Dialogue on 2-Color Number (4)

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	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.