Dialogue on 2 color number by Armahedi Mahzar

Part 1: 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

2X3=6 and Red 2 times Black 3 is equal to Red 6

2X3=6 and Red 2 times Red 3 is equal to Black 6

2X3=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= =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.

Part 2: 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 +

a a+c b

c+b

d a+d (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

d (ad) (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 âˆ’a such that a + (âˆ’a) = (âˆ’a) + a = Zero (âˆ’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.

Part 3 : 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. z

= 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.z2= (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

Îľ 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

Part 4: 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

d (ad) (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 X1 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

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