Re: The common usage of "nonnegative real number" is ludicrous.

On Mar 25, 6:22 pm, "RogerB" <rhberesf...@xxxxxxxxxxxxxx> wrote:
Dear Tim,
Sorry - I forgot to indicate that "~" means "is an equivalence
relation on". r~{p1,p2} translates as r=p1-p2 , -r={p2,p1},

This does not extend does it?
So when you write
r ~ { p1, p2 }
and then extend to
r ~ { p1, p2, p3 }
I cannot figure out how your definition works.

In the polysign system when we define
{ 1, 1 } = 0
we extend to
{ 1, 1, 1 } = 0
but at this point we have to accept that the components are
nonorthogonal and indeed even in the paired instance they were
nonorthogonal as well. The real number is just the interesting result
that the paired system collapses so a singular component value:
- 1 + 3 = + 2
whereas the triple does not generally:
- 1 + 3 * 4 = + 2 * 3
where the reduced form is to the right.

The raw components are magnitude under the polysign construction,
where magnitude is something more fundamental than the real number.
The continuum starts with magnitude, not the real number.

realzero~{pj,pj} i.e. a pair of identical primals equivalence to a
real zero. Similarly c~{p1,p2,p3,p4} means x+iy=p1-p3+i(p2-p4) and
{pj,pk,pj,pk} equivalences (when using the C4 table) to 0+i0, the
complex zero. I prefer not to scale my basis vectors, keeping them
orthogonal and normalised, avoiding the confusion (as I see it)
created by 0 = { 1.1387, 3.354608, 423.12345 }.

OK. That is clear and I agree though scaling is an area to grow into
above the bottom layer.
Your choices are toward collecting systems and so I thought perhaps
you had collectivized this.

Example. The primal 6-vector {3,6,1,2,4,5} 2-folds to {3-2,6-4,1-5}
or {1,2,-4}. This has two conserved "sizes", (1+2-4) =-1, and
((1-2)^2+(2-(-4))^2+((-4)-1)^2)/2=31.
It has a polar dual {-1,31, -ArcTan[Sqrt[27]/2]}. On multiplication,
the sizes multiply and the angles add.
Sizes are magnitude, and can be real, complex, terplex etc depending

Does size mean primal? If so then what I saw as a statement of support
for magnitude becomes contaminated. This statement above here does not
work for me. I put magnitude beneath the others as a fundamental from
which the others are constructed. Therefor it is not part of such a
progression. Instead I get one-signed numbers down in that slot, which
are radically different.

on the field. They can be negative, as in the example. Non-abelian
algebras can even have quadratic sizes that are negative. The Pauli 4-
element algebra separates time-like (+ve size) from space-like (-ve
size) vectors by the light-cone (0 size). Some physical entities
(time, mass, temperature?) appear to have primal measures. Others are
differences, and so cannot be primal - differentials are a key
example.

Differentials are a good topic. Does magnitude itself comes with
operators?
Superposition takes dimensionality and so I would argue that the sum
or difference (they are the same thing) does not occur until sign is
introduced. The product seems more difficult to dismiss at the primal
level since scaling is in effect what the primal seems capable of
doing. This is like saying continuum implies product yet that is not a
statement that I feel confident of. The product takes on other
mysterious behaviors that appear not to be part of nature as
superposition does. For instance we believe that the real line can be
used to represent qualities of physical space but when we look at
products on the real line we will see a broken symmetry that has no
direct correspondence to nature. I consider this an open problem and
beyond this accept that the scalar product is easily defined. Beyond
that the arithmetic product with its sign mechanics is also easily
defined.

That brings me to a challenge for you. We see within your notation a
collection of orthogonal two-signed systems. In particular
"c~{p1,p2,p3,p4} means x+iy=p1-p3+i(p2-p4)"
indicates this. Yet here we see two types of mechanics where the
polysign construction sees one type. You have the rules
(-a)(-b) = + ab, ...
and then you also have rules
(ia)(ib) = -ab, ...
These rules collapse to the same unitary rule
( s1 x1 )( s2 x2 ) = (s1 + s2) x1 x2
where s is sign and x is magnitude. The transform for the complex
values is involved and so I refer you to
http://bandtechnology.com/PolySigned/ThreeSignedComplexProof.html
The result is that rather than requiring these two seemingly
incongruent types we get a fundamental structure that yields the
complex type as a natural extension of the two-signed (real) type, or
rather both are elements of a family and are adjacent to each other
down in its early stages:
P1, P2, P3, ...
time, the reals, complex numbers, ...
0D, 1D, 2D, ...
All of these systems obey
Sum over s ( s x ) = 0 .
( s1 x1 )( s2 x2 ) = ( s1 + s2 ) x1 x2
where s is sign and x is magnitude. The sign sum in the product
definition is a modulo or radix type of summation. This is layed
clearly at
http://bandtechnology.com/PolySigned/PolySigned.html

Thank you for the Olariu reference - it looks meaty.

It is also identical to your math isn't it? He's got some projections
that get very close to the polysign, but his roundabout route is not
motivational. The systems (polysign and terplex) fly apart with the
orthogonality distinction.

Dear Hero,
I contend that numbers are not just pairs, ordered or not. Your
Bombelli example is a set of four numbers that multiply according to
the C4 group. There are many other possibilities. An Octonion is a set
of 16 primals or 8 reals that act as a vector in a non-associatve, non
commutative space with 16 directions or 8 dimensions. So is a split-
octonion. Clifford(p,q) algebras act in a space of 2^(p+q)
dimensions. Every group defines an algebra, but most are
uninteresting.
The sets need to be "indexed" rather than "ordered"; the
multiplication then gives the same result even if the multiplication
table is changed into an isomorph.
Thanks for the sunshine - it reached me this afternoon with a warm
east wind.
Roger Beresford.

We are humans doing math. The mathematician shoots for perfection and
must insist upon it. An errant system has no place in the structure.
Likewise a poorly built structure will be superseded. Whether the
existing mathematician who has imbibed in the supposed perfection can
open to new constructions is a humanistic problem.

-Tim

.

Relevant Pages

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