Definitions – General

The axiom definitions that follow derive from the 19 axiom definitions in Axioms for Real Numbers as interpreted for Wave Numbers.

1. The numbers 2?^? through 10?^? are defined by 2?^? = 1?^? + 1?^?, 3?^? = 2?^? + 1?^? etc. For example:
   • 2^{^} = 1^{^} + 1^{^}
   • 3^{^} = 2^{^} + 1^{^}, etc. and
   • 2^v = 1^v + 1^v
   • 3^v = 2^v + 1^v, etc. and
   • 2i^{^} = i^{^} + i^{^}
   • 3i^{^} = 2i^{^} + 1i^{^}, etc.
2. The decimal representations for other numbers are defined by the usual rules of decimal notation. For example:
   • 23^{^} is defined to be 2*10^{^} + 3^{^}
   • 23^v is defined to be 2*10^v + 3^v
   • 23i^{^} is defined to be 2*10i^{^} + 3i^{^} etc.
   • Note that the operator is a Counter  in these examples.
3. The additive inverse of a is ^–a and satisfies a + ^–a = 0 and whose existence and uniqueness are guaranteed by Axiom 11.
4. The difference between a and b is the Opposite Value defined by = a + ^–b, and is said to be obtained by adding the flip of b to a.
5. If a ≠ 0, the reciprocal of a in R1 and R2 is the Opposite Value a^-1 that satisfies a*a^-1 = 1^{^} where a^-1 = 1^{^}/a and whose existence and uniqueness are guaranteed by Axiom 12.  The reciprocal is a multiplicative inverse in that multiplying by the reciprocal of an Opposite Value is the equivalent of dividing by the Opposite Value. Reciprocals of expressions exist, such as (4^{^} + 3i^v)^-1 = (0.16^{^} + 0.12i^{^})
6. If a ≠ 0, the reciprocal of a in R3 is the Opposite Value a^-1 that satisfies a*a^-1 = 1‡a^‡a where a^-1 = (1‡a^‡a/a) and whose existence and uniqueness are guaranteed by Axiom 13. The reciprocal of an Opposite Value is a not a multiplicative inverse which do not exist in R3. Reciprocals of expressions do not exist in R3.
7. If b ≠ 0, the quotient of a and b, denoted by a/b, is the Opposite Value obtained by dividing a by b. For example, in R3: 5^{^}/2j^v = 2.5i^{^} =  5^{^}*1/2j^v = 2.5i^
8. An Opposite Value is said to be rational if it is equal to p/q for some integers p and q with q ≠ 0.
9. An Opposite Value is said to be irrational if it is not rational.
10. The statement a is less than or equal to b, denoted by a ≤ b, means |a| < |b| or |a| = |b|.
11. The statement a is greater than b, denoted by a > b, means |b| < |a|.
12. The statement a is greater than or equal to b, denoted by a ≥ b, means |a| > |b| or |a| = |b|.
13. The set of all ^{^} Opposite Values in a dimension is denoted by R?^{^}, and the set of all ^{^} integers by Z?^{^}.
14. The set of all ^v Opposite Values is denoted by R?^v, and the set of all integers by Z?^v.
15. All Opposite Values, with the exception of 0, are greater than 0.
16. An Opposite Value a is said to be nonvee if a = 0 or has a ^{^} sign
17. An Opposite Value a is said to be nonhat if a = 0 or has a ^v sign
18. If a, b and c are distinct Opposite Values with the same Opposite Sign and Opposite Type, an Opposite Value c is said to be between a and b if either |a| < |c| < |b| or |a| > |c| > |b|.
19. The square of an Opposite Value a is the Opposite Value a*a, denoted by a².
20. If S is a set of Opposite Values, an Opposite Value b is said to be the largest element of S for that Opposite Type and Sign if b is an element of S and, in addition, b ≥ x whenever x is any element of S for that Opposite Type and Sign. Thus there is a largest element of S for each Opposite Type and Sign. The term smallest element is defined similarly.
21. If S is a set of Opposite Values, an Opposite Value b is said to be the largest absolute element of S if b is an element of S and, in addition, |b|  ≥  |x|  whenever x is any element of S. The term smallest absolute element is defined similarly.
22. If S is a set of Opposite Values, an Opposite Value b (not necessarily in S) is said to be an upper bound for that Opposite Type and Sign in S if b ≥ x for every x with that Opposite Type and Sign in S. It is said to be a least upper bound for that Opposite Type and Sign in S if every other upper bound b^’ for S satisfies b^’ ≥ b. The terms lower bound and greatest lower bound are defined similarly.
23. If S is a set of Opposite Values, an Opposite Value b (not necessarily in S) is said to be an absolute upper bound for S if |b| ≥ |x| for every x in S. It is said to be a least absolute upper bound S if every other upper bound b^’ for S satisfies |b^’| ≥ |b|. The terms absolute lower bound and  absolute lowest bound are defined similarly.

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