July 22, 2024
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Axioms – Property of Order

Definition of the Property of Order

In Cecilia Hamm’s ‘Making Sense of Negative Numbers’ she states:

A: ‘For natural numbers, if a > b and a/b = c/d then c > d and there exists a number n > 0 such that a = nc and b = nd. Hence for any a, b, c ∈ N if a > b then ac > bc. Including zero in the domain and taking c = 0 will lead to a contradiction: if a > b then 0 > 0.’

B: ’Including negative numbers in the domain and taking a ≠ 0 and b = –a implies that if a > –a then ac < –ac. Taking c = -1 will lead to a contradiction: if a > –a then a(-1) < –a(-1), in short: if a > –a then –a < a. This contradicts the basic property of order.’

Contradiction in Classical Math not in Wave Numbers

Example for A

Classical Math

For A above in classical math let a = 3, b=2, c=6 and d=4:

  • 3 > 2                    a > b is true
  • 3/2 = 6/4             a/b = c/d is true
  • a = ½c and b = ½           
    • => there exists a number n = ½ where a = n*c and b = n*d

Now let a = 0, b = -2, c = 0, and d = 0   then a/b =0/-2, c/d = 0/0, so a/b = c/d = 0 is true.

Here a > b => c > d but 0 > 0 is not true leading to the contradiction. Division by 0 is undefined in classical maths.

Wave Numbers

For A above in Wave Numbers, let a = 3^, b=2^, c=6^ and d=4^:

  • 3^ > 2^                  a > b is true
  • 3^/2^ = 6^/4^       a/b = c/d is true
  • a = ½^c and b = ½^d         
    • => there exists a number n = ½^ where a = n*c and b = n*d

Now let a = 0, b = 2v, c = 0, and d = 0   then a/b =0/2v, c/d = 0/0.

However, if a = 0, then the statement a > b cannot be true as no Opposite Value is less than 0. So, this is not a valid example in Wave Numbers. Note also that division by 0 is undefined in Wave Numbers.

The statement ‘if a > b and a/b = c/d then c > d’ needs to be qualified in Wave Numbers to exclude b = c = 0.

Example for B

Classical Math

For B above in classical maths, let  a = 4, b = –a = -4, c = -1

  • if a > –a then ac > bc       
    • 4 > -4, so 4*-1 > -4*-1 (Not true)
  • If a > –a then ac > –ac but the example shows ac < –ac  ( 4*-1 < -4*-1)

This is what contradicts the basic property of order and is one reason for the introduction of absolute values.

Wave Numbers

For B above in Wave Numbers, let  a = 4^, b = a = 4v, c = 1v

In Wave Numbers only the magnitudes of Opposite Values can be compared using the less than concept. i.e. only their Counters can be compared. The absolute value symbol is used to refer specifically to an Opposite Values counter. So |4^| = |4v| and   |a| = |-a|. This means that the statement if |a| > |-a| can never be true. So, no contradiction of the basic property of order can occur.

Next: R1

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