Dialogue on n-Color Arithmetic (2)

Part 2:
3-Number Arithmetic

Ki Algo: I wonder, if you also went to other finitely populated numberland
Si Emo: Yes Grandpa, I have visited 3-number island. It is populated by 3-numbers.
Ki Algo: What are their melding rules
Si Emo: For addition the rules are simplified to this table

+	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

and for multiplication the rules are simplified to

.	0	1	2
0	0	0	0
1	0	1	2
2	0	2	1

Ki Algo: So they are similar to addition and multiplication modulo 3.
Si Emo: Oh!
Balanced 3-number
Ki Algo: I think the table will be more familiar if we replace 2 with -1
Si Emo: Well, the addition table will be replaced by

+	0	1	-1
0	0	1	-1
1	1	-1	0
-1	-1	0	1

Ki Algo:  What about the multiplication table?
Si Emo: The new multiplication table will be this table

.	0	1	-1
0	0	0	0
1	0	1	-1
-1	0	-1	1

Ki Algo: What is the arithmetic structure of the 3-numbers.
Si Emo: Because both multiplication and addition are commutative and associative, and the multiplication distributes upon the addition, and every nonzero number has an additive and multiplicative inverse, then it is a field arithmetic, similar to real number arithmetic.
Ki Algo: So, probably, it will be useful for engineers to build their coding system. To know n-number system more deeply you have to visit the islands in numberland which is populated by more numbers.