[MASTER INDEX] [DATE INDEX] [THREAD INDEX] [SUBJECT INDEX] [AUTHOR INDEX] |

[Date Prev] [Date Next] [Thread Prev] [Thread Next] |

Jan 30, 1994 02:40 PM

by eldon

Following are some thoughts on complex numbers. Consider it the first specualtion on the subject, and not a finished product ... Any comments helping fix up any math errors, should any exist, would be appreciated. I may have to rethink some of the symbolism after giving it some more thought ... ---- The topic of imaginary numbers deals with some fairly difficult symbolism, and requires careful thought to penetrate into its secrets. What does it mean to have a number that is multiplied by the square root of -1? Consider the special number *1*. Its square root is itself. In fact, no matter how many times we mulitply it by itself (what power that we raise it to) and no matter how many times we break it apart into another number to a certain power, we still end up with *1*. The number *1* represents complete balance. As a number or measure, it can be mulitplied by itself and remains the same. When we have 1 ft, and 1 1/sec, and combine them, we have 1 ft/sec. The *1* remains unchanged, although the attributes or nature of the being has changed, through the combination. There was something that was of the nature of "feet", and another thing of the nature of "per second", and in combination we have something different, we have "feet per second". We could say that *1* is a special number, a quantity that prevades being, that remains unchanged as we combine or break apart various qualities of existence. For positive numbers, we can take a square root, and find another positive or negative number that when multiplied together gives us our number. We can find that 2 * 2 = 4, that 2 is the square root of 4. And equally true, -2 * -2 = 4, that -2 is also the square root of 4. But consider a negative number. What is the square root of, say, -4? We know that 2 is the square root of 4, but how do we get the sign to come out negative, how do we find two numbers, that when multiplied, give us a negative number? There are so such numbers, and so a new type of number was invented, called an imaginary number. We would say that the square root of -4 is 2i, or the imaginary number 2. In considering mathematics, the fundamental philosophical truths come out of the simplest cases. As the mathematics gets more complex, we are expanding out into special cases of a simple truth. Until we get the simple truths clear, and develop certain core concepts, we are not ready for the further levels of meaning, we would end up bewildered! With imaginary numbers, the simplest origin is in the way that the signs of numbers change when we multiply or divide numbers. We need to ask how it works, what it means, and what we are doing by coming up with the imaginary numbers. We basically have, dealing with real numbers: (a) +N1 * +N2 = + N1N2 (b) -N1 * -N2 = + N1N2 (c) +N1 * -N2 = - N1N2 (d) -N1 * +N2 = - N1N2 That is, (a) two positive numbers, when multiplied, give us a positive number, (b) two negative numbers, when multiplied, also give us a positive number, and (c) and (d) a positive and negative number, when multiplied, give us a negative number. We need to examine each of these cases, to see what it means, and then to ask why we would need to break these rules, and to come up with a different kind of number. (a) The simplest case is when we combine two positive quantities. We have two existing beings, each with its own attributes and number, that combine to create a greater being, with their combined attributes and with a combined number. And that combined number is a measure of a higher space that is created by their combination. If we have Being A as a line of two inches, and Being B as a surface of four square inches, their combination gives us Being AB, with a volume of eight cubic inches. The number is now eight, and it refers to a greater space, to a volume, which arises from the combination of the respective qualities of inches and of square inches. (b) It gets more difficult when we combine two negative quantities. We have the absence or negation of two different entities, and combine those beings, or rather their negations. And the result of the combination is the same as if we had combined the two beings themselves. We end up with the same results in the multiplication when we combine two presences of beings as when we combine two absences of beings. We get the same higher space, with the same combined qualities, and the space exists, even though the two beings that make it up do not *actively exist* at the moment. In the first case, two existing beings combined to form a higher space that also exists. In this second case, we are still talking about two manifest beings. But they do not actively exist, they passively exist, they exist as *manifest negations*. They combine and somehow, because they are in harmony or synchronization, the higher space still comes into positive being. (c) and (d) The hardest situation to describe is when we combine an exiting being with the negation of another being. The qualities of the combined being come out the same, and the new space that is created has the same volume, but the combination of manifest negation with an manifest active existence somehow creates a higher space that is a negation. This is something to puzzle over. Basically speaking, when we multiply or combine two beings, the attributes always combine, and the numbers always combine, but the plus or minus nature of the combination may vary. It is possible to consider pure numbers, and multiple them, say 3 x 5 = 15, and to ignore the attributes. When we do this, we are still combining attributes: 3 (groups) * 5 (members) = 15 (group members) And once the combined being is created, the 15 group members, we can shift attributes among its component beings: 15 (group members) = 1 (groups) * 15 (members) This shifting of attributes comes from a division along different lines than the original mulitplication was done. Coming to imaginary numbers, we have a situation where we want to reverse the effects of combining positive and negative numbers in multiplication and division. We want two positive or two negative numbers, when multiplied, to give us a negative number. And we want a positive and negative number, when multiplied, to give us a positive number. Since we denote imaginary numbers with an *i,* we want: (a) +N1 * +N2 = - N1N2 (b) -N1 * -N2 = - N1N2 (c) +N1 * -N2 = + N1N2 (d) -N1 * +N2 = + N1N2 Real numbers deal with existing quantities in the world (positive) or their negations (negative). When we combine or multiply real or existing beings, we get a higher space or being that is active or an negation depending upon whether the component beings are in accord (both active or negations themselves). An imaginary number does not measure anything that exists, nor any comparison of existing things. We know that various problems in physics and mathematics require their existence. And we know that every measurement can have both a real and an imaginary value associated with it. We basically have two measures on an object. The real measure regards the manifest portion, and the imaginary measure regards that which is implied, but not manifest. The combination of the two measures, the real and imaginary numbers together, make a complex number. Complex Number = { Real Number, Imaginary Number } Every measurement could be considered a complex number. Every object has both a real part, a part that is either in active existence (positive sign) or passive existence (negative sign), and a part that can never be, a part that is in non-existence, or non-being. This second part could be called imaginary because it can only exist in our imagination, and not in any tangible object or the negation of any object. It can only be inferred by a necessity implied by what actually is. And its behavior is the opposite of the real. When two imaginary quantities combine we find that the resulting higher space actively exists when there is a lack of harmony or synchronization between the quantities. And the higher space passively exists or is in negation when there is harmony among its parts. As we undergo various transformations, as the numbers are combined in various mathematical operations, we find that a portion of any attribute of us comes into active existence (takes on a positive sign as a real number) or manifest negation (takes on a negative sign as a real number). And at the same time, there are changes in the the potential, the imaginary component, where it comes into its activity (positive imaginary number) or into its own negation (negative imaginary number). That is, a mathematical operation can increase or decrease both the real and imaginary part of a measurement. And when we consider a quantity, and choose to separate it into two, to divide out a part of it, to make it into a pair of opposites, we find various events happening. Consider taking out an actively-manifest part from an actively-manifest number: 8 / 2 = 4. We get the same, a remaining part that is also actively-manifest. But when we take out an negation from an actively-manifest part, we get a remaining negation: 10 / -2 = -5. We have turned something that was actively existing, and made it into two things that only exist as negations. When we consider imaginary numbers, when we take out a portion, we get something entirely different. For example, 14i / 7i = 2. When we break out an imaginary part from an imaginary number, we get a real, existing quantity. Perhaps the unmanifest is separating, and a portion remains unmanifest, but the rest is forced into existence? Eldon Tucker (eldon@netcom.com)

Theosophy World:
Dedicated to the Theosophical Philosophy and its Practical Application