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thoughts on imaginary numbers

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

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

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 (

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