Dialogue on 2-Color Numbers (3)

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. 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.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 e 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?