Dialogue on n-Color Numbers (3)

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

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

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.

Dialogue on n-Color Arithmetic (4)

Part 4:
Notes on 2-Number Arithmetic

Si Emo had a patience to wait to listen to Si Nessa's adventure with the Many-Colored Numbers. But to his surprise Ki Algo told him that not all arithmetic is about numbers
Si Emo:  Any other arithmetic?
Ki Algo: The mathematical modulo 2 arithmetic can be rewritten as qualitative verbal arithmetic. By replacing A with Even Number and B with Odd Number, we get the following addition table

+	Even Number	Odd Number
Even Number	Even Number	Odd Number
Odd Number	Odd Number	Even Number

and the multiplication table

.	Even Number	Odd Number
Even Number	Even Number	Even Number
Odd Number	Even Number	Odd Number

Si Emo: Oh, no. I have never thought about that.

Logical Arithmetic

Ki Algo: The other system is logical arithmetic that we get if we replace A with False, B with True and the operations + and * with XOR and AND respectively to get the addition table

XOR	False Statement	True Statement
False Statement	False Statement	True Statement
True Statement	True Statement	False Statement

and the multiplication table

AND	False Statement	True Statement
False Statement	False Statement	False Statement
True Statement	False Statement	True Statement

Si Emo:  My goodness. I thought arithmetic is just for numbers.

Ki Algo: Yes arithmetic is not about numbers. Polynomial can also has arithmetic. It's arithmetic is a ring, but sometimes the finite degree polynomials will have field structure arithmetic like real numbers. But, in general, the finite degree polynomial forms a ring: the ring of polyplex numbers.

Si Emo: Will you teach me the polyplex arithmetic?

Ki Algo: OK. It will help you to understand many-colored numbers. Wait till the next session of the dialogue.

Dialogue on n-Color Arithmetic (3)

Part 3:
The n-Number Arithmetic

Si Emo was astonished with his finding that small finite set of numbers has an arithmetic structure similar to the arithmetic of the infinitely large set of real numbers. But his grandpa Ki Algo enlightened his mind, by showing that such 2-number and 3-number fields is similar to modulo 2 and modulo 3 arithmetic of numbers. Let us listen to the continuing dialogue on n-Number Arithmetic.
Si Emo: Grandpa, I have visited the 4-Number Island and found out nothing interesting in it. It is populated by numbers 0, 1, 2 and 3. Its addition and multiplication rules is nothing but the rules for modulo 4 arithmetic. The table for addition is

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

and the table for multiplication is

.	0	1	2	3
0	0	0	0	0
1	0	1	2	3
2	0	2	1	2
3	0	3	2	1

Ki Algo:  What did you find in 5-number island?
Si Emo: Si Emo:  Nothing interesting in the island. It is populated by numbers 0, 1, 2, 3 and 4. And its arithmetic is similar to modulo 5 arithmetic.

n-Number Arithmetic

Ki Algo: Now, it seems that we can generalized this arithmetic of n numbers: 0, 1, 2, ....and n-1.
The system also defined by two operation: addition + and multiplication . 

a + b = the remainder of (a+b) when it is divided by n
a . b = the remainder of (a.b) when it is divided by  n
Si Emo: Is the arithmetic of n-number, for all natural number n, similar to real number arithmetic for all natural number n.
Ki Algo:  No!
Si Emo:  Why?
Ki Algo:  If  the number n is a nonprime number, then there is some strange property emerge for n is nonprime number which is not equal to powers of prime number
Si Emo:  Let me test it. See, 6 is non prime number. It is 3 times 2. For 6-numbers 2 and 3, they are multiplied to 0. Yes. It is so strange because, in the field of real numbers, if two numbers is multiplied to 0, then one of the two numbers must be 0.
Ki Algo: Mathematicians called 2 and 3 as Zero Divisors for 6-numbers. The existence of  zero divisors makes the arithmetic of 6-number not a field anymore. It's arithmetic is a Ring. Some mathematician call the Field as Division Ring.
Si Emo:  So the 10-number arithmetic is also not a Field, because 10=2.5, but 7-number arithmetic is a field like the 2-number, 3-number and 5-number arithmetic.
Ki Algo:  Yes. All p-number arithmetic, call it F_p, is a field if p is a prime number.
Si Emo:  OK. I know that 4 is not a prime number, but F₄ is a field
Ki Algo: Well! That's another matter. F_n is a field if n is a k-th power of a prime number p or n=p^k to prove that it is really a field is a tricky business, but it is similar in structure to the arithmetic of the  many-colored  numbers that live in the islands, in the lagoons of the Numberland, that Si Nessa visited yesterday. Let us listen to her story to Ni Suiti later.

Dialogue on n-Color Arithmetic (2)

Part 2:
3-Number Arithmetic

Ki Algo: I wonder, if you also went to other finitely populated numberland
Si Emo: Yes Grandpa, I have visited 3-number island. It is populated by 3-numbers.
Ki Algo: What are their melding rules
Si Emo: For addition the rules are simplified to this table

+	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

and for multiplication the rules are simplified to

.	0	1	2
0	0	0	0
1	0	1	2
2	0	2	1

Ki Algo: So they are similar to addition and multiplication modulo 3.
Si Emo: Oh!
Balanced 3-number
Ki Algo: I think the table will be more familiar if we replace 2 with -1
Si Emo: Well, the addition table will be replaced by

+	0	1	-1
0	0	1	-1
1	1	-1	0
-1	-1	0	1

Ki Algo:  What about the multiplication table?
Si Emo: The new multiplication table will be this table

.	0	1	-1
0	0	0	0
1	0	1	-1
-1	0	-1	1

Ki Algo: What is the arithmetic structure of the 3-numbers.
Si Emo: Because both multiplication and addition are commutative and associative, and the multiplication distributes upon the addition, and every nonzero number has an additive and multiplicative inverse, then it is a field arithmetic, similar to real number arithmetic.
Ki Algo: So, probably, it will be useful for engineers to build their coding system. To know n-number system more deeply you have to visit the islands in numberland which is populated by more numbers.