# 1. Properties Of The Binary Operators Ûp
And Ðown

## 1.1. Previous Definitions Of Ûp &
Ðown

All good miners need to know their tools - what they are capable of and where
they shouldn't be used. The same is true of Ûp
and Ðown

Here is my definition for the binary operators "Ûp"
and
"Ðown"

...(1)

Here
is my definition for the unitary operator "Ûp to the power of ** n**"
i.e.

...(2)

And
here is my definition for the "*Ûp Absolute*" value of x - I think I'll
call this the **Ûpsolute** value of x.

...(3)

# 2. List of Properties For Up
& Down

## 2.1 Commutative Property of Ûp &
Ðown

...(4)

## 2.2 Associative Property of Ûp &
Ðown

There are two possible combinations for the associative rule

..........................(5 a)

.........(5
b)

## 2.4 Distributive Property Of Ûp &
Ðown

There are four possible combinations for the distributive rule

....(6
a)

...(6
b)

....(6
c)

...(6
d)

## 2.8 The Existence Of An Identity "e"

...(7)

## 2.9 The Existence Of An Inverse

...(8)

## 2.10 The Exponent Rules For The Up-Exponent (An "Uponent"
perhaps?)

There are three possible exponent rules

.......(9
a)

...............(9
b)

....(9
c)

Here is a link to see some *proofs* on these group
properties Appendix

# 3. Complex Ûp and Ðown

We realize from the definition of
that

....(10)
for all real values of *x*.

This
is analogous to the familiar result that
**
**. However
people "invented" a number *j *such that *j *^{2}
= **-1**. I see no reason why I can't do the the same with the Ûp-Exponent. I will
therefore define a new number *p* such that

...(11)

If *j *is called an "imaginary
number" I think I'll call *p* a "*phantasy number*".

All *p* has to do is remain consistent within the rules of
mathematics to exist, but if it behaves in any way odd or different to other
numbers then it may indeed be a new number.

While
I'm at it, I don't see why my "phantasy number" can't be paired up
with conventional numbers in a similar fashion to *z* = x + *j
· *y.
I therefore propose a combination number* w* defined as

...(12)

If
"*z*" is called a "*complex number*" I think I'll
call *w* a "*profound number*"

# 4. Summary for Ûp and Ðown

In this web chapter I have listed the group properties of Ûp
and Ðown
. I have
included the *Ûp-Exponent*
or "*Ûponent*" properties and the "*Ûpsolute
value*" definition, analogous to the * absolute value*

I have also proposed the possible existence of a phantasy number *p*
in analogous definition the the * imaginary* number *j*. Just as
*
complex* numbers *z* are constructed from a real and an imaginary
part, I have proposed a *profound* numbers *w* constructed from
a real and a *phantasy* part.

No rules of arithmetic have been transgressed at this stage, nor have the
definition for * phantasy* and *profound* numbers followed any different reasoning or allowances
than those afforded to *imaginary* and *complex* numbers.

So I think all's fair in maths and conjecture!

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