Three-signed arithmetic : T space
Timothy Golden
Could anyone point me to some writings on this form of mathematics?
Or could you dispel its validity?
I will only define its construction.
There are three branches from an origin.
Each branch has its own sign.
I label these signs { -, +, * }.
Some numbers in this realm are { -1.234, +2.345, *3.456 }.
I call this space T.
Please do not think of these values in terms of vectors. One
proffessor I approached with this pointed me right back toward
vectors. These don't exist until you take a cross product.
T X T seems to be a competitor with R X R X R.
I find this attractive since a symmetrical basis is more believable.
There is also a natural rotation that reminds me of complex
arithmetic.
Operators must be created.
Thanks for applying yourself to this problem.
-Tim
Justin Davis
"Timothy Golden" <tttp...@yahoo.com> wrote in message
news:5c7f8ade.0307...@posting.google.com...
Yes, they must, otherwise this is R^3 with some extra symbols attached. What
is T x T? You seem to think that this is somehow different, so I don't see
how you can have no idea what the operations are.
Max
You cannot call this space T. The use of T for spaces has already
been established and is protected by international copyright. Your
posting is in copyright violation and you could be liable.
Other than that, I think this is a terrific discovery and is sure to
revolutionize mathematics. Once you create your operators, you will
have a system that will blow the real numbers out of the water. I
wouldn't be the least surprised if eventually you get the Nobel prize
for this discovery.
I especially like the idea of spaces competing with each other. That
is a new concept to me and one that is also sure make a deep
impression on mathematics.
By all means, the believability of the symmetric basis is an important
consideration. Also, this relates in a magnificent way to Nikola
Tesla, the man from Venus who invented three-phase power. Your new
system will be the mathematical framework into which Tesla's
discoveries can be molded.
Tesla himself had some difficulty with triphilia, obsession with the
number 3. My only concern at this point is that you could be subject
to this obsession as well, so do be careful if you find yourself
paying overmuch attention to the number 3. For example, when you are
writing with pencil and paper, do you feel a need to have 3 pencils?
If you should begin to suffer from triphilia, you would be well
advised to abandon this project, as tragic as that would be, because
your mental health is more important to you than mathematical
creations.
Why not call your space Y? That would seem to be an apt symbol, and
as fortune would have it, I happen to be the holder of the copyright
to that symbol, for my space Y, which, as it turns out, is a negative,
antisymmetrical space, and utterly useless, so I would be willing to
abandon it and I could sell you the copyright. We could negotiate as
soon as your operators are created.
Max, Mr. Maximum to the Max
Ernst Berg
Hello Tim
I think you are on to something.. I'm pondering the three thing
myself.
I have an example of three infinities NULL, Negative ( includes zero )
and Positive in dynamical systems. I question the nature of this and
wonder if in a dynamical way it is trinary logic.
However, my lack of math training is a handicap. I cannot compete.
Maybe others can work with you and I will read on.
-Ernst
Roger Beresford
> There are three branches from an origin.
> Each branch has its own sign.
> I label these signs { -, +, * }.
> Some numbers in this realm are { -1.234, +2.345, *3.456 }.
This is "ternagation", a property of "terplex algebra".
I mentioned it in ref [1]. Terplex is one of the Hoop
algebras that I developed and demonstrated in "Conservative Groups,
Moufang Loops, and Algebras" in the Mathematica Information Center
"Mathsource"[2]. It uses the cyclic 3-element C3 group
as multiplication table, so that the product of the "3-vecs"
A={a1,a2,a3} and B={b1,b2,b3} is
{a1 b1+a3 b2+a2 b3, a2 b1+a1 b2+a3 b3, a3 b1+a2 b2+a2 b3}
(you are right in saying that these are not vectors,
which is why I call them 3-vecs). Hoops conserve the
factors of the table determinant, in this case
s1=a1+a2+a3, s2=((a1-a2)^2+(a2-a3)^2+(a3-a1)^2)/2.
The multiplicative inverse of A is
{1/s1+2(2a1-a2-a3)/s2, 1/s1+2(2a3-a1-a2)/s2,
1/s1+2(2a2-a1-a3)/s2}/3. Like most hoops, terplex has
"partial-fraction divisors of zero".
The "natural rotation" that you mention arises from the
polar form {s1,s2,ArcTan[2a1-a2-a3,(a3-a2)Root[3]]}.
(This uses the Mathematica ArcTan[adj,opp] convention).
The first two polar terms are an offset and a squared
radius, which both multiply on 3-vec multiplication,
whilst the angles add. Powers and roots are defined, and
quadratics can be solved. The algebra works correctly
over the real and complex fields. I have yet to test it
over quaternions or octonions.
Terplex fits between Real algebra (based on C2) and
Complex algebra (based on C4, and having a circular polar
form). The K group generates an algebra with a hyperbolic
polar form. Combining this with terplex gives C3K,
isomorphic to the "Hypercomplex arithmetic" developed
by J.J. Hamilton[2], which includes half-spin quantum
operators. It has multi-phase sinusoidal "orbits" that
may be related to quarks (3,6,12-phase) and leptons
(4-phase).
I disagree with your suggestion that "each branch has its
own sign", preferring "each branch has its own
direction". Signs can then be applied to 3-vecs and the
elements can have ordinary signs from the appropriate
field. There are three signs, A+=A,Aleft={a2,a3,a1},
Aright={a3,a1,a2}, being rotations of the 3-vec. Adding
the three gives a zero for s2. This is analogous to the
real case, where A~{a1,a2}, Al=Ar={a2,a1}, adding to give
{a1+a2,a2+a1}, which equivalences to zero.
Be warned. No academic will commend you for dabbling in
these murky waters. I expect Uncle Al to make rude
comments about all this!
Roger Beresford.
References.
[1]http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=active&th=c5889ef5f336af86&rnum=5
[2]http://library.wolfram.com/infocenter/MathSource/4894/
[3]J.J.Hamilton,Hypercomplex numbers... J.Math.Physics,
38(10),Oct. 1997 p4914-4928.
Timothy Golden
space.
Let's admit that T X T cannot be a true competitor to R X R X R.
It doesn't contain enough information.
Let's fix up Y to be a variant of T.
In this new Y space we have the same three signs ( -, +, * ).
Consider every element in Y to contain all three signed magnitude
components.
For example,
y1 = ( -1.0, +2.0, *3.0 ).
These components are reducible by cancellation as long as there is
magnitude in each of them:
y1 = ( -0.0, +1.0, *2.0 ). [reduced form]
Where x is a magnitude (-x,+x,*x) it is (-0,+0,*0) (the origin) in its
reduced form.
Every reduced form number has at most two non-zero numerical
components since the third component has been reduced to zero.
Again, these components are not vectors. They are arithmetical until a
cross product is taken.
The information in an element of Y may be represented as two elements
in T. A natural pair of magnitudes generally exists.
The cross product Y X Y now contains more information, yielding a more
believable competitor to R X R X R.
tttp...@yahoo.com (Timothy Golden) wrote in message news:<5c7f8ade.0307...@posting.google.com>...
Timothy Golden
Hopefully the information in excess of R X R X R will come out as time.
Timothy Golden
So far this thing is really simple.
In T space there is just one magnitude with three signs.
It is interesting that you appear to have a pairing rather than three
symmetrical solutions. Perhaps Y space can help, but it is a tad more
complicated.
I think that there is a principle of accumulation.
This is built into the Y space further along this message thread.
How can a negative infinity include zero?
Perhaps you mean affinity rather than infinity?
Too much math training is a greater handicap...
Timothy Golden
I really appreciate your response.
But I would like to stick with the three signed approach.
I don't want to have a sub-sign.
It could be that what I am describing is much simpler than
what you think it is.
The T space is nothing more than the real numbers with one additional
sign.
One magnitude with three signs.
The imposition of the reals to geometrical space produces cartesian
geometry.
As such each sign does have a direction.
Imposing another sign atop these has never been done.
It would seem paradoxical to do this.
Thinking symmetrically it would be the same paradox in T-space.
I'm not trying to invalidate what you are thinking, just saying that
we are thinking of two different things.
I've looked at your references but have not yet read the paper on
hypercomplex numbers.
> I disagree with your suggestion that "each branch has its
> own sign", preferring "each branch has its own
> direction". Signs can then be applied to 3-vecs and the
> elements can have ordinary signs from the appropriate
> field. There are three signs, A+=A,Aleft={a2,a3,a1},
> Aright={a3,a1,a2}, being rotations of the 3-vec. Adding
> the three gives a zero for s2. This is analogous to the
> real case, where A~{a1,a2}, Al=Ar={a2,a1}, adding to give
> {a1+a2,a2+a1}, which equivalences to zero.
> Roger Beresford.
Justin Davis
> space.
> Let's admit that T X T cannot be a true competitor to R X R X R.
> It doesn't contain enough information.
> Let's fix up Y to be a variant of T.
> In this new Y space we have the same three signs ( -, +, * ).
> Consider every element in Y to contain all three signed magnitude
> components.
> For example,
> y1 = ( -1.0, +2.0, *3.0 ).
> These components are reducible by cancellation as long as there is
> magnitude in each of them:
> y1 = ( -0.0, +1.0, *2.0 ). [reduced form]
>
> Where x is a magnitude (-x,+x,*x) it is (-0,+0,*0) (the origin) in its
> reduced form.
> Every reduced form number has at most two non-zero numerical
> components since the third component has been reduced to zero.
> Again, these components are not vectors. They are arithmetical until a
> cross product is taken.
>
> The information in an element of Y may be represented as two elements
> in T. A natural pair of magnitudes generally exists.
>
> The cross product Y X Y now contains more information, yielding a more
> believable competitor to R X R X R.
>
What is {-0.0, +2.5, *5.3} x {-0.0, +1.3, *2.0}?
Timothy Golden
The signs and magnitudes of these two coordinates are free.
But R X R X R has three magnitudes. T X T will never yield three
magnitudes out of two, so no transform can exist.
Further along in this thread is a proposed Y space that rectifies
this.
An example value in Y X Y would be ( -0.0 +1.0 *2.3, -4.5 +6.2 *0.0 ).
Again, the signs and magnitudes of these two coordinates are free.
In Y there is a proposed reduction to cancel accumulation and hence
the coordinates that you see in this example are in their reduced
form.
In reduced form there are now four magnitudes. So clearly three can be
yielded in a transform.
Perhaps the excess information will allow for the cross product of
time.
I do struggle with the operators. They should not be arbitrary and
they must yield certain results. I am thinking of trying to go beyond
them straight towards calculus and get them back from there. I don't
know that an ordinary arithmetical product must even exist. It may not
be necessary. Much arithmetic on reals may simply be destroyed. Take
the inequality statement for example. It cannot be applied unless you
go to magnitudes. Should it be replaced with some complicated
threesome of inequalities? I would guess not. Perhaps inequalities
should only exist in the realm of magnitudes.
I don't really have a problem with the real numbers except when
thinking about the space that we live in. They have been burned into
all of our minds and are after all called "real".
Perhaps I will get to rename reality.
How about triality?
Timothy Golden
Very amusing stuff.
Is there really copyright law on the naming of spaces?
Can there only be 26 single lettered capitalized spaces?
That would really suck.
I can see the problem for mathematicians browsing journals.
Confusing one T space for another.
Thanks for offering up your Y space to me.
It is far superior to T.
I am sorry that yours didn't work out.
You don't seem like a negative person.
What is antisymmetry?
I think spaces do compete with each other in our heads.
I just hope that I haven't gotten it all backwards.
Transforms are probably better things to focus on.
Do you ever ponder R X R X R and time?
Also I don't just have three pencils...
I have three types of pencil.
Triphilia...
Here's to triality!
-Tim
MaxCun...@musician.org (Max) wrote in message
William Elliot
> I have played with this concept and would like to further explore it.
> Or could you dispel its validity?
> I will only define its construction.
> There are three branches from an origin.
> Each branch has its own sign.
> I label these signs { -, +, * }.
> Some numbers in this realm are { -1.234, +2.345, *3.456 }.
> I call this space T.
T = ({'-', '+', '*'} x { r in R | r > 0 }) \/ {0}
> Please do not think of these values in terms of vectors.
set theory product AxB = { (a,b) | a in A, b in B }, \/ union
> Operators must be created.
>
Do it.
Timothy Golden
---------------------
I propose the following sign functions in three-signed T space.
These rotational operators change the sign of a number.
In the table that follows "s" represents the sign of an element in T.
s | - s | + s | * s
~~~~~~|~~~~~~~|~~~~~~~|~~~~~~~
- | + | * | -
| | |
+ | * | - | +
| | |
* | - | + | *
For example:
- (-1.234) = +1.234.
+ (-1.234) = *1.234.
* (-1.234) = -1.234.
These are the first honest operators that I can come up with.
The three sign system obviously invokes rotational phenomena. Whereas
the reals use a negation to toggle back and forth in magnitude from
one extremity to another we now have more. The minus operator (-) will
toggle in one direction. The (+) operator will toggle in the other
direction, and the star operator goes all the way around to preserve
the original sign. These rotations are countable and can formulate an
integer counting system. The utility of this is not known to me.
Sorry I go so slowly.
-Tim
Timothy Golden
> What is {-0.0, +2.5, *5.3} x {-0.0, +1.3, *2.0}?
Hi Justin.
I'd like to just do an arithmetical product first.
The "x" in your notation I will treat like the question:
What is 7.0 x 5.0?
Where the answer would be 35.0 for real numbered values.
I feel reasonably sure that the answer is ( + 8.64 * 7.35 ).
I still am not entirely comfortable with this but will defend it:
Notation is a bit of an issue and I am sorry that I am changing it
around a little bit. I don't see the need for commas in an element.
Assuming we are doing an arithmetical product I also don't see the
need for the x.
I restate your question: "What is ( + 2.5 * 5.3 )( + 1.3 * 2.0 )?".
The elements in parenthesis are elements in Y.
Each element in Y is in effect two elements in T summed.
Using the math of rotational operators that I just put on this thread
and using the standard arithmetical laws (they do feel right for the
moment)
( + 2.5 * 5.3 )( + 1.3 * 2.0 )
= + + (2.5)(1.3) + * (2.5)(2.0) * + (5.3)(1.3) * * (5.3)(2.0)
= - 3.25 + 5.0 + 6.89 * 10.6
= - 3.25 + 11.89 * 10.6
= + 8.64 * 7.35 .
To extend this example in general some more notation might be helpful:
minus( y1 ) = the magnitude of the component of y1 in the minus
direction.
e.g. minus( * 3.4 + 2.0 ) = 0, minus( - 1.2 * 2.3 ) = 1.2.
Similarly for plus() and star().
Now the arithmetical product y3 of two values y1 and y2 in Y is:
y3 = ( - minus(y1)star(y2) - star(y1)minus(y2) -plus(y1)plus(y2)
+ minus(y1)minus(y2) + star(y1)plus(y2) + plus(y1)star(y2)
* minus(y1)plus(y2) * plus(y1)minus(y2) * star(y1)star(y2)
).
y3 will reduce to a clean value in Y.
Do you follow?
If you see any problems I hope you will let me know.
It's really not clear to me just how much of traditional mathematics
works with three-signed values. Thanks for making me exercise the
system.
William Elliot
> Rotational Operators
> ---------------------
> I propose the following sign functions in three-signed T space.
> These rotational operators change the sign of a number.
> In the table that follows "s" represents the sign of an element in T.
>
> s | - s | + s | * s
> ~~~~~~|~~~~~~~|~~~~~~~|~~~~~~~
> - | + | * | -
> + | * | - | +
> * | - | + | *
Same as addition of integers modulus 3.
+ 2 1 0
2 1 0 2
1 0 2 1
0 2 1 0
> For example:
> - (-1.234) = +1.234.
> + (-1.234) = *1.234.
> * (-1.234) = -1.234.
>
> These are the first honest operators that I can come up with.
It's not clear what T is, nor in this summary did you include any
description of T. I proposed ({-,+,*} x { x in R | x > 0}) \/ {0}
So would you give a construction of T, a mathematical expression,
in contrast to a verbal description prone to vagueness.
> The three sign system obviously invokes rotational phenomena. Whereas
> the reals use a negation to toggle back and forth in magnitude from
> one extremity to another we now have more. The minus operator (-) will
> toggle in one direction. The (+) operator will toggle in the other
> direction, and the star operator goes all the way around to preserve
> the original sign. These rotations are countable and can formulate an
> integer counting system. The utility of this is not known to me.
>
Huh? Why not just extend -r = -1*r and discuss -1, +1, *1 and is there an
operator a*b and if so, make that something else as * is already taken.
Also think about 0 = +0 = -0 = *0 and do you have convention, +r = r.
How is * parallel to -, which is the inverse of + ?
Robin Chapman
> On Thu, 10 Jul 2003, Timothy Golden wrote:
>
>> Rotational Operators
>> ---------------------
>> I propose the following sign functions in three-signed T space.
>> These rotational operators change the sign of a number.
>> In the table that follows "s" represents the sign of an element in T.
>>
>> s | - s | + s | * s
>> ~~~~~~|~~~~~~~|~~~~~~~|~~~~~~~
>> - | + | * | -
>> + | * | - | +
>> * | - | + | *
>
> Same as addition of integers modulus 3.
> + 2 1 0
> 2 1 0 2
> 1 0 2 1
> 0 2 1 0
And so the system is multiplicatively identical
to the union of the three rays through the origin
in the complex plane through the cube roots of unity.
Perhaps some day we'll get a notion of addition for
these. That would be fun :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"His mind has been corrupted by colours, sounds and shapes."
The League of Gentlemen
Timothy Golden
>
> It's not clear what T is, nor in this summary did you include any
> description of T. I proposed ({-,+,*} x { x in R | x > 0}) \/ {0}
I would say that this is valid.
But it is not fundamental.
I am averse to defining T in terms of R.
If you could define R in the same terms that you propose then I would
happily modify it to encompass T. But since such a definition of R
would read:
({-,+} x { x in R | x > 0}) \/ {0}
you would define R in terms of R and that would be a paradox.
Is there a clean mathematical basis for magnitude?
Or does magnitude always come from a distance function?
My gut feeling is that magnitude is a simplistic and clean concept and
so should be at the basis, not at a higher level.
> So would you give a construction of T, a mathematical expression,
> in contrast to a verbal description prone to vagueness.
T is one magnitude(an unsigned number) with one of three possible
signs (-,+,*).
Would an equivalent to the number line suffice?
Just draw a branch and label each extremity -, +, *.
Take a unit length and mark each branch from the vertex.
Plop a point anywhere on the lines and you can measure it.
The star symbol has been chosen because it has three lines
intersecting and so is the natural symbol to use for the next sign in
the progression -,+,?.
> Huh? Why not just extend -r = -1*r and discuss -1, +1, *1 and is there an
> operator a*b and if so, make that something else as * is already taken.
This is confusing me. Yes (-1)(*r) = -r, where r is a magnitude.
I understand that calling signs rotational operators is distasteful.
There is no conflict with using magnitude one with them.
Factors and multiplication work consistently:
-r = (r)( -1 ) = (r)(-1)(*1).
if (a)(b) = (r) then (-a)(-b) = +r, (*a)(-b) = -r.
Oddly, addition is a larger quagmire (see reply to Chapman, next in
thread).
> Also think about 0 = +0 = -0 = *0 and do you have convention, +r = r.
I think that an unsigned zero is fine and that it is equivalent to the
signed zeros. I see this as an exception and would like to see sign
preserved symbollically on all non-zero numbers. Therefore +r does not
equal r.
Unsigned number symbols should always be magnitudes.
> How is * parallel to -, which is the inverse of + ?
I'm not sure what you mean here.
I don't believe there is any parallel.
I see what you are getting at.
The sign operations yield rotational phenomena.
This gets close to the integer counting system, which could take
several forms.
To clear up your controversy lets define one form:
- rotates the sign + 1/3.
+ rotates the sign + 2/3.
* rotates the sign + 3/3.
Now, although the plus operator appears to be the inverse of the minus
operator the integer counting system has dispelled that notion.
Example:
(-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
I don't know that the integer system has any utility. It's just a way
of looking at the sign.
I'm sorry that this isn't more convincing.
The star operator in T appears unique as the plus operator in R does.
for t1 in T: *t1 = t1.
just as for r1 in R: +r1 = r1.
This is not just a convention.
When you view these postings do you view the entire thread?
Perhaps I am using the Usenet system differently than you.
I view these postings from google groups through a web browser.
This allows a view of an entire thread. Therefore I snip old info so
that there is less wading through redundant stuff.
Thank you for your response.
I do believe you understand very well the problem that I am working
on.
-Tim
Timothy Golden
> And so the system is multiplicatively identical
> to the union of the three rays through the origin
> in the complex plane through the cube roots of unity.
>
> Perhaps some day we'll get a notion of addition for
> these. That would be fun :-)
Strict summation in T is uncomfortable. This is why I proposed Y,
which is a sum of two Ts.
It may be that sum( t1, t2 ) is not always reducible.
On the real number line the choice is simpler and is always reducible.
If I choose to make sum( t1, t2 ) reducible then I break the link to
Y.
The crux of the matter lies in cancellation and superposition.
What is summation supposed to be?
Should sum( -2.3, *2.3) be zero?
Or should sum( -2.3, +2.3, *2.3 ) be zero?
I suggest the latter and so T space slides away in favor of Y space.
In this way T is just a stepping stone to Y.
Charles Douglas Wehner
> + rotates the sign + 2/3.
> * rotates the sign + 3/3.
>
> Now, although the plus operator appears to be the inverse of the minus
> operator the integer counting system has dispelled that notion.
> Example:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
There is a fundamental problem in the established mathematics of data
going missing. One example is the constant during a differentiation.
Where does it go?
The TRANSFINITE mathematics states that it becomes vanishingly small,
but if it could be scaled up by a special infinity it would return.
The COMPLEX mathematics states that it shifts SIDEWAYS in complex
space, and so leaves the real world. Multiplication by -i1 would bring
it back.
NEITHER is a total answer to the problem of vanishing data. Perhaps
there is the possibility of a new "Three-signed arithmetic" emerging
to tackle such problems.
However, at a first perusal I cannot quite see where this is heading.
It would be useful to have real-world problems solved or PARTLY-solved
by the "Three-signed arithmetic". Such "worked examples" would help
focus the mind, to see where we are heading.
On a mathematical page on my own website, I introduced the Eucalculus
- and promptly showed an electronic circuit whose "gain" can only be
explained by the "hump" in the Eucalculus curve.
That curve is itself no more than the reciprocal of Euler's Gamma
function - so it is an extension of mainstream maths.
In this present case, we seem to be leaving the customary definition
of + and - behind. Then we get:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
(1)(2)(1)(2) (multiplication) based on the old or the new maths?
This is not a criticism, just a question.
It would be useful to know where there is a complete synopsis of the
fundaments to be found - with no discussion. This would help in any
study of this system.
Charles Douglas Wehner
Charles Douglas Wehner
> + rotates the sign + 2/3.
> * rotates the sign + 3/3.
>
> Now, although the plus operator appears to be the inverse of the minus
> operator the integer counting system has dispelled that notion.
> Example:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
There is a fundamental problem in the established mathematics of data
going missing. One example is the constant during a differentiation.
Where does it go?
The TRANSFINITE mathematics states that it becomes vanishingly small,
but if it could be scaled up by a special infinity it would return.
The COMPLEX mathematics states that it shifts SIDEWAYS in complex
space, and so leaves the real world. Multiplication by -i1 would bring
it back.
NEITHER is a total answer to the problem of vanishing data. Perhaps
there is the possibility of a new "Three-signed arithmetic" emerging
to tackle such problems.
However, at a first perusal I cannot quite see where this is heading.
It would be useful to have real-world problems solved or PARTLY-solved
by the "Three-signed arithmetic". Such "worked examples" would help
focus the mind, to see where we are heading.
On a mathematical page on my own website, I introduced the Eucalculus
- and promptly showed an electronic circuit whose "gain" can only be
explained by the "hump" in the Eucalculus curve.
That curve is itself no more than the reciprocal of Euler's Gamma
function - so it is an extension of mainstream maths.
In this present case, we seem to be leaving the customary definition
of + and - behind. Then we get:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
(1)(2)(1)(2) (multiplication) based on the old or the new maths?
Charles Douglas Wehner
> + rotates the sign + 2/3.
> * rotates the sign + 3/3.
>
> Now, although the plus operator appears to be the inverse of the minus
> operator the integer counting system has dispelled that notion.
> Example:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
There is a fundamental problem in the established mathematics of data
going missing. One example is the constant during a differentiation.
Where does it go?
The TRANSFINITE mathematics states that it becomes vanishingly small,
but if it could be scaled up by a special infinity it would return.
The COMPLEX mathematics states that it shifts SIDEWAYS in complex
space, and so leaves the real world. Multiplication by -i1 would bring
it back.
NEITHER is a total answer to the problem of vanishing data. Perhaps
there is the possibility of a new "Three-signed arithmetic" emerging
to tackle such problems.
However, at a first perusal I cannot quite see where this is heading.
It would be useful to have real-world problems solved or PARTLY-solved
by the "Three-signed arithmetic". Such "worked examples" would help
focus the mind, to see where we are heading.
On a mathematical page on my own website, I introduced the Eucalculus
- and promptly showed an electronic circuit whose "gain" can only be
explained by the "hump" in the Eucalculus curve.
That curve is itself no more than the reciprocal of Euler's Gamma
function - so it is an extension of mainstream maths.
In this present case, we seem to be leaving the customary definition
of + and - behind. Then we get:
> (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
(1)(2)(1)(2) (multiplication) based on the old or the new maths?
Timothy Golden
> tttp...@yahoo.com (Timothy Golden) wrote in message news:<5c7f8ade.03071...@posting.google.com>...
> > William Elliot <ma...@xx.com> wrote in message news:<2003071022...@agora.rdrop.com>...
> going missing. One example is the constant during a differentiation.
>
> Where does it go?
>
> The TRANSFINITE mathematics states that it becomes vanishingly small,
> but if it could be scaled up by a special infinity it would return.
>
> The COMPLEX mathematics states that it shifts SIDEWAYS in complex
> space, and so leaves the real world. Multiplication by -i1 would bring
> it back.
>
> NEITHER is a total answer to the problem of vanishing data. Perhaps
> there is the possibility of a new "Three-signed arithmetic" emerging
> to tackle such problems.
>
> However, at a first perusal I cannot quite see where this is heading.
This will hopefully eventually replace space-time of classical
physics.
In a physical system the concept of differentiating out a constant
doesn't seem to be a problem. If you look at the velocity of a car it
doesn't matter where the car started travelling, if the resulting
velocity is the same for multiple cases in different positions, then
the velocities should match in the math. In this way the constant
could be seen as ethically positive and legititmates comparing the
velocity of cars at differing positions.
I think more interesting is that there does seem to exist a physical
fundamental from which the calculus operates, in this example it is a
spatial position, which has no further derivative according to
classical physics.
> It would be useful to have real-world problems solved or PARTLY-solved
> by the "Three-signed arithmetic". Such "worked examples" would help
> focus the mind, to see where we are heading.
I agree completely. I am working on it. I'm also hoping that someone
will come across this and find an application. I've discovered that
the Y-space covers the plane and can be graphed on a piece of paper. A
unit circle takes on foreign proportions and yet is still the unit
circle. I guess I can't demonstrate things like unit circles without a
website.
>
> On a mathematical page on my own website, I introduced the Eucalculus
> - and promptly showed an electronic circuit whose "gain" can only be
> explained by the "hump" in the Eucalculus curve.
>
> That curve is itself no more than the reciprocal of Euler's Gamma
> function - so it is an extension of mainstream maths.
>
> In this present case, we seem to be leaving the customary definition
> of + and - behind. Then we get:
> > (-1)(+2)(-1)(*2) = -4. (total sign is 7/3)
>
> (1)(2)(1)(2) (multiplication) based on the old or the new maths?
I suggest that the new math does still match the old in this way.
Isolating magnitude and sign should not be a problem.
I guess I am not justified in this example until the fundamentals are
in place.
This example accepts that a magnitude can be cast into T-space and can
be brought back out with the use of signed unit values in T.
Challenges arise when summation is incorporated into this. I've taken
a stab at that type of product for Y space, which you will find in
this thread up higher.
>
> This is not a criticism, just a question.
>
> It would be useful to know where there is a complete synopsis of the
> fundaments to be found - with no discussion. This would help in any
> study of this system.
Yes. This thread is not an effective presentation any longer. I guess
I just wanted somebody to shoot this down and it hasn't happened yet.
>
> Charles Douglas Wehner
Thank you for taking a look. I respect your input and will try to find
some more impressive results.
-Tim
Charles Douglas Wehner
> I just wanted somebody to shoot this down and it hasn't happened yet.
>
>
> some more impressive results.
>
> -Tim
I had a few thoughts on the subject. Have you heard of the QUATERNIONS
of William Rowan Hamilton?
If your +, -, and * are none-other than the i, j, and k of Hamilton,
then you have REDISCOVERED the Quaternions.
That's OK. I have often discovered something only to find that
somebody got there first. The effect on my mind is not SHAME, but
quite the REVERSE. I find that the great men of science thought along
the same lines as I do - so on those occasions when I am second, I
feel FLATTERED.
I have ALSO found things that nobody ever found before me. That is
because ones confidence grows with every discovery - new or not.
Do investigate the Quaternions:
http://mathworld.wolfram.com/Quaternion.html
Charles Douglas Wehner
Timothy Golden
I think this construction is more primitive than the quaternions.
I'm merely adding another sign to the real numbers and seeing what
happens.
Quaternions appear to contain lots of real numbers and so cannot be
equivalent.
Have you ever tried graphing in a plane like this?
+ plus pole
+
+
+
+
+
+ . . . . . . . p1 = - 7 + 4
+ .
+ .
. + .
. 0 - - - - - - - - - - minus pole
. * origin .
p3 . * .
. * .
* .
* .
* . . . . . . . . . p2 = - 9 * 6
*
*
*
* star pole
This is Y space as a plane. It is simpler than both cartesian 2D
and complex values in that it has just the three way branch instead of
a four way branch. As you can see parallelograms at angle 2pi/3
resolve the entire plane symmetrically. A reduced Y value always has
at most a pair of magnitudes. Every position in the plane can be
resolved.
The "dimensionality" of Y is quite a conundrum. Because
informationally there are two magnitudes informationally it is two
dimensional, yet the value is just one three-signed element in Y.
The question "why a pair?" should be in the mind at this point.
The answer is that in order to obtain a zero by cancellation in a
three-signed system we should require an equal magnitude for each pole
to provide that cancellation.
For a magnitude x
In R : - x + x = 0.
In Y : - x + x * x = 0.
In Y : y1 = - x + x is not zero!
We need y1 * x to get to zero.
If you believe that summation is superposition then I think you will
be convinced. The trouble is that we all think in context of "real
numbers".
Much of this thinking does not extend when more signs are used in the
construction.
I started this all in the context of the question "Why R X R X R X
t?".
This is the spacetime of classical physics which even string theorists
are not destroying. I consider Y X Y to be a competitor to R X R X R X
t.
But also if you add yet another sign ( four signs ) then the extension
would yield at most three magnitudes by the same law of cancellation.
And so four-signed arithmetic may also be a valid competitor to R X R
X R, especially if the cancellation effect is looked upon as
accumulation, which then provides a basis for time. I think it is wise
to continue opening up the three-signed can of worms before moving on
to the four-signed can. It is my hope that dynamics will be found in Y
which will surprise us.
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.03071...@posting.google.com>...
>
Charles Douglas Wehner
MISSING.
Alas! I search and search, but cannot find that e-mail. I must have
erased it by mistake.
If you wrote that e-mail, MY APOLOGIES. I did not mean to be rude -
just waiting until I had time for a fuller answer.
I have to answer it here.
The first question of data going missing and being "recovered" came
from this statement of mine:
>
> The TRANSFINITE mathematics states that it becomes vanishingly small,
> but if it could be scaled up by a special infinity it would return.
>
Here, we can draw a NUMBER LINE -3,-2,-1,0,1,2,3
and put the factorials above the positive part:
1,1,2,6
0,1,2,3
To find the factorial of 2 from that of 3 - which is 6 -
we DIVIDE by 3. It gives 2.
But we can always go back up again by multiplying by 3.
To find the factorial of 1 from 2!, we divide by 2.
We can also go back up.
To find 0! from 1! we divide by 1, and can go back.
To find (-1)! - note the brackets - we divide by 0.
ANYTHING divided by zero is said to be INFINITE.
Zero times infinite is said to be INDETERMINATE.
So we cannot go back up!!!!!
However, at the console we defined zero by typing it in.
We DO NOT KNOW WHAT IT IS, but we know WHERE IT CAME FROM.
We know therefore that (-1)! is 1 divided by CONSOLE-ZERO.
We call this CONSOLE-INFINITY.
As we still have console-zero on our number-line,
we can still multiply console-infinity by it.
So we can go back up.
(-2)! is -(console infinity)
(-3)! is +(1/2)(console-infinity)
&c.
And so, by strict definition of the nature of the zeroes and
of the infinities, we do not have to lose information by
defining it as "indeterminate".
This is one of my discoveries in the as-yet unpublished
"fantasy maths" - where "fantasy" means anything containing
easy-to-use numbers (FUN NUMBERS) and the "FOLLIES" zero and
infinity.
Another name is the TRANSFINITE MATHS.
However, as I made my discoveries by one route it was pointed
out to me by an eminent mathematician that Georg Cantor had
found something very similar a hundred years ago by another route.
The "FOLLIES" are almost the same as the "ALEPHS" of Cantor.
I am not ashamed. We sought in Nature, and we BOTH found.
I also stated:
> The COMPLEX mathematics states that it shifts SIDEWAYS in complex
> space, and so leaves the real world. Multiplication by -i1 would bring
> it back.
>
Here we could consider a cosine. It starts at 1 and descends.
However, I want you to imagine some "micro-polynomial" that, when
given the number 1 computes the cosine a trace further - that is,
0.999999999975625 or whatever.
Think of a "pico-polynomial" that is even smaller in its steps.
0.999999999999999999999999999999999
Think of it being applied as the cosine passed through zero at
the point Pi/2
But 2Cos(X) goes through zero just as 1Cos(X) and 3Cos(X) do.
And we are not allowed "special zeroes".
As the waveform creeps through zero, how does it "know" how to
re-emerge in such a way as to create a symmetrical cosine,
instead of Cos(X) above the zero and 2Cos(X) below?
In electronics engineering, we like to think of such a cosine
being part of a PAIR. The cosine is REAL, and is the SIGNAL -
but there is a hidden sine accompanying it. This we call the
PHASE.
The PHASE is hidden because it exists only in the IMAGINARY
world. However, it defines the SLOPE of the original cosine.
Thus, if the cosine began at Y=1, and the units of the X-axis
are RADIANS, the slope will be a downward slope of 1 as the
cosine goes through zero.
By means of its phase, a cosine can be allowed to pass through
zero without getting lost.
So - with RESTRICTIONS - we can use complex maths.
We use the Euler equation Exp(iX)=Cos(X)+iSin(X)
However, i means CURRENT in electronics - so we use j.
X is too vague, so we use omega-t
Omega is the angular frequency. t is the time.
So we write Exp(j.omega.t)=Cos(omega.t)+jSin(omega.t)
Here, the jSin(omega.t) is actually the INVERTED slope.
Yet it still defines the slope.
Such complications are necessary because the simple
mathematics will not always define what we want.
In the transfinite case, we have no need for the
imaginary numbers. In the complex maths we have no
need for the transfinite.
At least in theory.
Perhaps there are problems that can only be solved
by BOTH.
Then we have the "mad maths" of Boole. 1+1=1
At first sight, it is ridiculous.
Then somebody "explains" that 1 is TRUE.
He translates "TRUE AND TRUE IS TRUE".
Now he seems REALLY to have gone bananas.
Yet without Boole, there would be no computer and
no Internet for me to write this on.
The Quaternions are sometimes used in elementary-
particle physics.
I cannot ACCEPT a new mathematical system without
seeing its benefits.
I cannot REJECT a new mathematical system for
fear of rejecting something good.
If I had time to delve and delve, I might know
whether the "THREE-SIGNED ARITHMETIC" is useful
or not.
But it is not mine, and I am busy. I will, however,
if there is an adequate summary, look back now and
then to see if something important is emerging.
I reserve judgement.
Charles Douglas Wehner.
Justin Davis
exhaustive examples from scratch? My guess is that it is probably isomorphic
to something we're already familiar with, in which case there will be lots
of useful things you could draw on that already are known to understand how
T works.
Timothy Golden
Whereas the real numbers (R)can be defined as a magnitude with two
signs( -, + ), T is a magnitude with three signs( -, +, * ).
Examples of elements in T are:
- 1.23
+ 2.134
* 6.54
Now consider summation in T.
The sum of the example numbers reads:
( - 1.23 + 2.134 * 6.54 ).
It is a mistake to say that - 1.23 * 6.54 = * 5.31.
This style of cancellation will not work.
The order of operations would be too strict.
If we use the law:
Sum( t1, t2, t3 ) = Sum( Sum( t1, t2 ), t3 ) = Sum( Sum( t3, t1 ),
t2 )
we will find the problem.
Consider the following using the fraudulent style of cancellation:
( - 1.23 + 2.134 ) * 6.54 = ( * 6.54 - 1.23 ) + 2.134
( + 0.904 * 6.54 ) = ( * 5.31 + 2.134 )
( * 5.636 ) = ( * 3.176 ).
This is nonsense.
The problem occurred at the concept of cancellation.
Should * 1 - 1 = 0?
Should - 1 + 1 = 0 ?
Or should - 1 + 1 * 1 = 0?
The latter is the correct selelection.
For a magnitude x:
In R
- x + x = 0.
In T
- x + x * x = 0.
Some values are not reducible. Thich leads me to describing a new
space Y because Sum( t1, t2 ) is not generally in T. So even though
the subject in this thread is T space it is only a starting step to
get to Y space, which is general three-signed arithmetic.
Y is a general sum of T which is always reducible to at most a pair of
three-signed magnitudes. Now Sum( y1, y2 ) is always in Y.
This leads to ( - 1.23 + 2.134 * 6.54 ) = ( + 0.904 * 5.31 ).
where ( - 1.23 + 1.23 * 1.23 ) has been cancelled out.
This is all that I have time for right now. Arithmetical products
follow much more easily.
Max
> This is nonsense.
Yes, now you're talking sense.
Max de Macs
Timothy Golden
---------------------------------------
Here is the one law that may really break Y space:
Product( y1, Sum( y2, y3 )) = Sum( Product( y1, y2 ), Product( y1,
y3 ).
Allow * to represent the sum of two elements in Y since it preserves
sign and follows symmetrically to the addition operator in reals.
Allow arithmetical product to be as previously defined. These concepts
are quite symmetrical to the real number concept of opertations.
The above law can be restated as:
y1(y2 * y3) = y1y2 * y1y3.
If we invoke this law then the rules for arithmetical product and sum
that I have laid out might break down.
Let y1 = * 1.0 - 1.0, y2 = * 1.0 - 1.0, y3 = - 2.0 + 2.0 .
Invoking: (y1)(y2*y3) = (y1)(y2)*(y1)(y3)
(*1.0-1.0)(*1.0-1.0-2.0+2.0) =
(*1.0-1.0)(*1.0-1.0)*(*1.0-1.0)(-2.0+2.0)
(*1.0-1.0)(-2.0+1.0) = (*1.0-1.0-1.0+1.0)*(-2.0+2.0+2.0*2.0)
-2.0+1.0+2.0*1.0 = - 1.0 + 2.0
- 1.0 + 2.0 = - 1.0 + 2.0 .
Oops! That one worked.
Let y1 = * 1.0 - 2.0, y2 = * 1.0 - 1.0, y3 = - 2.0 + 2.0 .
Invoking: (y1)(y2*y3) = (y1)(y2)*(y1)(y3)
(*1.0-2.0)(*1.0-1.0-2.0+2.0) =
(*1.0-2.0)(*1.0-1.0)*(*1.0-2.0)(-2.0+2.0)
(*1.0-2.0)(-2.0+1.0) = (*1.0-1.0-2.0+2.0)*(-2.0+2.0+4.0*4.0)
-2.0+1.0+4.0*2.0 = -2.0+1.0+4.0*2.0
+ 3.0 = + 3.0 .
Oops! That one worked too.
Now prove that (y1)(y2*y3) = (y1)(y2)*(y1)(y3).
Define operators m(y), p(y), and s(y) which return the magnitude of y
for the signs minus, plus, and star respectively (e.g. m(-2.0*6.5) =
2.0.
y2 * y3 = - Sum(m(y2), m(y3))
+ Sum(p(y2), p(y3))
* Sum(s(y2), s(y3)).
(y1)(y2*y3) =
(
+ m(y1)(Sum(m(y2),m(y3))
+ p(y1)(Sum(s(y2),s(y3))
+ s(y1)(Sum(p(y2),p(y3))
- m(y1)(Sum(s(y2),s(y3))
- p(y1)(Sum(p(y2),p(y3))
- s(y1)(Sum(m(y2),m(y3))
* m(y1)(Sum(p(y2),p(y3))
* p(y1)(Sum(m(y2),m(y3))
* s(y1)(Sum(s(y2),s(y3))
)
= [ distributing the product through the sum and gathering like
terms...]
(
+ m(y1)m(y2)
+ p(y1)s(y2)
+ s(y1)p(y2)
- m(y1)s(y2)
- p(y1)p(y2)
- s(y1)m(y2)
* m(y1)p(y2)
* s(y1)s(y2)
* p(y1)m(y2)
+ m(y1)m(y3)
+ p(y1)s(y3)
+ s(y1)p(y3)
- m(y1)s(y3)
- p(y1)p(y3)
- s(y1)m(y3)
* m(y1)p(y3)
* p(y1)m(y3)
* s(y1)s(y3)
).
(y1)(y2) = [This matches the upper half of the equation above]
(
+ m(y1)m(y2)
+ p(y1)s(y2)
+ s(y1)p(y2)
- m(y1)s(y2)
- p(y1)p(y2)
- s(y1)m(y2)
* m(y1)p(y2)
* s(y1)s(y2)
).
(y1)(y3) = [This matches the lower half]
(
+ m(y1)m(y3)
+ p(y1)s(y3)
+ s(y1)p(y3)
- m(y1)s(y3)
- p(y1)p(y3)
- s(y1)m(y3)
* m(y1)p(y3)
* p(y1)m(y3)
* s(y1)s(y3)
).
This is adequate proof that the law
y1y2 * y1y3 = y1(y2*y3)
is holding.
Timothy Golden
I've got a proof that aligns well with standard arithmetic which is a
pretty good example of some three-signed arithmetic. It seems to be
holding up.
Timothy Golden
like the Mandelbrot plot for the complex plane. I was hoping for
something more interesting but I guess this does help to validate the
mathematics. The formula is:
y(n+1) = Sum( Product( y(n), y(n)), y(0))
Or:
y(n+1) = y(n)y(n) * y(0).
where the star represents summation.
The product is as I have laid out and the summation as well.
I have not bothered to do any transformation, choosing instead to
simply look at each third of the Y-plane at a time. If the magnitude
of any component exceeds 20.0 I call it escaped. Iterations to 100 are
adequate to get the shape. The lobes look nearly identical to my
memory of the Mandelbrot graph of the complex plane and all of the
fractal detail is there. I have zoomed in on a section and it has the
same lobed pattern on each sprig. The spike of the pattern comes out
centered in the plus-star third of the Y plane.
Is this meaningful to anybody?
Charles Douglas Wehner
> I have plotted the Mandelbrot function for Y and it is coming out much
>
> Is this meaningful to anybody?
The Mandelbrot images suddenly flooded onto the market in the middle
Eighties when some academics at Hamburg university mechanised their
production and captured the market.
There was hardly a computer exhibition that did not offer advertising
leaflets with spectacular surreal coloured images from the Mandelbrot
set.
Then the fad faded.
I myself made programs to create these images. I call them RECURSION
MAPS - and consider the Mandelbrot set to be just a subset of
recursion sets.
To make a Mandelbrot image, you imagine a plane X Y, and arrange for
the program to move in tiny steps along X (along the rows), and along
Y (from row to row), as with television scanning.
You use the COMPLEX MULTIPLY function.
To multiply, you arrange that the new X is X-squared - Y-squared and
the new Y is 2XY
We will see that this produces a "boring symmetry", so we also ADD 1
Now we check whether NEW X or NEW Y is "too big" - according to some
rule, such as above 256 (in a machine-code program) or 1000 or
whatever.
If not, we multiply again according to NEW X = X.X-Y.Y+1, NEW Y = 2XY
we check again, and repeat. This is the recursion.
When making a 256-colour GIF image, one would allow only 256
recursions. However, if one or other part had become "too big", we may
have exited from the recursion loop with a smaller number of counts.
The number of recursions achieved is plugged into the image at
position X,Y
The colour is chosen using "artistic skill and judgement".
Elsewhere on the newsgroups, we have been discussing the "boring
symmetry" of the imaginary axis of complex maths. This symmetry can be
seen in the Mandelbrot set - which is asymmetrical along the X-axis,
but otherwise symmetrical.
I substituted other functions for the COMPLEX MULTIPLY. One of these
was the Gamma function, for which I had ultra-precise coefficients.
New images emerged - but nothing so spectacular as to cause another
craze.
We cannot understand these images because of the iterative multiply.
Let us call it P+Q multiplied by P+Q.
This gives P-squared + 2PQ + Q-squared
When this is multiplied by itself, we get P-to-the-fourth + 4P-cubed.Q
+ 6P-squared.Q-squared + 4Q-cubed.P + Q-to-the-fourth
When this is multiplied by itself, we get CONFUSED.
This is CHAOS. The underlying idea is simple, but a first-order term
becoming a second-order, a fourth-order, a sixteenth-order quickly
exceeds the capacity of the mind to follow. We also have to visualise
the term exceeding a certain limit - and also translate the
recursion-number in our minds into a colour.
The Mayas might have found it other than confusing. They had not
invented the WHEEL, and yet they had a numbering system based on
wheels. If you have a wheel with five cogs meshed into a wheel of six
cogs meshed into a wheel of seven, you can uniquely define 210
separate numbers by the positions of the wheels.
The psychology is utterly alien to those brought up with decimal
numbers - but the Mayas had the finest astronomy.
So I look forward to a simple T-space solution to chaos.
Charles Douglas Wehner
Timothy Golden
But the graphics are inspiring.
Charles Douglas Wehner
> I think I can appreciate your criticisms of the meaning to the math.
> But the graphics are inspiring.
>
NOT a criticism.
> > So I look forward to a simple T-space solution to chaos.
CHAOS theory is indeed a field where a new "psychology" of mathematics
is needed.
The graphics are elaborate, convoluted, impenetrable. Given the
graphics, we cannot tell how they are made. Given the method, we are
surprised that something so simple produces such elaboration.
Nature is full of this - elaborate patterns created by simple means.
The skill that is sought is to find the ORDER within the CHAOS.
Charles Douglas Wehner
Timothy Golden
Iteration is always the key to it though.
And that is not unlike time, which I see a lot of people struggling
with on sci.physics.
I like to think in terms of accumulation, which is a more fundamental
principle than time. That's what makes the integral work. Puzzling
over the fundamentals it is difficult not to get stuck.
One funny thing about the poly-signed approach is that as the sign
increases by one so does the dimensionality. This assumes that you
allow for accumulation at a more fundamental level. As a three-signed
space covers a plane a four-signed space covers cubic volume. It is
just as intuitive as the graph of the plane I presented here on
2003/07/17.
Somehow the poly-signed approach is close to a cross product.
In RxRxR with unit vectors i, j, k we have:
(using the right-hand rule)
i x j = k
i x k = -j
i x -j = -k
i x -k = j
in poly-signed numbers we see the same thing happening with a product
of -1 arithmetically, except that the pattern can continue to the
equivalent result of i and -i as well.
In four-signed space X:
(-1)(-1) = +1
(-1)(+1) = *1
(-1)(*1) = #1 (symbol choice for next sign in progression)
(-1)(#1) = -1
This makes me question the necessity of the cross product for
poly-signed numbers. Perhaps you have some input on this line of
reasoning?
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.03072...@posting.google.com>...
Charles Douglas Wehner
systems.
Firstly, one can say that the "-" in existing systems is either UNARY
or BINARY.
Thus -5 is a UNARY use of the minus. 10 - 5 is a BINARY use. These are
standard terms.
Secondly, the unary form often IMPLIES a function. A function takes an
ARGUMENT and delivers a RESULT. So argument 5 with the negation
function gives -5.
The binary form always is a function, because you can see TWO
arguments shrinking into ONE result.
Functions have properties:
distributivity - can one do such as in conventional math when
5(4+3) = 5*4 + 5*3 ?
commutativity - can one swap arguments 5*4 = 4*5 ?
associativity - can groups be swapped (5+4)+3 = 5+(4+3) ?
Finally is there an IDENTITY FUNCTION.
For example, with factorials there are TWO:
1!=1
2!=2
Charles Douglas Wehner
Timothy Golden
Yes to all of your questions.
The associative, distrubutive and commutative properties do work for
Y.
I'm glad you're staying on with me here.
I have laid this out pretty well in the messages above this point.
I am sorry if my communication or this format is not efective.
Unary and Binary usages of the sign symbols are acceptable.
I would stress that there is no "10" in T.
"10" is a magnitude.
It should be +10, -10, or *10 in T.
as "10" is +10 in R I think you will understand this.
Think of the signs as you would think of them for real numbers, except
that there is one more of them. The star ("*") operator is symmetrical
to the "+" operator in reals.
In a message on 2003/07/10 I put out the arithmetical product rules
for sign.
Here are all the examples for a magnitude x:
-(-x) = +x.
-(+x) = *x.
-(*x) = -x.
+(-x) = *x.
+(+x) = -x.
+(*x) = +x.
*(-x) = -x.
*(+x) = +x.
*(*x) = *x.
There is a well founded pattern here.
The phenomena is rotational in nature.
The following operation is an example of summation or superposition:
* 1 + 2
Just as - 1 + 2 in reals is.
It's fairly easy to say that
* 1 * 2 = * 3
just as
+ 1 + 2 = + 3
and
- 1 - 2 = - 3.
T is a number system like R, just with an extra sign.
The above sums are graphically shown on the equivalent of the number
line, which in T has three extensions from the origin rather than two.
Here are some more examples of summation:
+ 1 * 2 - 3 + 2 - 2,
* 1 + 1 - 6 + 5 * 3 * 2,
+ 1.34 * 2.45 * 4.5 - 1.23.
In resolving sums I have discovered that they do not necessarily
reduce to one single magnitude! This is a result of the conception of
cancellation and superposition. If you attempt to resolve -1 + 2 to be
+ 1 you will find that the associative law breaks (see 2003/07/18
message). Cancellation only occurs when we have a component in every
sign within the sum (-x+x*x = 0). This is where the dimentionality
that I have mentioned is exhibited. For a three-signed space sums of
two signs do not resolve to a single magnitude and this has led to the
use of the term "Y-space" since I defined T-space to be a single
magnitude with three signs. In general sums are in Y, not in T.
For two elements y1 and y2 in Y their sum can be reperesented as
y1 * y2.
I am sorry if this is annoying because of the standard use of star for
products in reals, but symbollically in Y it is effective as sum.
Suppose y1 = - 3 + 2 and y2 = - 1 * 3.
y1 * y2 = ( - 3 + 2 ) *( - 1 * 3 ) = - 3 + 2 - 1 * 3
= - 4 + 2 * 3 = - 2 * 1 - 2 + 2 * 2 = - 2 * 1.
The - 2 + 2 * 2 at the end cancels out to zero.
Products are much easier. Just distribute the terms as for reals:
(y1)(y2) = ( - 3 + 2 )( - 1 * 3 )
= (-3)(-1) * (-3)(*3) * (+2)(-1) * (+2)(*3)
= + 3 - 9 * 2 + 6
= + 7 - 7.
The identity function; The factorial example is a little confusing to
me.
*1 is the identity value. (*1)(*1) = *1.
if we take an element y1 in Y and want (y1)(y1') = *1 then we must
find something like a complex conjugate:
(+1*1)(-1*1) = *1+1-1*1 = *1.
I have worked out division for specific examples. The method uses
linear equations by supposing that the answer is -a+b*c for a given
problem. Too much to print here.
I think you are like me in that it is much easier to create my own
math than it is to read someone else's. I really do appreciate your
questions and hope that you can see it. If not, make it up yourself
and I am almost certain you and I will speak the same language. There
are choices to make along the way but the choices must yield
consistency.
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.03073...@posting.google.com>...
> There are formal methods of logic for the analysis of new mathematical
> systems.
>
> Firstly, one can say that the "-" in existing systems is either UNARY
> or BINARY.
> Thus -5 is a UNARY use of the minus. 10 - 5 is a BINARY use. These are
> standard terms.
>
> Secondly, the unary form often IMPLIES a function. A function takes an
> ARGUMENT and delivers a RESULT. So argument 5 with the negation
> function gives -5.
>
> The binary form always is a function, because you can see TWO
> arguments shrinking into ONE result.
>
> Functions have properties:
> distributivity - can one do such as in conventional math when
> 5(4+3) = 5*4 + 5*3 ?
Yes!
>
> commutativity - can one swap arguments 5*4 = 4*5 ?
Yes!
>
> associativity - can groups be swapped (5+4)+3 = 5+(4+3) ?
Yes!
Charles Douglas Wehner
> -(-x) = +x.
> -(+x) = *x.
> -(*x) = -x.
> +(-x) = *x.
> +(+x) = -x.
> +(*x) = +x.
> *(-x) = -x.
> *(+x) = +x.
> *(*x) = *x.
> There is a well founded pattern here.
> The phenomena is rotational in nature.
Because this is a new math, I will not use the term "MULTIPLY".
Rather, I will say that the SIGNA (plural or SIGNUM, sign) are "COMPOUNDED".
It seems that the star is an IDENTITY FUNCTION when compounded.
From the above:
> -(*x) = -x.
> +(*x) = +x.
> *(*x) = *x.
Charles Douglas Wehner
Timothy Golden
Yes, star in three-signed is like plus in reals.
I would like to move ahead with this math but am stuck for the time
being.
I don't see how to do any calculus with it.
I try to think in terms of constructions for stability but I'm not
seeing anything new for this space except for the following:
(-1)^n [ -1 to the nth power ]
This yields the sequence { -1, +1, *1, -1, +1, *1, ... )
(+1)^n yields ( +1, -1, *1, +1, -1, *1, ... )
This makes me think that here are two stable oscillators which are
easily distinguishable and forever stable and extremely simple in
construction.
This is teetering much closer to the edge of complex arithmetic which
I hope you are thinking of.
You still aren't buying the dimensionality/cancellation argument are
you?
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.03080...@posting.google.com>...
Charles Douglas Wehner
> product or multiply are fine ways to look at the sign pattern.
> Yes, star in three-signed is like plus in reals.
SO USERS WILL HAVE TO BE WARNED THAT UNARY + IS NOT THE IDENTITY. IT
MIGHT HAVE BEEN BETTER TO PREPARE THE IDEAS BEFORE PUBLICATION. It is
hard to swap plus and star once one has already defined standards.
> I would like to move ahead with this math but am stuck for the time
> being.
> I don't see how to do any calculus with it.
>
> I try to think in terms of constructions for stability but I'm not
> seeing anything new for this space except for the following:
>
> (-1)^n [ -1 to the nth power ]
>
> This yields the sequence { -1, +1, *1, -1, +1, *1, ... )
>
> (+1)^n yields ( +1, -1, *1, +1, -1, *1, ... )
>
> This makes me think that here are two stable oscillators which are
> easily distinguishable and forever stable and extremely simple in
> construction.
> This is teetering much closer to the edge of complex arithmetic which
> I hope you are thinking of.
>
NATURALLY! A complex math without the "modular" restriction would be
nice.
> You still aren't buying the dimensionality/cancellation argument are
> you?
>
I haven't had the time to delve deep enough to understand it.
Rather, as I work on my OWN projects, I keep peeping in - trying to
pose questions about the CORE concept.
For example, in nuclear physics one has the QUARKS - up, down, left,
right, strange and charmed. It would be nice if, say, + define the
up/down; - defined the left/right and * defined the strange/charmed.
However, they don't, do they?
The singularity at zero (the origin) is where THREE signa branch out.
The number-line is no longer a line.
I am trying to see if there is anything useful there - forever mindful
of the fact that math must be USED. It must serve a PURPOSE. It is a
TOOL.
One does not have to solve all the problems of the world - just show
the way IN to the solution of something that cannot be solved, or
cannot be solved easily, by other means.
Charles Douglas Wehner
Timothy Golden
I hope it will be helpful to you.
The identity sign is * for T and + for R.
Though this is annoying there is reasoning behind it.
I made the symbolism aesthetically meaningful.
- is the first sign. (one line)
+ is the second sign. (two lines)
* is the third sign. (When I write it by hand it is three lines)
And for 4-signed I have chosen "#" which becomes the identity value.
I'm sorry if you don't care for this. However as you see the integers
assigned to these signs are accurate mnemonics to how the signs
operate. In T: - moves one branch over. + moves two over. * moves
three over. In 4-signed ( X ) the same is true but there is # to move
four over.
This logic works backward to the reals also.
I think it is extremely important that you try to see the
dimensionality phenomenon. It is really fundamental and comes purely
out of superposition and cancellation, or summation. I believe this to
be at the core of any application that this math will yield.
In R: -1.5 + 2.0 = -1.5 + 1.5 + 0.5 = 0 + 0.5 = 0.5.
You could look at this as cancellation or superposition or as
summation.
The reason that I broke the math down to - 1.5 + 1.5 was to provide a
zero.
If we are to do the same in T then - 1.5 + 1.5 * 1.5 will be zero.
This is proper cancellation.
If you go down the road of thinking that * 1.5 + 1.5 could be zero you
immediately get stuck at the associative law:
( - 1.5 + 1.5 ) * 1.5 = ( * 1.5 - 1.5 ) + 1.5
* 1.5 = + 1.5
Nonsense!
The problem with the above math was the assumption that the new number
system matched the reals in the way of always providing cancellation
between any two values.
Proper cancellation requires an equal component in every sign:
- 1.5 + 1.5 * 1.5 = 0.
This result causes the dimentionality phenomenon.
And now:
- 2.5 + 3.0 * 1.5 = - 1.0 + 1.5.
This is possible if you accept that
- 2.5 = - 1.5 - 1.0
and
+ 3.0 = + 1.5 + 1.5.
In three-signed arithmetic general sums are two-dimensional.
Likewise in four-signed space they are three-dimensional.
The sign phenomenon happens to allow proper graphical analysis.
The plane is perfectly resolved in three-signed math:
+ plus pole
+
+
+
+
+
+ . . . . . . . p1 = - 7 + 4
+ .
+ .
. + .
. 0 - - - - - - - - - - minus pole
. * origin .
p3 . * .
. * .
* .
* .
* . . . . . . . . . p2 = - 9 * 6
*
*
*
* star pole
The plane exists as thirds rather than quadrants.
Likewise R x R x R is perfectly resolved with a four-signed system and
quadrants.
I understand the the geometrical argument is hard to believe, but
informationally it is easy to believe. The transforms clearly exist
and they exist symmetrically and geometrically.
One big question I have is whether the cartesian product is even a
necessity in poly-signed arithmetic. Instead the increase in sign can
be used.
We generally accept that we exist in three-dimensional space. It's
replacement is four-signed space. How do you get to four signs?
Perhaps by coming up from one sign. So in effect we might accept a 4 x
3 x 2 x 1 construction.
The 3 x 2 part might model quarks effectively and composes them with a
volume.
This is very abstract but it seems to be the level you are thinking
on.
This is where I would like to go but I am not stuck on trying to come
up with quarks. I'd like to find some source of stability that I can
believe in mathematically.
I really appreciate your attention on this.
Even if this flops at least it is brain food for you.
If you can make it flop I would really appreciate it.
I'm struggling with where to go from here.
You're statements are always encouraging.
Thanks.
I'd like to email you some of my Mandelbrot results images.
They are in windows .bmp format right now but I could convert them.
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.03080...@posting.google.com>...
> SO USERS WILL HAVE TO BE WARNED THAT UNARY + IS NOT THE IDENTITY. IT
> MIGHT HAVE BEEN BETTER TO PREPARE THE IDEAS BEFORE PUBLICATION. It is
> hard to swap plus and star once one has already defined standards.
>
> NATURALLY! A complex math without the "modular" restriction would be
Charles Douglas Wehner
> I hope it will be helpful to you.
> The identity sign is * for T and + for R.
OK - that is no problem.
One problem is that in the standard math we have UNARY and BINARY + as
well as UNARY and BINARY -
Now we will require UNARY and BINARY *
That implies a NEW function - NEITHER an addition nor a subtraction.
The A*B function.
I can imagine your bank manager writing to you "DEAR MR. GOLDEN, YOUR
ACCOUNT IS NEITHER IN CREDIT (+) NOR IN DEBIT (-), BUT IN STAR (*) -
HOW DO YOU PROPOSE TO REMEDY THE SITUATION?
In addition, in any new math every function has to be rigidly examined
as regards commutativity &c. For example, the Boolean AND and OR,
together with NAND, NOR, XOR and XNOR are ALL commutative.
In conventional maths, + and * are commutative whilst - and / are not.
We need a lot of precise definitions of this system before we can see
whether it is of use.
Regarding Quarks &c. - I am just looking loosely for places where a
weird math might fit.
Charles Douglas Wehner
Timothy Golden
mechanics.
Subtraction is still summation, just of a negative sign.
i.e. in R: 4 - 5 = (+4) + (-5) = -1.
In three-signed, star (*) is the general summation operator.
The plus operator in R and the star operator in Y both preserve the
sign of the values they are applied to. There is no actual third
operator. It is just sign modification.
Suppose y1, y2 in Y.
y1 * y2 means the sum of y1 and y2.
y1 - y2 = y1 * (-y2) which means the sum of these values with y2's
sign modified.
Suppose y1 = - 2 + 3 and y2 = + 2 * 4.
y1 * y2 = - 2 + 3 + 2 * 4
= - 2 + 5 * 4
= + 3 * 2.
Because -(y2) = -( + 2) -( * 4 ) = * 2 - 4:
y1 - y2 = - 2 + 3 * 2 - 4
= - 6 + 3 * 2
= - 4 + 1.
And y1 + y2 = - 2 + 3 - 2 + 4
= - 4 + 7.
It believe it is best just to use the term summation and let the sign
operators do their thing. It's all consistent and the concern over
unary and binary goes away.
> One problem is that in the standard math we have UNARY and BINARY + as
> well as UNARY and BINARY -
>
> Now we will require UNARY and BINARY *
>
> That implies a NEW function - NEITHER an addition nor a subtraction.
> The A*B function.
>
> I can imagine your bank manager writing to you "DEAR MR. GOLDEN, YOUR
> ACCOUNT IS NEITHER IN CREDIT (+) NOR IN DEBIT (-), BUT IN STAR (*) -
> HOW DO YOU PROPOSE TO REMEDY THE SITUATION?
>
> In addition, in any new math every function has to be rigidly examined
> as regards commutativity &c. For example, the Boolean AND and OR,
> together with NAND, NOR, XOR and XNOR are ALL commutative.
>
> In conventional maths, + and * are commutative whilst - and / are not.
But they have simple conversions to the straight product and sum:
for a, b in R
a / b = a(1/b) = (1/b)a.
a - b = a + (-b) = (-b) + a.
This method is applicable in three-signed as well.
The reciprocal value should be *1.
> We need a lot of precise definitions of this system before we can see
> whether it is of use.
>
> Regarding Quarks &c. - I am just looking loosely for places where a
> weird math might fit.
>
> Charles Douglas Wehner
Maybe one way to convince you is to complicate sign modification by
nesting more signs:
In R:
-(+(-(-1) = -1
In Y:
-(+(-(-1) = +1
Working in R you count around the number of poles you pass and the
last one you come to is the sign of the number. The same is true in Y.
In effect the sign operator should be viewed as a discrete rotational
operator.
Charles Douglas Wehner
I introduced the terms "MONAL" and "BINAL" elsewhere - from which
"TERNAL" &c. follow. This was in another context.
BINAL would be a term for "REQUIRES TWO" numbers, where BINARY means
"HAVE TWO".
Consider a NEGATION FUNCTION in Three-signed arithmetic.
If -, +, and * are mapped onto a piece of paper (for our ease of
understanding), they produce - as you showed - three NUMBER-LINES from
one ORIGIN.
Thus, a position on a number-line is MONAL (requires ONE specification
from the origin).
The NEGATIVE of that position will be in the SPACE between the OTHER
two signa, and will be BINAL (requires TWO specifications).
As - is not longer the negation operator, we have to use some jargon
such as "NEG".
Perhaps NEG(+5) gives -5,*5
Perhaps NEG(+5) gives -2.5,*2.5
Perhaps NEG(+5) gives -2.5(root3),*2.5(root3)
YOU, Timothy Golden, are making the rules - and I am not fussing over
the exact rules. Instead, I am showing that there is a ONE
parameter/TWO parameter transmutation going on when negating.
Odd!
Charles Douglas Wehner
Timothy Golden
Yes. I agree with what you are saying and I would choose your first
supposition for NEG().
I do not like the word "negative" as a concept since the construction
is already sign based and the negative sign has taken on its own form
of negation.
I would think that the NEG function you are thinking of returns a
value in Y such that the sum of the original and the result are zero:
Sum( y, NEG(y) ) = 0.
Or:
y * NEG( y ) = 0.
If y is + 5, then NEG(y) is - 5 * 5, since - 5 + 5 * 5 = 0.
You are definitely getting the concept.
As you say:
> I am showing that there is a ONE
> parameter/TWO parameter transmutation going on when negating.
If you consider summation of the simplest values( like *1 and +1 )
you will find that they do not condense to one simple value in T.
The expression * 1 + 1 is not reducible.
However any three termed sum is always reducible:
- 2 + 3 * 4 = + 1 * 2.
+ 4 + 5 - 2 = + 9 - 2.
- 8.12 + 3.10 * 5.14 = - 5.02 * 2.04.
+ 2 + 3 + 6 = + 11.
This then leads to the Y space, which is at most a pair of
three-signed values in T summed.
This has been defined much earlier in this thread.
So your NEG() function also needs Y to work in.
charle...@hotmail.com (Charles Douglas Wehner) wrote in message news:<e349085c.0308...@posting.google.com>...
Timothy Golden
For any y in Y:
* y + y - y = 0.
Timothy Golden
following:
x = s - 1/2( p + m )
y = sqrt(3)/2( m - p )
where x and y are the traditional cartesian coordinates and s, m
and p are the star, minus, and plus components of a value in three
signed space.
I have written some code that performs the conversion and I have
checked products against their x + iy form and get complete agreement!
For two values y1, y2 in Y the product y1y2 is the same as if I
convert them to z1, z2 in C and compute the product z1z2.
This means that angles add in the product.
I believe that this is quite a discovery.
Simply put, the complex plane is equivalent to three-signed space as I
have defined it.
Timothy Golden
By choosing the star branch to be the real axis the product arithmetic
works out. The transform from three-signed Y to a complex C is:
s(y) - (1/2)(m(y) + p(y)) + i(sqrt(3)/2)(m(y) - p(y))
Where y is in Y and m(),p(),s() return the minus,plus and star
components of their argument.
These equations come from pure graphical analysis of the three-signed
branches being at angles 2pi/3. The components resolve via a 30-60-90
triangle.
- minus pole
-
-
-
-
-
- . . . . . . . p1 = * 7 - 4
- .
- .
. - .
. 0 * * * * * * * * * * star pole
. + origin .
p3 . + .
. + .
+ .
+ .
+ . . . . . . . . . p2 = * 9 + 6
+
+
+
+ plus pole
It can be shown that (y1)(y2) = y3 is equivalent to the product of the
transformed values (a1+b1i)(a2+b2i).
Proof of Equivalent Product Between Y and C
-------------------------------------------
The following transform allows the conversion of a value in Y to a
value in C:
z = s(y)-(1/2)(m(y)+p(y)) + i(sqrt(3)/2)(m(y)-p(y)).
This is a geometrical analysis of the three-signed plane suprposed
with the complex plane with the star pole aligning with the positive
real axis. The star component matches the real and the minus and plus
components derive from right triangles.
To simplify the notation I will simply use s,m,and p to represent the
three components. I will also use q3 to represent the square root of
three.
All of the math for this proof is in the real domain.
Now prove:
y1y2 => z1z2
We have
z1 = s1-(1/2)(m1+p1) + i(q3/2)(m1-p1)
and
z2 = s2-(1/2)(m2-p2) + i(q3/2)(m2-p2)
Their product is then:
(s1-m1/2-p1/2 + iq3m1/2-iq3p1/2)(s2-m2/2-p2/2 + iq3m2/2-iq3p2/2)
= s1s2 -s1m2/2 -s1p2/2 +s1m2iq3/2-s1p2iq3/2
+ -m1s2/2 +m1m2/4 +m1p2/4 -m1m2iq3/4 +m1p2iq3/4
+ -p1s2/2 +p1m2/4 +p1p2/4 -p1m2iq3/4 +p1p2iq3/4
+ +m1s2iq3/2 -m1m2iq3/4 -m1p2iq3/4 -m1m2(3/4) +m1p2(3/4)
+ -p1s2iq3/2 +p1m2iq3/4 +p1p2iq3/4 +p1m2(3/4) -p1p2(3/4).
Gathering terms with precedence in the order of s,m,p:
= s1s2 -s1m2/2 +s1m2iq3/2 -s1p2/2 -s1p2iq3/2
+ -m1s2/2 +m1s2iq3/2 +m1m2/4 -m1m2(3/4) -m1m2iq3/4 -m1m2iq3/4 +m1p2/4
+m1p2iq3/4 -m1p2iq3/4 +m1p2(3/4)
+ -p1s2/2 -p1s2iq3/2 +p1m2/4 +p1m2(3/4) +p1m2iq3/4 -p1m2iq3/4 +p1p2/4
-p1p2(3/4) +p1p2iq3/4 +p1p2iq3/4
And reducing cancellable terms:
= s1s2 -s1m2/2 +s1m2iq3/2 -s1p2/2 -s1p2iq3/2
+ -m1s2/2 +m1s2iq3/2 -m1m2/2 -m1m2iq3/2 +m1p2
+ -p1s2/2 -p1s2iq3/2 +p1m2 -p1p2/2 +p1p2iq3/2
The product y1y2 has the following properties based on its definition:
m = m1s2 + p1p2 + s1m2.
p = m1m2 + p1s2 + s1p2.
s = m1p2 + p1m2 + s1s2.
Converted to a complex value it is:
+m1p2 +p1m2 +s1s2 -(1/2){m1s2 +p1p2 +s1m2 +m1m2 +p1s2 +s1p2 }
+i(q3/2){m1s2 +p1p2 +s1m2 -(m1m2 + p1s2 + s1p2)}
= m1p2 +p1m2 +s1s2 -m1s2/2 -p1p2/2 -s1m2/2 -m1m2/2 -p1s2/2 -s1p2/2
+m1s2iq3/2 +p1p2iq3/2 +s1m2iq3/2 -m1m2iq3/2 -p1s2iq3/2 - s1p2iq3/2
Now gathering terms from the above to better match the complex product
further up:
= +s1s2 -s1m2/2 +s1m2iq3/2 -s1p2/2 - s1p2iq3/2
-m1s2/2 +m1s2iq3/2 -m1m2/2 -m1m2iq3/2 +m1p2
-p1s2/2 -p1s2iq3/2 +p1m2 -p1p2/2 +p1p2iq3/2
This exactly matches the reduced form of the complex product and
represents the equation of the product of two three-signed values y1y2
(where s1,m1,p1 represents y1 and s2,m2,p2 represents y2 converted to
the complex representation.
Sums are also equivalent. That is:
y1 * y2 => z1 + z2.
The proof is trivial.