
Origins of complex integration

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Copyright 2008 by K. Ferlic, All Rights Reserved  
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Complex integration arises from the fact there are certain mathematical problems that could not be solved with what are called real numbers. They have no solution if only real numbers are used. However, if we use what are called complex numbers, numbers composes of a real and an imaginary part, and enter the imaginary plane of numbers we can make an the unsolvable problem solvable. In doing so, we get a real part and an imaginary part to the solution. What is interesting is that the real part of the solution to complex problem is the real solution to the unsolvable real problem. In other worlds, by entering the imaginary plane, using the complex numbers having a real and an imaginary part. We can solve problems that would otherwise be unsolvable. The concept of complex integration in our creativity provides for a variety of concepts that build on the same type of logic and thinking that gave rise to imaginary numbers and complex integration that can be used with our creative imagination to solve otherwise insolvable problems. Two discussion are provided here, one, "Origin of complex numbers" is more mathematical in content about a way to see mathematical origin of complex numbers. The other "Complex integration and creativity" is about how this concept parallels what can be done in creativity to solve seemingly insolvable problems. Origin of complex numbers The term complex integration has its origins in a phenomenon of mathematics involving what are called complex numbers or numbers composed of a real part and an imaginary part. As such they are not simple counting number and hence the term "complex." The real part of a complex number is the ordinary numbers one encounters such as 1, 2, 3, π, etc that we use every day in the way we use them. The imaginary part is something different. The term "imaginary" numbers have their origin in a mathematical operation which at one time seems to have no real counterpart as to what it represented. In particular, these were numbers that were somehow related to, or resulted from, the square root of negative numbers. The square root of the number negative one does not exist as an ordinary number. To take the square root of negative one is to really take the square root of a negative sign. That is "what entity" time itself equals a minus one or the minus sign. In looking for mathematics to represent operations in real life, the square root of the minus sign simply had no counter part. Hence the numbers were called, imaginary. The following discussion of the origin of complex number is not necessarily historically accurate but serves as a pedagogical tool move into the concept of complex numbers. From the author’s understanding, the evolution of mathematic was faced what seemed to be an insurmountable problem at one time. The development of the issue is presented as it could have evolved in the following scenarios. Certain mathematical operations would result with the need to take the square root of a negative number. The question was what does one do with, or what is the square root of negative number? In particular, the square root of 1. Numbers, such as 1, 2, 3, etc., although symbols themselves, were at one time were seen as very real since they could be mapped to a physical form or a physical process such as counting. They were number that could be associated with a position on an number line as in the Figure entitled, "Real Numbers" Similarly, one could associate the number "1" with a rock or small pebble that represented a particular sheep. A second rock or pebble could be associated with the number "2" and could represent a second sheep and so on, such that I could have a pile of pebbles, each representing my individual sheep. A one to one mapping between the rocks and my sheep. Obviously, it was much easier to carry pebbles around representing the sheep rather than carrying the sheep. Using the pebbles, one could trade and sell sheep and not have to carry the sheep around to the bank all the time. (Obviously, the beauty and size of the pebble was exactly representative of the beauty, size and health of the particular sheep it represented) Since both the sheep and the pebbles had an identity that was tagged to a number that could be counted, and both the pebble and sheep were real, obviously the number used for counting was somehow also real. Eventually pebbles were replaced with money and we come to value the money more than what the money represents, in this example, the sheep, but that is another story. In this story, the loss of one of the sheep, for example selling one of the sheep, required removing a rock from my pile. It could be seen as subtracting a sheep or adding a "negative" sheep to cancel one of the "positive sheep" (i.e. adding a 1). However we look at it, the rock, and the sheep it represented no longer existed since it no longer exists in my rock pile. So naturally negative numbers represented some thing that was no longer existing. Numbers that represented things that no longer existed, or were missing was not too bad to comprehend . But what was hard to comprehend was this square root of a negative number? What was the square root of something that did not exist? The issue itself was one only of perception. We only had to define the numbers in the reverse order for them to make sense. For example, to a bandit, all the animals not in his possession were a loss or negative hence the entire heard of sheep owned by the shepherd was nonexistent to the bandit. So when the bandit took a sheep, he now had one (+1) and the shepherd could be seen as less positive and more negative by one (1). To the bandit, only the sheep he did not possess would be seen as not existing. This bring up two interesting thoughts. First, although this issue here seem rather trite to most of us today, especially to those who have studied mathematics, it was a very real problem to very early mathematicians and it is representative of the evolution of concepts. All observation of reality results in similar evolutions. If we look carefully we will see there is an ongoing expansion of our consciousness and understanding such that our concepts become more and more "sophisticated" and evolved. However each current concept is an expansion of the earlier concept. It includes and encompasses in some way, that which has gone before it. Much like pealing the layers of an onion. The second thought is that much of our view of reality depends on what perspective we take. In our sheep example, to both the shepherd and the bandit, sheep not in their personal possession could be seen as not existing. It makes no sense to attempt to keep track of something they did not own. The sheep still exist, but not in the realm of where the shepherd or the bandit needed to personally keep track of them. There is a perception of something existing for that which we "own" or hold on to and possess. The same is true about our reality. That which we hold on to in some fashion, we possess and feel it determines our reality. That which is beyond our conscious control we frequently think we have no influence so it does exist as part of our reality. If however, we step out of our role, for example we cease to become the shepherd or the bandit, we find all is connected and the total number of sheep never changed. Returning to our imaginary number, the issue was solved when someone said the equivalent of let us considered there exists an number, a quality or relationship called "i" (i for imaginary) were "i" multiplied by "i" equals 1 or "i" is the square root of 1. What is interesting is the numbers were initially called imaginary for when viewed from the perspective of taking the square root of a concept (i.e., negative), that "i", it truly is imaginary. Creating the imaginary number "i" allowed the concept of a "complex" number to come into existence where a complex number has a real portion and this imaginary portion related to the "square root of the negative." Using imaginary numbers provided a new dimension to the real number as diagramed in the Figure "Imaginary numbers" As seen below, one interpretation which can be given to the imaginary number is that there is an infinite of graduations between any positive and negative number. Over time, the imaginary portion could be related to real phenomena in certain applications like phase angles as diagramed in the Figure "Phase angle" in electricity. In this electrical application, the imaginary portion represent the phase of an alternating electrical current. The phase can best be explained as relating to the timing between two alternating currents and describing how well two alternating current start and stop their cycle relative to each other. It describes how the currents are, or are not, at the same point, at the same time. Or in other words, it describes if the currents are together in time, or out of time, with each other. It is interesting that the imaginary number "i" came to be associated with the phases angle and time. If you think about it, it makes perfect sense. A negative operations is just the exact opposite of a positive operations. The ‘i" can then be seen as how much of a given action is opposite a given reference action. That, how much a given action is out of phase with a reference. For lack of better words, it represents or is a measure of the shading or the gray area between that which is defined as "black" and its opposite in the duality, "white." Alternatively said, black is 180 degrees opposite white. Anything between 0 and 180 degrees is a shading away from which toward black. The number "i" when used to measure the phase angle is related to how much out of phase we are with our reference direction. Since opposites exist 180 degrees apart, the "i" can also be said to be a measure of how much out of phase we are with the half of the duality we chose as our reference. What this also means is that for any action where we want things to be black and white the question becomes how much is the action we take in phase, or out of phase, with the half of the duality that we choose as reference. Here the element of choice enters the picture in two ways. One is we choose one half of the duality (black or white) as our reference. The second element of choice is we can move either toward black or white. The element of choice introduces a gray area. It is a gray area between to opposites of a duality. Both literally and figuratively we create a whole new dimension of existence in the gray area between the opposites of duality. In many ways, our reality, the reality we created every day, exists in the dimension between two halves of a duality. For each and every polarity of opposites exist a new dimension. What we fail to realize is that from an energy standpoint, this dimension between opposites is real and we can plug parts of our spiritual, emotional, spiritual and physical energy into it. Complex integration and creativity Now the really fascinating thing about complex numbers is that there are mathematical operations that cannot be done if one does not use the complex number. In the case of electrical currents, the issue involves the phase or timing of the current. In other operations the issue is not as clear what the complex number would represent. What is most interesting is that when we do use complex numbers for solving an insolvable problem for mathematical operation with only real numbers, for example, insolvable integrals in integral calculus, we can throw the imaginary part of the answer away since it is not necessary for our solution. Conceptually this amounts to going outside of the dimension in which the problem exists, going to another realm, and working a solution. Bring that solution back into the dimension in which the problem exists, discarding the part of the solution that is applicable to the other dimension, and live with the solution that couldn’t or wouldn’t exist if we stayed in the dimension we started in. Conceptually, this is quite a fascinating journey especially when one considers this is exactly the same description people use to describe the journey of a shaman during soul retrieval. They leave this dimension to go to another realm. They go into the realm that can best be described as residual emotional energy. They retrieve or obtain something (described as part of the soul or part of one’s spirit) from this other realm bringing it back into this dimension to give a different way of being in this dimension. The key here is that there exists other realms that cannot be fully comprehended or explained from a physical world view in which we can go, work solutions to our problems, and bring back these workable solutions into our physical existence. For all practical purposes, these other realities are out of time or out of phases with the normal world we see and experience. It is suggested that these realms are only different forms or aspects of energy and nothing really mystical and beyond our understanding. They can be understood by understanding the "as above, so below principle." It is to realize these other dimensions are the same as our but only different. (Sorry, we do live in a realty of paradoxes. Everything, everywhere is exactly the same, just different. Look at it as electromagnetic radiation. Each frequency wave operates the same as any other frequency. But, because their frequencies are different the two frequencies seem to operate differently. ) The discussion here on complex and imaginary numbers is an example of a mental reality or a mental dimension embedded within the dualistic concept of positive and negative numbers. There is an entire mathematical world that exists in which one can enter and return to work mathematical problems which provide workable solutions for the physical world. Similarly, there exists and entire world within the physical duality of separating positive and electric charges from each other. We would not have anything that runs on electricity if we did not enter the duality and utilize the principles of positive and electric charges. In analogous fashion, there is an entire emotional world between the polarity of friend and enemy, or love and hate, that can be entered and returned to work problems that have physical world solutions. Similarly, there is an entire spiritual world that exists between the dualistic polarity of God and Satan. What we fail to realize is that energy is energy. In the same way we store energy in the physical world, convert energy into mass and mass into energy, and use energy to change our physical world, we do, and can do the exact same thing with mental, emotional and spiritual energy. It is just that the form of the energy is different. As one moves into the spiritual realm away from physical form the energy takes on a form more and more like what we would call consciousness. However, it is still a form of energy and nothing else. All the discussion we see and hear about in spiritual teachings and in any form of healing about alignment and balance is really no more that bringing the utilization of energy in one’s every day life consistent with the energy arrangement that will give the individual the experiences they incarnated to have. Energy in motion is the experiencing of reality. Our life experiences routinely spread across our physical boundaries into the other realms of emotional, mental and spiritual energy and life is meant to be that way. However when we deny the existence of these other realms, we deny that portion of our selves that exists or goes into these other realms that allow us or help us to create the experience of the physical we came to have. This is why a shaman in a soul retrieval goes back and regains portion of the soul left behind after some experience. It is not unlike leaving that gas we need in our car’s fuel tank back at the gas station in the underground gas tank. We were suppose to stop and get gas but you got distracted and forgot to fill up the tank. It is the same with life experience. We stop to get something that we need to obtain in a particular experience. But, we get distracted and leave a part of what we need behind in that experience. We then must go back at a later time and get it. We need to retrieve it and bring that energy or consciousness forward into the now.
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