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COMPLEX NUMBERSFlowers

Co-written with Nahid Sahel Gozin

 

Over the past centuries, classical physics has served as an effective theory to explain the nature. However, today many puzzles and paradoxes have emerged that cannot be explained within the context of classical mechanics.

 

Consciousness also remains a mystery. Findings in this area over the past century defy the principles of contemporary neuro-science as well. New revelations have painted a more detailed picture of the universe and present an alternative reality that we cannot explain within the traditional axioms. A brand new science is needed to explain the unexplained.

In these articles, I will present an alternative model and offer explanations for existing paradoxes based on this new concept. In this model, the observable universe is just a portion of reality. Here, I include an unconventionally defined singularity to the scenario as the complimentary portion. According to the Big Bang theory, singularity is the zero-size entity that initiated our universe.

In order to better follow this model, it is helpful to review the concept of Complex Numbers. I will try to explain the concept in layman’s terms. Alternatively, the reader may choose to skip the math equations and just look over the derivations made. Doing so will not prevent comprehending the concept.

I will start by explaining the basic principles of complex numbers. This will be followed by my derivations of the complex number mathematics as it applies to physical world. The interpretations and assertions made are not necessarily accepted in contemporary physics. The analysis is derived and defined on the context of this model.

Complex number can be shown in a Cartesian plane, where the X-axis represents the real value and the Y-axis denotes the imaginary part of a parameter.

Imaginary Numbers

Imaginary numbers were discovered by Omar Khayyam, an Iranian poet and mathematician, when he introduced the quadratic equations. They are the square root of negative numbers (√-n). Known mathematics could not offer a solution for this problem, because no number could be squared to a negative number (e.g., -1). Until the sixteenth century, mathematicians avoided imaginary numbers like the plague. However, when Italian mathematicians were working with cubic equations, they realized to solve them sometimes they needed to use these imaginary numbers.

Since these numbers do not point to objective aspects of physical element, they are called imaginary numbers, and are shown by the symbol i. The square of this imaginary number would be a negative number (e.g., -1).

I take imaginary numbers to represent factors that are in a sense hidden in our classical underestanding of reality. Or better, qualities that do not possess objectivity. However, these numbers denote potentiality. If we want to bring them out of obscurity, then we need some manipulation. By squaring them, we turn them to a real number that would disclose them in our objective reality and can contribute to the real number equations.

It should be noted that while any object contains measurable parameters, it also has numerous essences that are not computable or observable. We may assume that imaginary numbers represent qualitative aspects of an object. These qualities are the base for potentialities, which in turn can evolve the object to a new state.

Since neither objective quantities nor subjective qualities could be ignored, it was decided that a combination of real number and imaginary number is essential to explain any entity. Therefore, complex numbers are considered fundamental to mathematics. (Mind you, I adopt the view that mathematics itself is merely an invention to simplify our understanding of physical reality) Therefore,

Complex number = [ x (real number) + iy (imaginary portion)]


Argand DiagramIn 1806, Jean-Robert Argand, trying to give geometrical visualization to complex numbers, suggested the diagram: 

 

In 1799, Gauss proved the fundamental theorem of algebra using complex numbers. Nowadays, complex numbers are used throughout mathematics and have many applications in modern science. In this spirit, numbers are rightfully reintroduced as follows:

N = x + 0i

Where N denotes any parameter, x is the real value, and i conveys the imaginary aspect of the parameter. In the Argand diagram, real numbers are zero thickness points occupying a one-dimensional geometric line. Therefore one can conclude the following:

 

Assertion C1: In a sense, pure real numbers do not portray the total reality.  

Real values occupy a geometric line (zero thickness) inside the diagram. We must include an imaginary number value (along the Y-axis) as well to encompass reality in its totality.

periodic complex number

In diagram 2, the polar version of Argand diagram is shown, where r = |z|. r is called the absolute value, or modulus. The angle a = arg(z) is called the complex argument of z.   Therefore, every point in the diagram can be shown as follows:

Z = x +iy = r (cos,a + i  sin,a), where x = r cos a, is called the real part and iy = i sin a represents the imaginary part.

Rotation and circling of the modulus in the plane creates a sine wave motion. Mind you that in quantum theory every particle has a wave like nature as well.

The complex number system is the mathematics of our most accurate discipline in physics, quantum mechanics. Roger Penrose points out, “These strange numbers also play an extraordinary and very basic role in the operation of the physical universe at its tiniest scales.”

Mind you that in the Schrodinger's equation, that supposedly represents every state of the world, ψ includes an imaginary portion as well. 

In this context, I take real numbers to represent observable properties of elements and imaginary numbers to symbolize the out-of-sight portion of reality.

   
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