Sun 24 Feb 2019 07:32:30 AM EST
Real and Complex Numbers from the same axioms
Consider the behavior of the reals:
- 1 + 1 = 0
or more generally - a + a = 0 .
If we introduce a third sign above and beyond minus and plus; say star: - 1 + 1 * 1 = 0 .
Clearly the three-signed system and the two-signed system are different entities and so a value such as
- 3.2
is limited in meaning until its containier is specified, for the meaning of the ‘+’ symbol is simply sign two:
- : 1 : minus,
- : 2 : plus,
- : 3 : star,
: 4 : sharp,
…
We need not even go into sign four here, except for the fact the generalization of sign is in fact general and so we can discuss 21-signed numbers if we wish. To achieve the goal of complex numbers we need only the third sign, and rather than digress into the full generalization I should remain at three-signed numbers (P3) so that the neophyhte can learn.
Whereas P2 (the two-signed real numbers) have a modulo-two behaved product:
-
- = + ; 1 + 1 + 2
-
- = - ; 1 + 2 = 1
-
- = + ; 2 + 2 = 2
it should be clear already that the P3 system will be modulo-three behaved:
- = + ; 2 + 2 = 2
-
- = +
-
- = *
-
- = -
-
- = -
-
- = +
-
- = *
and so to those who are large minded enough to grant the generalization of sign its wings we should admit that the identity has shifted to star in P3, whereas it was plus in P2. Though notationally this would seem undesirable, by remaining within these bounds of ordinary nomenclature we get a system which is notationally consistent with the existing notation. To rectify this we should introduce a zero sign ‘@’ which need not shift and maintains the familiar modulo arithmetic which carries such a zero result.
For this introductory attempt we will ignore the zero sign as most have proven not to be quite that large minded. Now if you are insulted by this language already it is likely time for you to turn away. Be gone then.
- = *
Well; hooked you are then who have gotten this far, and the promised land of the title of this thread can be yours, but surely P3 has no resemblance to the complex plane of old. Well, let us consider that the real line (P2) was composed of two rays equally opposed to each other such that
- 1 + 1 = 0 .
In effect this law of balance requires that geometry, so what about P3? On a piece of paper simply draw out three rays from an origin each 120 degrees from its neighbors and label them -, +, *. Using vectors trace one unit in the minus direction(from any place on the paper really), thence one unit in the + direction, thence one unit in the * direction, and you will land exactly where you started. P3 develops the plane via its strict balance - 1 + 1 * 1 = 0 .
If you’d like to challenge this you’ll need another instance of a geometry which obeys this first law of balance. Thus far I know of none. In effect already with one single law we have developed a numerical system which inherently carries geometry that is general dimensional. We see in the sign product the rotational nature of its mathematics, so should it really surprise you that P3 is equivalent to the complex numbers? No. The full build is at my website
http://bandtechnology.com/PolySigned
which carefully lays out all of this through product, sum, proof of equivalence of P3 to the complex plane, and on into other details.
The ramification that I am focused on this morning here is simply to popularize the fact that the generalization of sign is possible; is fully discovered; and that its extensive yet fundamental nature are in fact an upset to existing mathematics so base that most are repulsed why? It is because we have been trained upon the real number as fundamental in every field from mathematics to physics even on into relativity theory. We bump into those mysterious complex numbers not just in mathematics but as welll in engineering and in physics, where they provide solutions that are otherwise even more mysterious. P3 are every bit as fundamental as are P2. Their little sister P1 gets just this littly bitty mention where it fits right in with P1, P2, P3, P4, P5, and the rest.
Here in polysign land the whole family are algebraically behaved geometries built from the marriage of a discrete component (sign) and a continuous component (magnitude) married together in the form
s x
where s is sign and x is magnitude. We ought not to build complicated things from other complicated things, particularly when such refined solutions as this exist. Sadly mathematicians have made a religion of the real number that has gripped humanity for four hundred years. Now it is time that we make a break with our primitive ancestors. It is time for a new universal geometry and algebra. It is the time of the polysign numbers.