General Axioms

The following axioms are adapted from the 15 axiom statements in  Axioms for Real Numbers , specifically interpreted for Wave Numbers.

Wave Numbers assumes that the following statements are true. Here a, b, c and d represent arbitrary Opposite Values.

1. (Existence)
   • There exists a set R? consisting of all Opposite Values for the dimension ?. It contains a subset Z? ⊆ R? consisting of all integers for the dimension ?.
2. (Closure of Z?)
   • If a and b are integers, then so are a + b and a*b.
3. (Closure of R?)
   • If a and b are Opposite Values, then so are a + b and a*b.
4. (Commutativity in R1 and R2)
   • a + b = b + a
   • ab = ba
   • |ab| = |ba|
5. (Commutativity in R3 and higher dimensions)
   • a + b = b + a.
   • a*b ≠ b*a
   • |a*b| = |b*a|
6. (Associativity in R1 and R1)
   • (a + b)+ c = a +(b + c) and (a*b)c = a(b*c) for all Opposite Values a, b, and c in R1 and R2.
7. (Associativity in R3 and higher dimensions)
   • (a + b)+ c = a +(b + c) for all Opposite Values a, b, and c in R3 and higher dimensions.
   • (a*b)c ≠ a(b*c) for all Opposite Values a, b, and c in R3 and higher dimensions.
8. (Distributivity)
   • a(b + c) = a*b + a*c for all consolidated Opposite Values a, b, and c. Note a consolidated Opposite Value means that any Opposite Values of the same Opposite Type have been added together.
9. (Zero) 0 is an integer that satisfies (a + 0) = a = 0 + a for every Opposite Value a.
10. (One) 1 is a Counter that is not equal to zero and satisfies (1 * a) = a  for every Opposite Value a.
11. (Additive inverses)
    • If a is any Opposite Value, then there is a unique Opposite Value ^–a such that a + ^–a = 0. This number is the Opposite Value with the same Counter and just the Opposite Sign changed.
12. (Reciprocals in R1 and R2)
    • If a is any nonzero Opposite Value, then there is a unique Opposite Value a^-1 such that a*a^-1 = 1^{^}.
13. (Reciprocals in R3)
    • If a is any nonzero Opposite Value, then there is a unique Opposite Value a^-1 such that a*a^-1 = 1‡a^‡a.
14. (Trichotomy law)
    • If a and b are Opposite Values, then one and only one of the following three statements is true: |a| < |b|, |a| =|b|, or |a| > |b|.
15. (Closure of R)
    • In R1 and R2, if a and b are Opposite Values of the ^{^}/^v Opposite Type and ^{^} Opposite Sign , then a + b and a*b are of ^{^}/^v Opposite Type and ^{^} Opposite Sign. For example: 6^{^}*3^{^} = 18^{^}
    • In R3, if a and b are Opposite Values of the same Opposite Type, then a*b is of the same Opposite Type and Sign as the Operand. For example: 6^{^}*3^v = 18^v, 6i^v*3i^{^} = 18i^{^}. If a and b are Opposite Values of the same Opposite Type and Sign, then a + b is of the same Opposite Type and Sign
16. (Addition law for inequalities)
    • If a, b, and c are Opposite Values and |a| < |b|, then |a| + |c| < |b| + |c|. For example in R3:
      • a = 6^{^}, b = 8i^{^}, c = 3j^v.
      • |a| + |c| = 9, |b| + |c| = 11,  9 < 11.
17. (The well ordering axiom)
    • Every nonempty set of ?^? integers in R? contains a smallest element.
18. (The least upper bound axiom)
    • If S is any nonempty set of Opposite Values in R? and S has an upper bound for an Opposite Type and Sign, then S has a least upper bound for that Opposite Type and Sign.

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