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 $\frac{a} {b} = \frac{c} {d}$ then $c > d$ and there exists a number $n > 0$ such that $a = nc$ and $b = nd$ . Hence for any $a, b, c \in$ 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 $0 > 0.$ “

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

The statement if $a > b$ and $\frac{a} {b} = \frac{c} {d}$ then $c > d$ needs to be qualified in Wave Numbers to exclude the cases where $b = 0$ or $d = 0$ .

B: “Including negative numbers in the domain and taking $a \not = 0$ 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.”

In Wave Numbers, only the magnitudes of Opposite Values can be compared using the concept of “less than.” This means only their Counters can be compared. The absolute value symbol is used to refer specifically to an Opposite Value’s Counter. For example, $|4\hat{ }| = |4^v|$ and $|a| = |^-a|$ . Therefore, the statement $|a| > |^-a|$ can never be true. This ensures that no contradiction of the basic property of order can occur.

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
= ½ and = 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
= ½^{^} and = ½^{^}
- => there exists a number $n$ = ½^{^} where $a$ = $n$ * $c$ and $b$ = $n$ * $d$

Now let $a$ = 0, $b$ = 2^v, $c$ = 0, and d = 0 then $a$ / $b$ =0/2^v, $c$ / $d$ = 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 > – then >
- 4 > -4, so 4*-1 > -4*-1, -4 > 4 (Not true)
If $a$ > – $a$ then $ac$ > – $ac$ but the example shows $ac$ > – $ac$ is not true ( 4*-1 < -4*-1, -4 < 4)

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^{^}, $b$ = $a$ = 4^v, $c$ = 1^v

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^{^}| = |4^v| 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.

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Axioms – Property of Order

Definition of the Property of Order

Example for A

Classical Math

Wave Numbers

Example for B

Classical Math

Wave Numbers

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