Definition of the Property of Order
In Cecilia Hamm’s ‘Making Sense of Negative Numbers’ she states:
A: “For natural numbers, if and
then
and there exists a number
such that
and
. Hence for any
N if
then
. Including zero in the domain and taking
will lead to a contradiction: if
then
“
However, in Wave Numbers, if , then the statement
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 and
then
needs to be qualified in Wave Numbers to exclude the cases where
or
.
B: “Including negative numbers in the domain and taking and
implies that if
then
. Taking
will lead to a contradiction: if
then
, in short: if
then
. 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, and
. Therefore, the statement
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 = 3,
=2,
=6 and
=4:
- 3 > 2
>
is true
- 3/2 = 6/4
/
=
/
is true
= ½
and
= ½
- => there exists a number
= ½ where
=
*
and
=
*
- => there exists a number
Now let = 0,
= -2,
= 0, and
= 0 then
/
=0/-2,
/
= 0/0, so
/
=
/
= 0 is true.
Here >
=>
>
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 = 3^,
=2^,
=6^ and
=4^:
- 3^ > 2^
>
is true
- 3^/2^ = 6^/4^
/
=
/
is true
= ½^
and
= ½^
- => there exists a number
= ½^ where
=
*
and
=
*
- => there exists a number
Now let = 0,
= 2v,
= 0, and d = 0 then
/
=0/2v,
/
= 0/0.
However, if = 0, then the statement
>
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 >
and
/
=
/d then
>
’ needs to be qualified in Wave Numbers to exclude
=
= 0.
Example for B
Classical Math
For B above in classical maths, let = 4,
= –
= -4,
= -1
- if
> –
then
>
- 4 > -4, so 4*-1 > -4*-1 (Not true)
- If
> –
then
> –
but the example shows
< –
( 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 = 4^,
=
= 4v,
= 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 || = |-
|. This means that the statement if |
| > |-
| can never be true. So, no contradiction of the basic property of order can occur.
Next: R1
Previous: Ratios