Complex Numbers

In order describe the presented  model in the Universal Theory, review of the concept of complex numbers is needed. To review the model, the reader is referred to www.universaltheory.org. We first review the basic principles of complex numbers. Our physical interpretation of different elements in complex number mathematics will be followed in this chapter and following chapters as per indicated.

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand).

The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the polar coordinates, are-- r = |z|, called the absolute value or modulus, and f = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have
  
Z =x +iy =r (cosf +i   sinf)
  
x= r cosf  is called real part
  
iy = i rsin f is called imaginary part. So complex numbers are a combination of purely real and purely imaginary numbers.

In sixteenth century Italian mathematicians frequently encountered with square root of negative numbers (√-1) in their calculations. Known mathematics could not offer a solution for this problem. Eventually, an imagined number was chosen which its square would end up to -1. This was called imaginary number and is shown by symbol i.

Later on it was noticed that a combination of real number and imaginary number is essential to explain the fundamentals of mathematics. this combination is called complex number.

Complex number = ( x + iy )

In 1806, Jean-Robert Argand, trying to give geometrical visualization to complex numbers suggested the diagram below;

Argand Diagram

In 1799 Gauss proved the fundamental Theorem of Algebra using the complex numbers. Nowadays, the use of complex numbers pervades all of mathematics and its applications in science. Even real numbers are rightfully reintroduced as;

x + 0i

One will assume, if imaginary numbers are so fundamental in mathematics, it should posses a physical reality. As 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." 56 p 67

We measure the elements of space time by real numbers. The concept of complex number implies that any of these elements should have an imaginary dimension in their nature. In another words, every quantitative measurable in universe also contains a qualitative non measurable aspect on it.

In 1707 Abraham De Moivre found the similarity between complex numbers and trigonometry. These numbers are following the same rules applied to trigonometric calculations. For example, when we square a complex number we double its phase (angle).

Imaginary numbers are sometimes called magic. One of the strange characteristics of these numbers is the fact that in De Moivre diagram any real number coupled with (multiplied by) them will be reduced to zero.

As shown in diagram above when we multiply any real quantity by i, its real value ( Its real number coordinate value) is reduced to zero. Its algebraic will be written as;

(X+0i) i = Xi + 0(ii) = Xi

As it is shown the real value disappears and pure imaginary value illustrates. In trigonometry we can show the fact as;

X = r Cos a, since we took a = 90 and, Cos a = 0 then X=0. So in each period the real value hits the zero line (Y) twice.

Although real numbers field may create the illusion of continuity, complex number system show us that continuity of real number breaks down intermittently. We can also show this fact by evaluation of function of (x) in any equation. We take y = xIxI as an example.

plot of fx = xIxI ref. # 56 p109

For any other function of finite real numbers we can come to a derivative which shows lack of smoothness and continuity.

We can take y = 1/x as another example. It is of interest that always discontinuity happens around point zero.

Plot of 1/x Ref 56 p 111

So real number quantities inherently are not smooth or continuous(holomorphic). If continuity is desired we have to incorporate the imaginary number and concept of complex numbers into equation and rewrite the y = 1/x equation as y = 1/z. where z = x + ib.

Also any point in the domain can be considered point zero (cross section of coordinates). That is because we can choose any other point in the domain and shift the point zero to that point and use Cauchy formula in the origin shifted form.

n!/2pi ∫ f(z)/(z-p)ⁿ⁺¹dz = f(p),

And the nth-derivative expression

n!/2pi ∫ f(z)/(z-p)ⁿ⁺¹dz = f ⁽ⁿ⁾(p),

"Thus complex smoothness implies analyticity (holomorphicity) at every point of the domain." Ref 56 p128. This is important for us when we define singularity and its relation with space-time in the next chapter.

On the other hand, the complex number equation Z = R [cos a + i sin a] indicates that these numbers also have a periodic nature. So they loose their real number value and hit zero twice in each period. We take the periodic nature and intermittent appearance and disappearance of real value of measurables as sign that every measurable in our universe has a discrete nature. This will include matter, time or space.

For example in above diagram, if x-coordinate denotes the mass of particles, somewhere in its endeavor the mass gradually loose its value and disappears. On the other hand if x indicates dimension and distance, because of the function of the complex system, it has to disappear and reappear during each period. This is the basis for our assumption that space and time are discrete and not continuous.

In this model we take the imaginary number (i) as a factor which represents the singularity effect on different phenomenon. Point zero on the other hand represents singularity.

Another interesting characteristic of imaginary numbers is the fact that although they are influencing the real numbers in equations, they normally do not mix up with them. In a complex number we normally have to deal with each portion separately. For example for addition we write the equation as;

( 6  + 3 i ) + ( 5  + 2 i ) = 11 + 5 i

It means that real numbers and imaginary numbers are two separate entities.

Summary

The concept of complex number opens are eyes to the combination of qualitative and quantitative characteristics of the elements present in the universe. Although the mathematics of these numbers are highly developed the physical interpretation of complex system are not fully understood. In this model we hypothesize some physical interpretations for the elements of the system and will examine if these interpretations will offer solutions to existing paradoxes.

56- The Road to Reality, Roger Penrose, Jonathan Cape, London, 2004