Dialogue on n-Color Numbers (1)

Part One: Re-Viewing 2-Color Numbers

In the last dialogue on n-number we found out that they arithmetically are field like real numbers. Maybe you wonder if there are many colored n-number too, and are they also a field. Before we answer the question, It is probably helpful if we are exploring the possibility many colored real numbers as n-color number with n>2. To seek the answer for this mysterious possibility, I think it is good if we look into two color number in a different perspective: the perspective of polyplex number. Let us eavesdrop the dialogue between Ki Algo and his grandson Si Elmo on 2-color numbers.

2-Color Numbers as Duplex Numbers

Ki Algo:   Do you remember about 2-color number that Si Nessa found in Numberland
Si Emo: Of course. There are two kinds of 2-color numbers. The Black-Red numbers and the Black-Pink numbers
Ki Algo:  Do you that they actually are Counter-Complex Numbers and Complex Numbers respectively.
Si Emo: Yes I do.
Ki Algo:  Well, both the Counter Complex number and the Complex number is actually binary or two dimensional number. I will call binary number as a duplex numbers.
Si Emo: What is duplex number?
Ki Algo:  Well a duplex number is the set of all polynomial that will give the same rest polynomial when divided by x²+px+q.
Si Emo: Wow. The set has infinite number of elements.
Ki Algo:  Yes, but all the polynomial set can be represented by the similar rest polynomial.
Si Emo: Because the dividing polynomial is a quadratic or degree two then the rest polynomial is degree one or linear function such as 2 +5 x.
Ki Algo:  Yes only two coefficients for each rest polynomial.
Si Emo: There are infinitely many linear polynomials.

Algebraic Operations

Ki Algo:  Yes. There are uncountable infinite number of real numbers, but  they can be added and multiplied to each other to get another real number which is the member of the same set. Mathematicians called the set as closed to two fundamental operation addition and multiplication
Si Emo: how can we add two polynomials?
Ki Algo:  By adding their coefficients for x⁰=1 and x.
Si Emo: So (1 + 2 x) + (4 + 3 x)=(1+4) + (2+3) x = 5 + 5 x. That's easy, but how can we multiply two duplex numbers.
Ki Algo:  By multiplying both polynomials and then dividing it with x²+px+q to get the rest of the division. The rest polynomial is the result of the duplex multiplication.
Si Emo: That's a long work.
Ki Algo:  Yes you can get the same result by multiplying the both duplex number and then replacing x² in it with (-px-q) in the result.
Si Emo: OK. That's easy. (1+2x)(4+3x)=4+11x+6x²=4+11x+6(-p-qx)=(4-6p)+(11-6q)x. So every multiplication is depending on p and q.
Ki Algo:   Yes. But you can do your multiplication more easy with with the following table multiplication of units

*	1	x
1	1	x
x	x	px+q

Si Emo: OK. But now, I see that for complex numbers we have p=0 and q=-1. For counter-complex number we have p=0 and q=1
Ki Algo:  Your right. Here are their unit multiplication tables

*	1	1
1	1	1
1	1	-1

for complex number rewritten as Black-Pink numbers and

*	1	1
1	1	1
1	1	1

for counter-complex number rewritten as Black-Red numbers.

3 Kinds of 2-Color Numbers

Si: Emo: So the Black-Red numbers and the Black-Pink number is just 2 kinds of duplex numbers. Is there any other 2-color number as a kind of duplex numbers.
Ki Algo:  As you know it, the characteristic polynomial of general duplex number is x2+px+q. Now you can rewrite x²+px+q as (x - 1/2 p)² - 1/4 p² + q.  So x²+px+q= 0 can be written as  y² - D = 0 where y=x+1/p and D=1/4 p² - q or y² = D.

Si Emo: And then?
Ki Algo:  If D is negative then the duplex number system is characterized by x²+px+q has the same arithmetic rules as the complex number or Black-Pink Numbers.
Si Emo: What if D is positive?
Ki Algo:  If D>0 then  the duplex number system is characterized by x²+px+q has the same arithmetic rules as the counter-complex number or Black-Red Numbers.
Si Emo:  OK, but what if D is zero?
Ki Algo:  If D=0 then the duplex number system is characterized by x²+px+q has the same arithmetic rules as the dual number.
Si Elmo: What is dual numbers?
Ki Algo:  Dual number is two dimensional number where its non-real unit is squared to zero. If we call the squared to zero unit as Brown Unit then It has the unit multiplication table

*	1	1
1	1	1
1	1	0

Si Emo: Oh, What a weird number. Brown Unit squared to zero. So all brown numbers are squared to zero.
Ki Algo:  Yes. To answer your question: there are three kinds of duplex number system:

1. the complex numbers,
2. the counter-complex numbers, the complex numbers and
3. the dual numbers.

So there are only three kinds of 2-color number system:

1. The Black-Pink Numbers
2. The Black-Red Numbers
3. The Black-Brown Numbers

Si Emo: Are they similar to real or black number arithmetic?
Ki Algo: No, only the first one. The other ones are plagued by zero powered number like 3 and zero divisors like (2+2) and (5-5)
Si Emo: Oh yes, 3*3=0 and (2+2)(5-5)=0. It's so sad that they are so pathological. I wonder if 3-colored number is also pathological.
Ki Algo:  Wait until next session of our dialogue.
NOTES ON THE DIALOGUE

Many color number is many dimensional number. Many dimensional numbers are numbers which have to be described by more than one real number.  The maximum dimension which is taught in high school is two. The two dimensional number is the complex numbers which is discovered by the Italian physician  Gerolamo Cardano in 1576 .. The complex number is the combination of  real numbers and imaginary numbers.

Imaginary numbers is the imaginary unit multiplied by real numbers. The imaginary unit i is a number which can be squared to negative unity or -1. Usually the square of any real number, positive and negative, is also a positive real number. So it is difficult to conceive a number which can be squared into a negative number. That's why mathematicians called them, following the famous Rene Descartes (1596-1650) , as imaginary numbers. But "imaginary" is just a name. Rather than dubbed it as "imaginary" number, we can also dubbed it as "pink" number such as it is used by the imaginary dialogue of Ki Algo and his relatives in Arma's blog http://integralist.multiply.com.