Armahedi Mahzar © 2008

Prologue on a Dialogue

When I am retired as a physics lecture in 1999, I started exploring the cyberspace to

compare my hypothesis of prequarks which was proposed in the newsgroup.

Eventually I found a great website of Tony Smith, a lawyer who happened to be a

very bright amateur physicist, who discussed many new kinds of number more

general than complex numbers called hypernumbers discovered by the Charles

Muses.

In those webpages I read the name of Kevin Carmody who is the owner of the

hypernumbers egroup. I join the egroup and becoming friend to a number of

mathematicians by hobby, one of them is the Marek Czech wonderboy. What so

wonderful about Marek is that he discovered another system of generalized complex

numbers he called polyplexes.

In his polyplex system, complex numbers are just special types of duplex numbers

characterized by quadratic equations. Duplex numbers is a combination of real

numbers with kinds of non-real numbers : the counter imaginaries and the duals.

Later on I have an idea to represent polyplexes with many colored numbers. 2-colored

numbers is the representation of duplex numbers to economized the notation.

In multiply.com I write a dialogue on the many color numbers between Ni Suiti and

her husband Ki Algo and their grandchildren: Si Emo and Si Nessa. There are some

notes about the dialogue on 2-color numbers

1. Si Nessa discovered that there are two kinds of 2-color number. One type is 2-color

numbers with Red number unit which is squared to unity and the other type is 2-color

numbers with Pink number unit which is squared to minus unity. The Red 1 and the

Pink 1 is respectively represent counter-imaginary unit and imaginary unit i of the

counter-complex and complex numbers system.

2. Color symbology is more economic because we do not have to use new letter

symbols for new non-real units. We are just coloring the real number symbol to

represent the non real number. Of course the colors chosen is arbitrary. This color

symbology will make the hypercomplex number system more intuitive and

comprehensible for primary school students.

3. It can be proved that Black-Red numbers or countercomplex number, while does

not form a field arithmetic, can be regarded as direct sum of two fields, each is

isomorphic to real number field, form by by real multiples of idempotent because the

the two idempotents are zero divisors. That's why it does not have many interest

from mathematician. On the other side, the black-pink numbers or complex numbers

are very useful to physics and engineering.

4. But it can also be proved that complex and countercomplex number system is just

two of three binary or duplex number systems in which non-real units is following

quadratic equation. The other one is dual number system where its non real number

unit d is squared to zero. This number system can be realized as another 2-color

number, say Black-Brown number, with multiplication checker board.

where the white box represents zero.

5.In the mythical Numberland, Black-Brown numbers more likely to populate the

Dichromic Zero province that has not been visited by Si Nessa. The Dichromic

provinces are in the many dimensional number country: Polychromic. The

polychromic country is the mythical symbol for higher dimensional number domain:

the hyper-complex numbers. Discussion of hypercomplex number can be found in

hypercomplex@yahoogroups.com maintained by Jens Koeplinger. Anybody who are

interested to many dimensional numbers can join the egroups to discover how binary

numbers itself is just special kind polyplex numbers that was discovered by my

creative friend Marek Ètrnáct.

6.The Black-Red or Counter-complex numbers and Black-Pink or complex numbers

are only parts of the more mysterious hypernumbers discovered by late logician and

philosopher Charles Muses that was discussed in hypernumber at yahoogroups.com

maintained by myself. I created the egroup to continue the discussion in

hypernumbers@yahoogroups.com that was extinct because it was closed by the

owner Kevin Carmody, an informal student of Charles Muses, has been redirect his

attention from the study of hypernumbers to the study of science of consciousness in

the context of Maharishi Mahesh Yogi Transcendental Meditation in the now extinct

scienceofconsciousness@yahoogroups.com.

Thank you

Arma

CHAPTER ONE 2-Color Numbers

Ni Suiti was in the veranda when her granddaughter Si Nessa come to tell her

experiences when she went to Numberland brought by her aunt Mak Retitia. She told

Ni Suiti that the Numberland is a wonderful land. In one region of the island he found

out that in there there are colorful numbers.

Si Nessa: Grandma, I have never known that numbers in the amazing Numberland

where numbers have colors.

Ni Suiti: Do you mean that we can find numbers with the same value but has

different colors?

Si Nessa: Yes, but more interesting is their properties. When two numbers meld into

another number then they become another colored number.

Ni Suiti: What is the color of the new number?

Si Nessa: They get a new color or stay in their original color according to the way

they meld: addition or multiplication.

Ni Suiti: What is the rules for color change for the addition?

Si Nessa: If the colors are similar, the result is similar color

Ni Suiti: So, red one plus red three is red five.

2 + 3 = 5

Black two plus Black one is Black three.

2 + 1 = 3

But what if we add two numbers of

different colors?

Si Nessa: It yields a duet which is a pair of numbers of different colors

Ni Suiti: How many kinds of colored number are there in Numberland?

Si Nessa: I was in two colored number region of the land. They call it Bichromic

province which is populated by two kind of number: Black and Red. But there are

many other regions which is occupied by numbers with more than two colors. I have

never been there.

Ni Suiti: Now what is the multiplication rules of of two colored numbers?

Si Nessa: That's really simple. Black number does not change the color of the

number it multiply. Red number change it.

This rule can be simplified to:

Similar colors are multiplied to Black

Different colors are multiplied to Red.

Ni Suiti: That means

Black 2 times Red 3 is equal to Red 6

2 X 3 = 6

and Red 2 times Black 3 is equal to Red 6

2 X 3 = 6

and Red 2 times Red 3 is equal to Black 6

2 X 3 = 6

Si Nessa: That's correct. But Dichromatic province is more populated by pairs of

numbers of different colors called duets.

Ni Suiti: I wonder what are the rules of duet melding

Si Nessa: If it is addition, then any colored number of the first duet will add to the

member of the other duet of the same color. In short: Equal colored number add up

the same color

Ni Suiti: You mean

2 + 3 +

1 + 2 =

3 + 5

Si Nessa: You are right grandma.

Ni Suiti: Now how can we multiply two duets?

Si Nessa: We multiply every colored number of the first duet to every colored

number of the second duet and add all the results up. In short: Add up all possible

multiplications

Ni Suiti:

(1 + 2) X (3 + 2) =

=1 X 3 + 1 X 2 + 2 X 3 + 2 X 2 =

= 3 + 2 + 6 + 4 =

= 3 + 4 + 2 + 6 =

= 7 + 8

Si Nessa: Right again Grandma

Ni Suiti: What about negative numbers

Si Nessa: They're all has the same rules.

Ni Suiti: So we can have Red minus 2 = minus Red 2 = - 2

Si Nessa: Good Grandma!

Ni Suiti: So we will make a colored number dissapear if we add it to its negative.

Red 3 + Red minus 3 =

Si Nessa: Yes. That's the magic of colored numbers

.

Ni Suiti: In summary,

For single color number addition of similar color singlet yields similar colored

singlet.

Different color numbers add to a duet.

Adding colorful number duets is adding their members colorwise.

Multiplying by black singlet does not change color.

Multiplying by red singlet changes the color.

Mutiplying duets is adding all the multiplication of of their colored members.

Si Nessa: Good Grandma. But, sorry I have to go home now. Because I have to

prepare my self for tomorrow Journey to the another province of the Numberland: the

Bichromic Two. Bye, now.

CHAPTER TWO 2-Color Arithmetic

Ni Suiti was in in the company of his husband, Ki Algo, the grandfather of Si Nessa. She told him about the discoveries of their granddaughter in the Bichromic

One. She like to know his opinion to his granddaughter's discoveries. Then Ni Suiti told Ki Algo about the rules of color transformation due to the arithmetic operations

as it was told by Si Nessa

Ki Algo: I am surprised, but I think the table for addition is the following

+ c d

a a + c a + d

b c + b (b +

d)

Ni Suiti: That's cool.

Ki Algo: From the table it can be shown that the multiplication has the following

property:

If a, b and c are colored number then

(1) a + b = b + a

(2) a + (b + c) = (a + b) + c

Ni Suiti: So the ordering of the addition does not matter.

Ki Algo: The multiplication for colored number singlets is summarized in the

following table

X c d

a ac (ad)

b cb (bd)

Ni Suiti: That's also cool.

Ki Algo: From the table it can be shown that the multiplication has the following

property:

If a, b and c are colored number then

(1) ab = ba

(2) a(bc) = (ab)c

(3a) a(b+c) = ab + ac

(3b) (a+b)c = ac + bc

Ni Suiti: So the muliplication is indifferent of ordering of terms and it is both left

and right distributive to addition.

Ki Algo: It can easily proven that the 2-color number system has a multiplicative

unit: Black 1 or 1

1 (x + y) = (x + y) 1 for any duet x + y

Ni Suiti: So is both left and right unit.

Ki Algo: I can also prove that there also have an additive unit: Zero.

Zero = x - x which has the following property

Zero + a = a + Zero = a

Ni Suiti: I think zero is colorless

Ki Algo: From the table I can derive the formula for multiplying two colored number

duets

(Black a + Red b)(Black c + Red d) = Black (ac+bd) + Red

(ad+bc)

If the duet x + y is abbreviated as (x, y) ,then the rule of multiplication is

(a,b)(c,d) = (ac+bd, ad+bc)

Ni Suiti: Simple formula to represent the long table. But the wonderful colors is

lost. What a pity.

Ki Algo: Conclusively, the 2-color numbers form what the mathematician called

Ring. Of course The mathematician Ring is not some thing you can wear in your

finger, it is a collection of numbers with two compositions (+ and .) which follow

certain axioms.

Dichromic numbers form a Ring because for all 2-color numbers a, b and c follow

the following eight Axioms

Four Axioms of Addition

(R1) (a + b) + c = a + (b + c) ( the addition + is associative)

(R2) Zero + a = a (existence of identity element for addition)

(R3) a + b = b + a (+ is commutative)

(R4) for each 2-color number a there is a 2-color number −a such that a +

(−a) = (−a) + a = Zero

(−a is the additive inverse element of a)

Two Axioms of Multiplication

(R5) (a . b) . c = a . (b . c) (the multiplication . is associative)

(R6) 1 . a = a . 1 = a (existence of identity element for multiplication)

Two Axioms of Distribution

(R7) a . (b + c) = (a . b) + (a . c) (left distributivity of multiplication)

(R8) (a + b) . c = (a . c) + (b . c) (right distributivity of multiplication)

Ni Suiti: Wow. That's right but I lost the visual beauty of the colored. numbers.

Ki Algo: Yes, but now you gain the beauty of logical consistency.

CHAPTER THREE Strange Numbers

Ni Suiti was so bewildered by Ki Algo exposition of Ring as the arithmetic structure of 2-color numbers. She thought there is nothing strange with that at all. All the Ring axioms are also followed by real numbers. So real numbers arithmetic is also a Ring.

Ni Suiti: I suspects that the 2-colored numbers has similar arithmetic as the real

numbers.

Ki Algo: Oh, no. There are duet numbers which is squared to themselves. z2

=

z

Ni Suiti: I think that is not so. Real number arithmetic has those too. Zero and Unity

is such a number

Ki Algo: Well the 2-color numbers have other numbers squared to themselves

beside them.

Ni Suiti: What numbers?

Ki Algo: They are z1= 1/2 + 1/2 and

z2= 1/2 - 1/2

Ni Suiti: My goodness. There are two of them.

Ki Algo: Mathematicians called the number as Idempotent number. Idem means

equal, potent means power. Because if you power them with any number then the

results will be equal to themselves. zn

= z with n any integer.

Ki Algo: OK you know now that there are two really duet numbers that square

themselves to themselves. Now try to multiply them to each other.

Ni Suiti:

z1.z

2= (1/2 + 1/2)(1/2 - 1/2)=Zero

Oh! It is very strange. In 2-color arithmetic, zero is equal to multiplication of two non

zero 2-colored numbers. No nonzero real numbers will multiply themselves to zero.

Ki Algo: They called by mathematician as Zero Divisors. In fact there are

infinity of zero divisors. All multiple of z1 and z2 are zero divisors. (3 + 3)(5

- 5)=Zero for example. The existence of strange numbers, Idempotents and

Zero Divisors, shows us that 2-color arithmetic is not similar in structure to real

number arithmetic.

Ni Suiti: OK, I am wrong. The arithmetic of 2-Color Numbers is not similar to the

arithmetic of the real numbers. They have more idempotents and infinity of zero

divisors.

Ki Algo: Actually, mathematicians called the arithmetic of real number as Field

and the arithmetic of 2-color number as commutative Ring with unity (which is

Black 1 as unity). A Field is a commutative Ring with unity containing no Zero

Divisor.

Ni Suiti: So, the 2-ColorNumber algebra is unique because it has unique structure as

the ring with infinite zero divisor and a pair of idempotent.

Ki Algo: No, it's not unique. The Ring of 2-Color Numbers has similar arithmetic

structure to counter-complex numbers with two units 1 and where both

units are squared to one. Each of them equivalent to 1 and 1 . Other arithmetic

similar in structure to the 2-Color arithmetic is the Group Algebra based on the

2-element reflection group.

Ni Suiti: Anyway, I think all 2-color Numbers has common arithmetic property.

Ki Algo: I do not think so. Please wait for Si Nessa after her travel to Bichromic

Two and beyond. See what she found there.

Ni Suiti: Ok. We will see who is right. You or me?

Note:

I am not a mathematician, just a retired physicist. Exploring new kinds of number is just my hobby. Please correct me if I am wrong. Thank you. Arma

CHAPTER FOUR: Arithmetic Similarity

Si Nessa had returned from Bichromic Two which is a province in

Numberland. Bichromic Two is populated by 2-color numbers consisting black and pink numbers. Bichromic One, that she had visited before, is a province populated by 2-color numbers consisting of black and pink numbers. Si Nessa found out that both regions have similar rules of composition except

for the multiplication rules for the same colored numbers. The pink number times a pink number is a negative black number, while she know before that the red number times a red number is simply black number. That's why she called the Black-Pink number is a twisted 2-color number. She told her grandma Ni Suiti about her findings in the company of her brother Si Emo and her Grandpa Ki Algo:..

Ni Suiti: Nessa, your discovery interesting, but I'll let you know that your grandpa Ki

Algo reformulate my verbal rules for 2-color number multiplication with the

following table.

X c d

a ac (ad)

b cb (bd)

Si Nessa: Wow, it is difficult for me to memorize.

Ni Suiti: For you, if you remember distribution axiom, I will simplify your

grandpa's table by changing all letters with number 1, then the multiplication table

can be simplified into

X 1 1

1 1 1

1 1 1

in more simplified form

1 1

1 1

Si Nessa: Yes. That is a simpler table.

Ni Suiti: We can simplify the table more, by drawing just a 2 x 2 checker board with

just two colors.

For black-red number, the table will be represented by

Si Nessa: That's beautiful and very easy to memorize. Now what about the Black-

Pink numbers that I found in Bichromic Two.

Ni Suiti: The multiplication checker board for Black-PinkNumbers is

o

o

where the white ring is representing the minus sign.

Si Nessa: How can I use the wonderful table

Ni Suiti: We can replace the formula (a + b)(c + d) = (ac+bd)+(ad+bd)

in this simple steps

Make a multiplication checkerboard

Put column (a,b) on the left of the table

Put row (c,d) on the top of the table

Multiply the elements of row and the column and

multiply them with the sign in the suitable board little boxes

Add up all the elements of the table using the rule of addition.

Ni Suiti: I think this algorithm is easier for people's mind

who is stronger in intuition, like me, rather in logic, like Ki Algo.

Your Grandpa's algebraic formula is suitable for left-brainer

my diagramatic algorithm is suitable for right-brainer.

Si Emo: OK my brain is like grandpa's. For me the algorithm is too complicated.

How is about that Grandpa?

Ki Algo: Good. Your grandmother Ni Suiti has make the 2-color number

multiplication more visual. I will reformulate your grandmother's checkerboard with

numeral and letters. Let us symbolized black box with 1, the red box with the

symbol e and pink box with the simbol i

Si Emo: Grandma's multiplication black-red checker board will be symbolized by

the following table

1 e

e 1

and Grandma's multiplication black-pink checker board will be symbolized with the

following table

1 i

i -1

Si Nessa: Oh! So simple table. That's really a very simple table.

Ki Algo: Formulated as such symbolic table, mathematicians will directly know that

Black-Red numbers is nothing but another form of hyper-complex numbers and

Black-Pink numbers is nothing but another form of complex numbers.

Ni Suiti: In my words. Hyper-complex numbers is nothing but another form of

Black-Red numbers and complex numbers is nothing but another form of Black-

Pink numbers.

Ki Algo: In other words the arithmetic system of black-red numbers is similar to the

arithmetic of hyper-complex number ring and the arithmetic system of black-pink

numbers is similar to the arithmetic of complex number field. Mathematicians found

out that all the field axioms for the real number arithmetic are also followed by

complex numbers arithmetic.

Ni Suiti: Why is that black pink numbers form a field arithmetics?

Ki Algo: No zero divisors exist in its arithmetic due to the presence of minus sign in

its unit multiplication table.

Ni Suiti: OK. Now, by using my two color checkerboard we can teach the complex

and hyper-complex arithmetic to primary school kids as 2-color number arithmetic.

Ki Algo: That's a great idea. Hopefully teachers will take your advice.

CHAPTER FIVE 2-Number Arithmetic

Si Emo came to his Grandpa Ki Algo telling him his adventure in the strange islands

of Numberland. Each island is populated by finite collection of numbers. Each

number in the island adds or multiplies to other number getting other number:

1-Number Arithmetic

Ki Algo: What is the strangest island that you have visited.

Si Emo: It is the smallest of the islands is populated by single number.

Ki Algo: What number is that?

Si Elmo: It is zero which is symbolized by 0.

Ki Algo: What is so strange about that?

Si Elmo: The single zero that I met in the told me that in the past there are many zeros

lived in the island. But because every time they're melt to each other by addition and

multiplication, the island population is reduced. It is because 0 + 0 = 0 and 0 x 0 =

0. So every time they meld a zero is diminished. That's why the population now is

only one number: zero

2-Number Arithmetic

Ki Algo: Surely you have a boring adventure in the land of 1-number. What is your

next adventure

Si Emo: I visit the land of 2-number. The population is the numbers 0 and 1. They

have a rule of melding by addition similar to the rule of melding by multiplication for

2-color numbers that Si Nessa had found out. If you add zero to any number, you get

the same number. It is similar to zero in our real number arithmetic.

Ki Algo: Well it seems that they follow the rule of modulo 2 addition

Si Emo: What is modulo 2 ?

Ki Algo: If you have a number X, you can subtract the number by 2 repetitively until

the rest is equal to a number smaller than 2. The rest number is called X modulo 2.

For example 7 modulo 2 = 1 and 4 modulo 2 is 0.

Si Emo: I got it. So the table for the addition is

+ 0 1

0 0 1

1 1 0

Ki Algo: What about 2-number multiplication rules

Si Emo: The rules can be simplified to the following table

. 0 1

0 0 0

1 0 1

Ki Algo: Obviously, it is the rule of multiplication of numbers modulo 2. Because

now we have only 2 numbers, let us call the arithmetic of the two numbers as 2-

number arithmetic.

Si Emo: Is the 2-number arithmetic a field?

Ki Algo: Yes, it is called finite field by mathematicians. Electric engineers now use

it to code their messages across the noisy channel of communication.

CHAPTER SIX 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.

CHAPTER SEVEN

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.

CHAPTER EIGHT

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 Fp, 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. Fn is a field if n is a k-th power of a prime

number p or n=pk 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.

APPENDIX A

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.

APPENDIX B

Notes on 3 colored number

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