-color number using the concept of polyplex number.

3-Color Number Multiplication table

Ki Algo: Now, let us talk about 3-color number
Si Emo: Do you mean a number like 1 + 2 + 3 ?
Ki Algo: Yes. It can also be represented by polynomial 1 + 2 x + 3 x²
Si Emo: That means it is a rest polynomial when a polynomial is divided by degree 3 polynomial. It is a number that is called 3-plex number or terplex number
Ki Algo: Good, you remember lesson number one perfectly. Now, let us investigate 3-plex or terplex numbers generated by the dividing polynomial x³-1.
Si Emo: That means we have multiplication table for mononomial units like this

*	1	x	x2
1	1	x	x2
x	x	x2	1
x2	x2	1	x

Ki Algo: That's good. Let us represent the mononomial with colored ones. For example x with orange one 1and x2 with green one 1
Si Emo: OK the multiplication table are now like this

*	1	1	1
1	1	1	1
1	1	1	1
1	1	1	1

Ki Algo: How can you use it to find the multiplication of  (1+2-3)*(2-3+4) for example
Si  Emo: I just put the first number 1+2-3 in the leftmost column and the second number 2-3+4 to the uppermost row and multiply each element of the leftmost column with each element of the uppermost row like this

*	2	-3	4
1	2	-3	4
2	1	-6	8
-3	-6	9	-12

and then I add up the elements of the 3x3 white matrix colorwise like this
(+2+8+9) + (-3+1-12) + (+4-6-6) = 19 - 14 - 8.
So  (1+2-3)*(2-3+4)  =  19 - 14 - 8
Ki Algo: Good you're already understand the arithmetic of terplex or 3-color number

Self-Powered 3-Color Number

Ki Algo: There is a 3-color numbers which are powered to themselves
Si Emo: What is the self powered number?
Ki Algo: It is 1/3 + 1/3 +1/3
Si Emo: Let me check it using the multiplication table

*	1/3	1/3	1/3
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9
1/3	1/9	1/9	1/9

I will collect the same color numbers and add them up
(1/9 + 1/9+1/9) + (1/9 + 1/9 +1/9) + (1/9 + 1/9 +1/9) = 1/3 + 1/3 +1/3.
Yes, it is self-squared.(1/3 + 1/3 +1/3)² = 1/3 + 1/3 +1/3
So, it is also self-powered (1/3 + 1/3 +1/3)ⁿ = 1/3 + 1/3 +1/3 for any n
 Ki Algo: You can also checked this number 2/3 - 1/3 -1/3
Si Emo: OK

*	2/3	-1/3	-1/3
2/3	4/9	-2/9	-2/9
-1/3	-2/9	1/9	1/9
-1/3	-2/9	1/9	1/9

the same color number will add up like this
(4/9 + 1/9 + 1/9) + (-2/9-2/9+1/9) + (-2/9+1/9-2/9) = 2/3 - 1/3 - 1/3
so it is self-squared
(2/3 - 1/3 - 1/3)² = 2/3 - 1/3 - 1/3
and consequently it is also self-powered
(2/3 - 1/3 - 1/3)ⁿ = 2/3 - 1/3 - 1/3 for any n
Ki Algo: Yes, it is.
Si Emo: Are there any more self-powered 3-color numbers?
Ki Algo: Yes, but they are not interesting because it is so obvious. They are 1 and 0
Si Emo:  Not interesting at all. They inherited from Black Number

3-Color Zero divisors

Ki Algo: Now about your question in the first dialogue. Is terplex number system a field?
Si Emo:  Well, is it?:
Ki Algo: No, it has infinitely many zero divisors.
Si Emo: Show me two of them!
Ki Algo: You can multiply the two self-powered 3-color numbers. Check it up!
Si Emo: I will put  1/3 + 1/3 +1/3  in the leftmost column part and  2/3 - 1/3 -1/3 in the uppermost row part of the 3-color multiplication.

*	2/3	-1/3	-1/3
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9
1/3	2/9	-1/9	-1/9

I'll collect numbers of the same color  and add it up.
(2/9 -1/9 -9) + (-1/9 + 2/9 -1/9) + (-1/9 -1/9 + 2/9) = 0
Wow,
(1/3 + 1/3 +1/3)*(2/3 - 1/3 -1/3)=0
So, self-powered 3-color numbers are zero divisors
Ki Algo: Yes, you can also try to multiply  1/3 + 1/3 +1/3 with a + b + c with a+b+c=0
Si Emo: OK. I'll try it up.

*	a	b	c
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3
1/3	a/3	b/3	c/3

I'll collect numbers of the same color  and add it up.
(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)+(c/3 + b/3 + a/3)=(a/3 +c/3 + b/3)+(b/3 + a/3 +c/3)1+(c/3 + b/3 + a/3)1=0 because a+b+c=0
Oh my goodnes! There are infinite number of 3-color numbers zero divisors. It's fascinating.
Ki Algo: Yes, 3-color number system is not a field. But like Black-Red or Counter-Complex numbers it is a direct sum of two fields anyway.
Si Emo: Oh yeah? What fields are they?
Ki Algo: Wait until our next dialogue.

Notes on the Dialogue

3-color number system is a generalization of complex number system.

1. It is called 3-polyplex numbers by the Czech Marek 17 in his theory of polyplex numbers in http://tech.groups.yahoo.com/group/hypercomplex/
2. It is called terplex numbers by the British Roger Beresford in his theory of Hoop Algebras in here
3. It is called tricomplex numbers by the Rumanian physicist Silviu Olariu in his theory of n-complex numbers in his paper http://front.math.ucdavis.edu/0008.5120
4. It is called 3-numbers by the Russian group who study the polynumbers  in http://hypercomplex.xpsweb.com/page.php?lang=en&id=148