Dialogue on 2-Color Numbers (2)

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

+	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.