THE LOGIC OF THE SQUARE ROOT

ABSTRACT

Let us start with the square root of a positive number. Previously it should be pointed out that the square root of a negative number, receive another treatment in the science of mathematics today, since it is considered an operation "impossible". As you know, the square root of a number enables two possible outcomes: one with a positive sign, and another, of the same value as the previous, but with a negative sign. This is what most fits in mathematics the concept of polarity, or conjugated states.

So if the result of this operation, is squared (which is the inverse operation call empowerment), with any of the two signs of the result, you get the original positive number.

Symbolically : Sq. Root (+A) = + or – Sq. Root ( A ) .

Making SR ( A )= R

Therefore : (+R). (+R) = +A and (-R) . (-R) = +A

Let us now a little bit of history in the development of mathematics. Since its discovery in the sixth century B. C. , man has perfected the use of irrational numbers (which are in type square root), always assuring in its awareness of the concept that the square root of a positive number was the only one that could lead to real solutions. Funnily enough, it is always identified with one of the two poles of the result, denying the simultaneous consideration of both.This despite the fact that the rule of signs that is the basis of mathematical operations, says:

+ . + = +
- . - = +
+ . - = -

Where as seen, appears the explicit consideration of both simultaneous signs, in the third line. This happened because, in the inverse operation, call of empowerment, to reach the original number, (positive), the result of the root is seen with a single sign to be squared, by definition of empowerment. The operation of empowerment considers the number with your sign as a whole. Therefore, you cannot have opposite signs at the base of the power.

In the XII century, the brahman Bhaskara expressed this as follows: The square of a positive number, such as a negative number, it is positive; and the square root of a positive number is twofold, positive and negative; there is not the square root of a negative number, since this is not a square. Same as the word implies, empowerment is to highlight one of the aspects of the fact conjugate, turning to the "forgot" the other.

But the limitations imposed by the mathematicians to the empowerment transaction do not justify the fact to divest happily one of the parts of the rule of the signs, because this is a database of mathematical constructions. In fact, if the operation of empowerment is the inverse operation of the operation of square root, and in the latter, the result is a number with two signs, is not clearly defined in science, the reason why these signs cannot be considered simultaneous.

What is clear is that the latter is allowed by the third rule of the signs, and therefore must be applied in this particular case, as well as normally applied in the product of two different numbers, with different sign.

Determining that itself is clearly involved with the cubic or higher roots odd exponent , as shown below:

                      Cubic root (- A) is negative , and the operation of empowerment that allows return to (- A) is as follows , considering only the sign of the result :

                     Step 1 ) ( - ) * ( - ) = ( + )

                     Step 2) (+) * ( -) = (- )      In this step the simultaneous consideration of both signs can be seen , allowing display it as a natural transit of empowerment.

Along with the course of history, in the XVI century, Cardano and Bombelli put wax in this detachment, inventing, through a mathematical artifice, the imaginary number (to distinguish it from a real solution) by sealing as well, at least the consciousness of the western man, in a single pole position until our days. This artifice was to enter a letter: i, imaginary, so that i=root ( - 1), with which hides the "problem" that causes the negative sign.

 Thus the result of making the root of a negative rational number is expressed: Root (- A) = i . R, where R is the square root of the positive number A. In this way was originated a new branch of mathematics that was called "the complex number", which, by analogy, it is as if you have installed a "complex" in the scope of this science. Such complex consists in choosing a sign of the lies and hide the opposite sign.

Serve then the treaty here to define a novelty in the field of mathematics: there is a real solution for the square root of a negative number. It is an irrational number with both signs considered simultaneously, which makes this solution indeterminate.
In symbols: Root (- A) = + and - ( R ) because + ( R ) x - ( R ) = - A , what leads to the desirability of adopting the same current denomination, but giving the i the meaning of undetermined: Root (- A) = i . R This is a key aspect of this logical content, because it is clear the need to conserve the complementarity of the two solutions to sustain consistency, but with indeterminacy in its manifestation.

This suggests to us that there is a field of  logic reality  where things must not appear yet defined, but that does not deny its intelligibility as a pair of conjugate solutions.
What will be the implications of this discovery in mathematics, it is a fact to investigate. What is certain is that it could generate an innovative movement in consciousness.

In the figure you can read the following:

* The logic of square root supports the relationship between two fields opposing and complementary .

* From the  negative field, the solution is exclusionary: Indeterminacy (there is no conscious choice), because the root is simultaneously + and -.

* From the positive field, the  solution derived in three options: two of empowerment of both signs separately, and an option with both simultaneous signs (there is conscious choice), the latter generates uncertainty in the conscience, and is the only option to connect to the opposite field. 

* It appears that the positive field has correspondence with the conscience, because there is  intelligible option.

* It appears that the negative field has no correspondence with the conscience, because there is no option. The conscious choice is denied and therefore there is correspondence with the unconscious.

* To move from the rational field conscious to the unconscious, it is necessary that  consciousness go away  from the rational position, like in the case of emotion. 

* Any deployment irrational from the positive rational field is assembled in their two opposite aspects, despite the fact that the operator will reject or to forget about one of them, so that this attitude is not progressive. The progress is with the advance in consciousness, both simultaneous aspects.

* The final connection with the rational complementary field, requires, in the awareness, the maintenance of a dynamic contradiction.

* In the negative rational field  dialectic exists by necessity, complementing the rational positive, from the unconscious.

* The progress of the conscience, holding the contradiction, is epistemological because it seeks to expand its expertise. By complementarity, the indeterminacy of the deployment of irrational -A, is ontological.

* Given an evident unpredictability  for the connection of consciousness with the field ( -A), because the conscious does not define this, it is necessary to accept the existence of an "ontological determination" to this fact.

* It could be said that the positive is what appears or is known, what is determined, while the negative is what denies its appearance or what should remain undetermined.

Possible Applications:
1) Physics: Reviewing the EPR`s Paradox and Bell`s Theorem. Anti-matter.
2) Psychology: The known and the unknown as opposite pair together with conscious and unconscious.

LA LÓGICA DE LA RAIZ CUADRADA

Partamos de la raíz cuadrada de un número positivo. Previamente se debe aclarar que la raíz cuadrada de un número negativo, recibe otro tratamiento en la ciencia matemática actual, ya que es considerada una operación “imposible”.

Como se sabe, la raíz cuadrada de un número habilita dos resultados posibles: uno con signo positivo, y otro, del mismo valor que el anterior, pero con signo negativo. Esto es lo que más se adapta en matemáticas al concepto de polaridad, o estados conjugados.
De manera que si el resultado de esa operación, se eleva al cuadrado (que es la operación inversa llamada de potenciación), con cualquiera de los dos signos del resultado, se obtiene el número positivo original.

Simbólicamente: Raíz (+A) = + ó – Raíz (A), o bien + ó – R, donde R es Raíz (A)
De modo que :+ R . + R = + A o también (– R ) . (– R) = + A
Hagamos ahora un poco de historia en el desarrollo de las matemáticas. Desde su descubrimiento en el siglo VI A.C., el hombre ha perfeccionado el empleo de los números irracionales (que son los del tipo raíz cuadrada), asegurando siempre en su conciencia el concepto de que la raíz cuadrada de un número positivo era la única que podía dar lugar a soluciones reales. Curiosamente se identificó siempre con uno de los dos polos del resultado, negando la consideración simultánea de ambos. Esto a pesar de que la regla de los signos que está en la base de las operaciones matemáticas, nos dice:

(+) x (+) = (+)

(–) x (–) = (+)

(+) x (–) = (–)

Donde como se aprecia, aparece la consideración explícita de ambos signos simultáneos, en el tercer renglón.
Esto sucedió porque, en la operación inversa, llamada de potenciación, para llegar al número original, (positivo), el resultado de la raíz se considera con un único signo para ser elevado al cuadrado, por definición de potenciación. La operación de potenciación considera al número con su signo como un todo. Por lo que no puede haber signos opuestos en la base de la potencia.
En el siglo XII, el brahman Bhaskará expresaba esto de la siguiente manera:
El cuadrado de un número positivo, como el de un número negativo, es positivo; y la raíz cuadrada de un número positivo es doble, positiva y negativa; no existe la raíz cuadrada de un número negativo, puesto que éste no es un cuadrado.
Como la misma palabra lo indica, potenciar es resaltar uno de los aspectos del hecho conjugado, pasando al “olvido” el otro.
Pero las limitaciones impuestas por los matemáticos a la operación de potenciación no justifican el hecho de desprenderse alegremente de una de las partes de la regla de los signos, porque esta es una base de las construcciones matemáticas. De hecho, si la operación de potenciación es la operación inversa de la operación de radicación, y en esta última el resultado es un número con dos signos, no está claramente definido en la ciencia la razón por la cual esos signos no pueden considerarse simultáneos. Lo que si está claro es que esto último está permitido por la tercera regla de los signos, y por lo tanto debe aplicarse en este caso particular, así como normalmente se aplica en el producto de dos números distintos, con diferente signo. Determinación que sí está claramente en juego con la raiz cúbica o de exponente impar mayor, como se muestra a continuación:
                      La raiz cúbica de (- A), es negativa, y la operación de potenciación que permite volver al número origen (- A) es la siguiente, considerando sólo el signo del resultado:
                                   Paso 1)    (-) * (-) = (+)  
                                   Paso 2)    (+) * (-) = (-), que es el signo del número origen. En este paso se aprecia la consideración simultánea de ambos signos, lo que permite mostrarla como un tránsito natural de la potenciación.
Siguiendo con el curso de la historia, en el siglo XVI, Cardano y Bombelli le ponen lacre a este desprendimiento, inventando, mediante un artificio matemático, el número imaginario (para distinguirlo de una solución real) sellando así, al menos la conciencia del hombre occidental, en una posición monopolar hasta nuestros días.
Este artificio consistió en introducir una letra: la i, de imaginario, de manera que i=Raíz (–1), con lo cual se oculta el “problema” que ocasiona el signo negativo. Así el resultado de efectuar la raíz de un número racional negativo se expresa: Raíz (– A) = i . R, donde R es la Raíz cuadrada del número positivo A.
De esta manera se originó una nueva rama de las matemáticas que se denominó “del número Complejo”, que, por analogía, es como si se hubiera instalado un “complejo” en el ámbito de esta ciencia. Tal complejo consiste en elegir un signo del radicando y ocultar el signo contrario.
Sirva entonces lo aquí tratado para definir una novedad en el campo de las matemáticas: existe una solución real para la raíz cuadrada de un número negativo. Es un número irracional con ambos signos considerados simultáneamente, lo que vuelve dicha solución indeterminada.

En símbolos:

Raíz (– A ) = + y – ( R ) porque + ( R ) x – ( R ) = – A

Lo que conduce a la conveniencia de adoptar la misma denominación actual, pero dando a la i el significado de indeterminado: Raíz (– A ) = i . R
Este es un aspecto clave de este contenido lógico, porque surge claramente la necesidad de conservar la complementación de las dos soluciones para sostener la coherencia, pero con indeterminación en su manifestación. Lo que nos sugiere que hay un campo de la realidad lógica donde las cosas no deben aparecer todavía definidas, pero que no niega su inteligibilidad como par de soluciones conjugadas.
Cuales serán las implicancias de este descubrimiento en matemáticas, es un hecho a investigar. Lo cierto es que podría generar un novedoso movimiento en la conciencia.

Una posible manera de tratarlo se muestra en el siguiente esquema:

En el cuadro se puede leer lo siguiente:

*La lógica de raíz cuadrada admite la relación entre dos campos racionales opuestos y complementarios.
*Desde el campo negativo la solución es excluyente: Indeterminación (No hay opción consciente), porque la raiz es simultaneamente + y -.
*Desde el campo positivo la solución deriva en tres opciones: dos de potenciación de c/u de los signos, y una opción con ambos signos simultáneos (Hay opción consciente), esta última genera incertidumbre en la conciencia, y es la única opción para conectar con el campo opuesto.
*Se deduce que el campo positivo tiene correspondencia con la conciencia, porque hay opción inteligible.
*Se deduce que el campo negativo no tiene correspondencia con la conciencia, porque no hay opción. La elección consciente está negada y por lo tanto hay correspondencia con lo inconsciente.
*Para pasar del campo racional consciente al inconsciente, es necesario irracionalizar la posición consciente racional.
*Todo despliegue irracional desde el campo racional positivo está ensamblado en sus dos aspectos opuestos, a pesar de que el operador rechace u olvide uno de ellos, por lo que esta actitud no es progresista. El progreso se da con el avance en la conciencia, de ambos aspectos simultáneos.
*La conexión final con el campo racional complementario, requiere, en la conciencia, el sostenimiento de una contradicción dinámica.
*El campo racional negativo existe por necesidad dialéctica, complementando al racional positivo, desde el inconsciente.
*El avance de la conciencia, sosteniendo la contradicción, es epistemológico porque busca ampliar su conocimento. Por complementación, la indeterminación del despliegue irracional de –A, es ontológica.
*Dada una imprevisibilidad evidente para la conexión de la conciencia con el campo (-A), porque el darse cuenta no lo define esta, es necesario aceptar la existencia de una “determinación ontológica” para tal hecho.
*Se podría decir que lo positivo es lo que aparece o se hace conocido, lo que se determina, mientras que lo negativo es lo que niega su aparición o lo que debe permanecer indeterminado.


Posibles aplicaciones:
1) En la Física: Complementación con la paradoja EPR y el teorema de Bell. Antimateria.
2) En Psicología: Lo conocido y lo desconocido, como par de opuestos, aplicado al par conciencia- inconsciente.
Nota: este trabajo se complementa co:
http://pascal142857.blogspot.com.ar/2011/09/determinacion-fisica-de-la-raiz.html

AIR ENERGY CONVERSION

BY: RAFAEL R. TERESCHUK, ENGINEER, B. AIRES,ARG.


INTRODUCTION:
At present, it is possible to observe the following structural deficit, of social character, referred to energy availability, and to the hazardous effect that conventional sources of energy produce in the atmosphere.
-Conventional sources (fossil), were suitable to develop a human paradigm: “to obtain useful energy it is necessary to burn up something”, paradigm coming from the Fire Era.
-In this way, man has experienced with several sources, adopting finally three: carbon, petroleum and gas, because of its heating power. These basic sources conform the platform that support mechanical and electrical generation around the world.
-As we know, these sources are limited, and at the present consumption rate, known reserves will be exhausted in short. Furthermore, this situation will be harmful for the petrochemical industry, which produces an enormous variety of products on which human activity depends.
-Combustion products from these sources, concentrated on the Earth atmosphere, are producing ecological problems with unforeseeable consequences.
-Concentrated energy resources in a few nations, added to inevitable scarcity due to its limitation, will produce important lack of equilibrium of nations, with unforeseeable results.
-Water scarcity, that people are beginning to talk, could substitute with sea water through de-salination process, but it is so expensive at present that it becomes inconvenient, in practise.
-The present generation and distribution of electricity, is not sufficient to supply all the human requirements, limiting human development, in a progressive sense.
-The use of alternative sources like solar and eolic, is very limited and dependent on available sunlight and wind.
-Great dams and nuclear plants produce irreversible ecological problems.
-Bio-fuel requires such a large cultivation area, that could affect production and price of foods.
-As we can notice, the structure of energy availability, on which humanity bases its development, is at risk, and could produce great social problems in the near future.

THE PROJECT:
As a consequence of fifteen years labour partially dedicated to search in eolic energy, surged an idea, that was registered, and would permit equipment production to converse air energy into useful mechanic and electric energy.
The observation of conventional eolic equipment performance, plus the observation of Magnus effect in rotating cylinders submitted to an air flow, allowed the elaboration of a project finally rewarded at Lujan`s University, Argentina (Dec 2001). This produced deepened understanding on air energy and the means to access it.
It was possible to apply the simple idea that birds and planes, fly without the need of natural wind. It is not necessary to expect that winds blow.
The observation of technical characteristics of commercial planes, lead to develop this idea that I am applying in a prototype, which has the following characteristics:
- It is different from eolic conventional equipment, where it is necessary a natural wind.
- Here a forced wind develops Magnus forces on rotating cylinders, integrating same wing plane effects, but with cylinders around a center, in a circular design.
- Wing plane effect is a lifting force, equal or greater than plane`s weight, obtained with its motors force that is as much the third part of its weight. (Like all people can know in the web about Airbus or Boeing planes). This important difference between lifting and dragging forces revel an additional energy coming from static pressure in the air, useful to plane lifting in the fly.
- Lifting force in a cylinder gets a maximum, for the greatest relation established between circulatory velocity flow and the translational one. This implies maximal torque in the beginning of rotation movement of the system.
- Converts static air energy, that by its nature, pictured by State Equation of Physics, is a way of renewable and available energy, inner related to solar radiation.
The project focus is then, the construction of an idea`s prototype, and the corresponding verification tests, taking the basis traced by a math model, that has been realized.

If this focus result could be attained, it will be possible to get a solution to social-energetic problems described above.

PAST HISTORY: A German scientist called Magnus discovered a new effect in Fluid Mechanics. Working for the German army in the solution of the unforeseen problem surged with canyon bullets running on transversal wind, that displaced bullets away from its objective, Magnus discovered the cause. It is due to interaction between wind and rotation of cylindrical body of the bullet.
Important experiences were realized with rotating cylinders.
In Germany, in 1924, two cylinders of 18 m high and 3 m diameter where used to replace the sails of a ship. Those cylinders, which were ten times greater in exposed area than the sails, moved the ship with same power. This experience was called Flettner`s ship.
Project Madaras was another case, designed to generate electricity by means of a ring of cylinder-cars running on rails, moved by wind.
National Advisory Committee for Aeronautics in the U.S.A, searched the Magnus effect on cylinders in their laboratories to verify the Flettner`s ship experience, at 1925. The conclusions were the following:
“1. The air forces obtainable by a superposition of a circulatory flow upon the one arising from translation, of a doubly symmetric body, are several times greater than have ever been observed on any un-symmetric body.
2. Lift increases with circulation (circulatory flow), although the law connecting the variables is not definitely established or the limits of application known.
3. The rate of increase of lift with rate of revolution seems practically independent of the shape of rotating body, provided it is symmetrical about both axes in its plane of rotation”.
(N.A.C.A., Technical Note Nº 209, 1925, U.S.A.)

IMPORTANT CONCLUSIONS: NACA`s conclusions suggested that rotating cylinders submitted to an air flow can produce a relation between obtainable lift force vs. delivered force, much greater than 1, and similar to the produced in airplanes, with the advantage that this relation can be regulated by cylinder rotation.

PROJECT DESCRIPTION:
The originality of this design is based on the realization of a device made of cylinders distributed around the center of a rotating platform which support the cylinder. Each of them can rotate around his own axis producing a rotating flow of air that combines with a translational forced flow from the axis of platform, flowing through the free space between the cylinders, which is auto-sustained by the rotation of the system. By the Magnus effect, this combination can generate transversal forces to the radius of platform, on each cylinder, producing a resulting mechanical moment on the system axis, which can be used to move a standard electric generator connected to this axis.

Experiences made by NACA (NASA, at present), 1925.
CL: Lift coefficient. Proportional constant in lifting force equation.
CD: Drag coefficient. Proportional constant in dragging force equation.

Important difference can be appreciated between these two coefficients.
The mechanic work realized by wind of velocity V, represented by drag coefficient CD, is several times less than obtained work, represented by lift coefficient CL. On the other hand, both works develops in transversal directions, eliminating the possibility of mutual collaboration.
Necessarily, as can we note, this implies the participation of surrounding air to generate lifting force, and taking account that wind`s work is acting like excitment work for the conversion process, combined with friction`s rotation work, that joined, can be less than lifting work, depending on rotation`s velocity w.r on cylinder.
On this way, a free mechanic energy can be produced from static mass of air, for human use.

ADVANTAGES FROM THIS PROJECT:
-Reduction of fossil fuel consumption.
-Reduction of present CO2 emissions, diminishing the risk of climatic change.
-Availability of abundant, renewable and immediate energy for all.
-To Reduce the cost of energy and obtain free availability, yet in the mountain like in the ocean.
-Human mass incorporation to the benefits of energy, reinforcing their activity.
-To reduce international stress coming from appropriation of natural sources of energy.
-To produce potable water from sea water.
-To reserve gas and petroleum for petrochemical industry.
-To eliminate the construction of great nets of energy distribution.
-To create a new paradigm: there is only one source of natural energy that congregate us.

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