July 22, 2024
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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
    • 2v = 1v + 1v
    • 3v = 2v + 1v, 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^
    • 23v is defined to be 2*10v + 3v
    • 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^ + 3iv)-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‡aa/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^/2jv = 2.5i^ =  5^*1/2jv = 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 ab, 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 ab, 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 and b are two 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, bx 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 bx 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 bb. 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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