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.