If you are like me and sometimes have one of those days with nothing better to do then a good substitute for real work is to start inventing. Now this can either be in a physical location and dealing with physical things or it can be in the workshop of the mind.

The easiest and most pliable of objects to invent with are of course numbers.
However what exactly constitutes such an invention - what requirements for
validity do we have. For example, in the physical world, someone could claim to
have invented a new substance and call it "*Obscurocom*" and claim some
mystical properties. But would such an invention be valid?

First of all,

Obscurocom must also exist in the physical world and not just be an idle
figment of imagination (i.e. available for display, measurement and
criticism by others) | |

Obscurocom must obey (i.e. be consistent with) known physical laws or
point to new physical laws resulting from novel behavior (I guess
"Blubber" would have suggested such a need) | |

Obscurocom must present properties that currently available substances do
not offer, i.e. it is either disjoint from known substances or known
substances are a subset of it |

*Obscurocom* need not be of any practical use of course (at least for
now).

These criteria are necessary to avoid unsupportable claims and time wasting duplication. The same criteria must apply to the invention of new numbers. These must exhibit new properties and remain consistent with standard mathematical properties. For example, they must obey commutative, associative, distributive operations with addition, subtraction, multiplication and division. They must have an inverse. In other words they must obey the properties of a "group". Numbers that do not comply with such operations are probably useless (although things that are not numbers, e.g. matrices, may be allowed to avoid some criteria based on their practical usefulness)

Let us examine some simple analogies. In the beginning we had integers, 1,2,3... etc. These could be added and multiplied etc in groups and also in different ways according to the rules of algebra. The invention of a "negative" allowed -X to be different from the operator "-" acting on "X". In some way the negation formed a new number as opposed to an operation on a number.

Then people thought about "*rational numbers*". The consisted of a
ratio of two * integers* e.g. ½, ¼, ¾ etc. These were obviously not
"*integers*" but could form *integers*. For example 3 = ^{12} /
_{4}. Therefore *
rational* numbers were "new" and* integers* became a subset of
* rational*
numbers.

** Hippasus,** a student of Pythagoras started to ponder the "square root of two" - was this
number

√2 · Q = P

Then he reasoned that if both sides were equal he could square both sides

2 · Q^{2}
= P^{2}

Anything multiplied by 2 is even so P^{2}
must be even.

Since odd numbers have odd squares (e.g. 11^{2} = 121) and even
numbers have even squares (e.g. 8^{2} = 64) it followed that P must also
be even.

Since P is even we must be able to divide P by 2 and get another integer k = P / 2 i.e. P = 2 · k

Therefore
2 · Q^{2} = (2 · k)^{2}

So
that
2 · Q^{2} = 4 · k^{2}

And so further that Q^{2}
= 2 · k^{2}

So now, as we reasoned before that P^{2} was even, we must
conclude that Q^{2} is also even

Since Q^{2} is an even square, as before, Q must also be even.

Well what have we here now? P turned out to be even and now it seems that Q is also even. So both P and Q must have a common factor of 2.

But recall at the beginning - we explicitly demanded that P and Q had no common factors but here they are caught out with a common factor of 2!

This contradicts the original bargain, so assuming the opposite didn't work. * Hippasus*
concluded that √2 couldn't possibly be rational as P / Q was not "

From this contradiction, the initial assumption that √2 could be
expressed as a* rational *number is false; that is to say, √2 is __irrational__

This proved that was a new number but it still obeyed all the rules of algebra.

The term "*irrational*" was coined to represent such numbers.
Pythagoras was infuriated at his student ** Hippasus** - in his mind
all numbers should have nicely perfect values. How dare

So
it seemed, like the layers of an onion, * rational* numbers became a subset of
*
irrational* numbers. For example, any integer can be expressed as the square root
of its square (of course) but many square roots cannot be expressed in terms of
integers or *rational* numbers composed of integers.

The
search for new numbers continued and along came "pi" π
and the "exponential" **e**, the "golden ratio" φ
≡ ½ · (1 + √5) etc.

As
if this was not enough, what was the square root of a negative number? For
example what strange numerical creatures lurked inside the solution to ** x^{2}**
<

I hope this historical background provides a useful framework in which to search for new and novel numbers - perhaps unlike fish they are out there to be mined in them there hills of mathland.

Just as a gold miner needs a pick-axe and shovel, a miner for numbers also needs suitable tools. The mainstay operators are typically addition, subtraction, multiplication and addition. Further to this swaggering complement are the rules of algebra.

However are such tools good enough for the job at hand? After all, today's mining relies on heavy machinery. This performs the same basic functions as a pick axe and shovel but on a much greater scale. Further, otherwise untraceable veins of wealth so deeply buried can only be extracted by enhanced mechanical means.

So it may be with the four basic binary operators. This is why I propose to add an extra two. I call these "Ûp" (i.e. ) and Ðown" (i.e. ¯) and are defined quite simply as

or equally in web page form

**
x**** y** ≡ exp{ Log

and

*
x*
¯

It is perhaps instructive to remember those old slide rules that used to be used in the 1960's before the calculator. These multiplied two numbers X and Y as follows

X · Y = 10^{Log{X} + Log{Y}}

Similarly the slide rule divided two numbers X and Y using

^{X }/ _{Y}
= 10^{Log{X} - Log{Y}}

This operation is perfectly valid so long as X and Y are not zero, although they can be negative to allow operations with complex numbers.

The use of "Base 10" is a bit clumsy in mathematics and most people
prefer
the use of the exponential and natural logarithms. The "binary
operators" Ûp and Ðown I propose to use are therefore not exotic and the
addition **"+"** is simply replaced with a multiply **"·"** and the
subtraction **"-"**
with a divide **"/"**. The only mathematical constraint is that y cannot
have a value equal to 1
for Ðown as this results in a divide by zero!

For completeness I will also allow unitary operators defined the same way

and note that

and just mention an equivalent "absolute value" unitary operator

with the result that

This is exactly analogous to

**
**

These are the extra tools with which we will hunt. However, if you wish to see some more of the properties of Ûp and Ðown before our numerical excavation then why not check out this link Group Properties

Armed with our new tools we begin the great search for potentially new and
novel numbers subject for subsequent analysis and conjecture. In another
numerical mine in a time long ago an "imaginary" number "** j**"
was unearthed and became unified with ordinary numbers to cerate a complex
numerical Gemini

* j ^{2}* = -1

It was thought at the time that all known solutions had to be zero or positive. How could such a solution exist with a negative sign?

Not surprisingly, two twin solutions are possible

** (+ j )^{2} = -1**

and

**(- j )^{2} = -1 **since

The positive solution is selected by convention but both solutions for ** j ^{2} = -1**
are valid.

By analogy
**³
1** for any real ** x**
and
is used to represent the unitary operator "Up
squared". To recap

*
*

Since **[**Log_{e}{ ** x **}

*
*

There should be no solution for ** p** given our assumptions so far but our
question now is, if

Perhaps ** p** could exist in the well known complex number domain? Let us express
a potential solution in polar form, i.e. we propose

** p** = | p | · e

This would infer the following sequence of outcomes

** p**
= ( | p | · e

= exp{ **[** Log_{e} { ( | p | · e ^{j · Ø} ) **]**^{2}
}

= exp{ **[** Log_{e} { | p | } + j · Ø **]**^{2}
}

We therefore must have this result

exp{ **[** Log_{e} { | p |
} + j · Ø **]**^{2} } = ^{1}/_{e}

Taking natural Logarithms of each side now

**[** Log_{e} { | p | }
+ j · Ø **]**^{2} = -1

Taking the square root of both sides

Log_{e} { | p | } +
j · Ø = ±* j*

Both **real** and **imaginary** components of each side *must* be
equal, therefore

Log_{e} { | p | } = 0

i.e. | p | = 1

so that p = ±1

and Ø = ±1

Well isn't this interesting. The proposed solution ** p** exists and
has

** *** p* = ±e

Is "** p**" a new number? It certainly has valid solutions
that are consistent with algebra. However it has

Does this represent a "new property" - have we mined out a new numerical substance?

Let us for now assume that other numbers lack four-fold "valuation". Let us propose an equally odd-ball compound defined as

* *

where x and y represent ordinary numbers, non zero, real or complex. (This is
an equivalent form to z = x + ** j** · y - the same operator
precedence of multiplication over addition, "Up" over multiplication
etc, is hopefully an obvious convention)

Wow! Perhaps I may have just discovered a new number - I'd better hurry up and give it a name!

If "** j**" is called an

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

That *phantasy* numbers and *profound* numbers exist is certain as
I have not broken any arithmetic rules in their discovery. The ** Conjectural**
is whether they are in fact different from existing numbers

That last mathematical mining operation may still have some remaining items of interest still deeply buried. I think I will return. I thought I glimpsed something interesting nearby

exp{ **[** Log_{e} { |
p | } + j · Ø **]**^{2} } = ^{1}/_{e}

for ** *** z*
≡ p · e

I
had to wonder if the phantasy number ** p** was not alone. What if
"generic" complex numbers

Both r and Ø will be defined as real numbers with the angular tern
constrained to ** 0** ≤ Ø <

I substituted this polar representation for ** z** into the definition for

Let's now express Ψ in conventional polar form

We see that
Ψ
has
a magnitude equal to exp{ **[** Log { ** r** }

[Log {}r]^{2}³0

for all real values of ** r**. Now lets look at the range of the
angular contribution from Ø. Since we defined

These two observations tell us that the magnitude of Ψ must be greater than

for all complex values of * z ≡ r · *e

Once again I had to ask, what's inside
this * forbidden zone* of ** ^{
}**? This is no different
than asking "what numbers are inside the

Let us return to our result for the magnitude of Ψ i.e.

**
|Ψ| **=** exp{ [** Log { ** r**
}

Surely we should not constrain our search for new numbers to such a restricted angular range? Let us first consider negative angles e.g.

-**2** · π
≤ Ø < 0 i.e. **4**
· π^{2}
≥ Ø^{2} > 0
- the same range for Ø^{2} as before but just in a "different
order"

Clearly no opportunity exists to enter the forbidden zone with negative angular values of Ø.

However, what about the numbers that turn? Why should we assume that angular
values with additional integer multiples of 2 · π
should be equal? Perhaps they are not, at least in the *forbidden zone*.
For example, what if we let ** z** take a

** 2** · π
< Ø ≤ **4** · π

Since the **[** Log { ** r**
}

** 4** · π^{2}
< Ø^{2} ≤ **16** · π^{2}

Clearly we have now entered the first layer of the forbidden zone of and the magnitude of Ψ is now outside a much smaller second forbidden zone defined by

Well isn't this also interesting. Now we have a number that has a **multi
turn **property. Each solution for a forbidden zone is different and
any number of forbidden zones clearly now seem definable

So do these "*multi-turn*" numbers really exist? Just like the
layers of an onion so are the forbidden zones of

In this web chapter we have journeyed on a mathematical mining expedition. Did we discover the new deep and buried numbers we searched for?

I introduced two new binary operators Ûp and Ðown ¯ as well as a unitary Ûp-exponent operator to help us on our search. These were based on well known exponential and natural logarithm functions and parallel the use of old slide rule log-antilog approach to perform multiplication and division tasks. This extends the set of binary operators to

(+, -, ·, /, , ¯) and and the set of unitary operators to ( #

, # ).^{n}

These could be extended indefinitely I guess e.g.

*
*

and perhaps some subsequent use might be found for this.

In the same way that "** j **" was proposed to represent
one of the solutions to the otherwise absent members of

* *

i.e.

I would have
thought * "what's good for the goose is good for the gander"*,
so if an

The solution "** p** " remains consistent with well known
algebraic rules but has

Further we looked deeper into the mother load of potential new golden numbers
(hopefully not fool's golden numbers) and proposed solutions to other regions
where none should be found. In this case we looked into a circular forbidden
zone of **
**for even more numerical treasure. By analogy once more, this is no
different for the proposal that ** j^{2}** =

Once again we observed odd-ball properties. This solution set Ψ
appears to be differentiated inside ever decreasing circular regions i.e. *forbidden
zones,* depending on its * winding number*. Perhaps Ψ
could be called a

Do other numbers exhibit such differentiation? Surely Ψ
has as much "right to exist" as ** p** just as

Well this has been a really fun web chapter to write. I don't make any claims
of course as my web chapter on the search for inventing new numbers is yet
another entry in my *Conjecturals* devoted to wonder, fascination and
thought ☺

I wonder how much I could get for ** p** or Ψ
on Ebay?

I also have to wonder what would happen to me if Pythagoras were to read my web chapter here today?

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