Dialogue on n-Color Arithmetic (1)

Part 1:
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.