General Complex Numbers....vs.........Specific Complex Numbers by Robt W. Brown
The following is a 'general form' that all numbers from the Complex Number Set can take:
a + bi
If: a= 2 b= 1.4142135
we can say:
a + bi = 2 + 1.4142135i..................eq. 101
It might appear that we have now defined a 'specific complex number' in eq. 101. But I contend that we still have a 'complex number' of generality. We know this about the number: 1. It will graph as a single point on the 'complex plane'. 2. It is a complex number that can be part of a multitude of geometrical sequences.
Let's now suppose we setup a 3-axes system, for graphing complex numbers: a= axis of reals.........horozontal axis #1 b= axis of imaginaries.............horozontal axis #2 y= axis of rank...............vertical axis
With this system of graphing rectangular coordinates the "complex plane" can be considered as the (a-b horozontal plane).
Then to explain this idea further, let's consider this rule for ranking terms:
Rule #1: The rank (y) of any term in a geometrical sequence is always equal the logarithm of the related log base (B).
According to this rule we now can say this about our number:
Let's also set the Domain of (y) as the 'real number set'.
At this point it becomes obvious that we initially had 2 unknowns in the LHS of the equation 201. This means (B & y) could be ANY combination of real numbers. So let's simply set the rank of the Number/Term (y) at zero:
Then with eq. 301 let's try and determine the 'related log base' of our complex number:
B= (2 + 1.4142135i)^1/0............
Unfortunately, we find the power of our complex number is 'undefined' because:
So, we cannot arrive at a 'related log base' in this special instance! Therefore, we must consider this new rule:
Rule #2: Any complex number that graphs as a point upon the 'complex plane' has a rank of zero (y= 0). Therefore, any such number cannot not be a "specific number". Namely as number that is derived from a 'specific log base'. It is therfore a 'general complex number'.
Now let's give an example of a 'specific complex number':
(3 + 1.7320508i)_1
we see that in the above number, we have attached a rank to the number and the rank equals one: y= 1 a= 3 b= 1.7320508
Therefore, if we graph the above number on the 3-axes system proposed, the resulting point will NOT be on the complex plane. (Because any point on the complex plane maintains a constant coordinate...y= 0.)
Surprisingly we find that the Log is the original number it's self: Log Base= 3 + 1.7320508i
B^1= 3 + 1.7320508i or: B^y= a + bi
Since the related Log base equals = the number in question, we can state another rule:
Rule #3: When the rank of a complex number equals one (1)... this indicates that the related log base of the number is the same number to the power of (1). Which means, the Log Base IS precisely the SAME number.
So, in this 3 coordinate system of Numbers & Ranks we can can differentiate between 'general complex numbers' and 'specific complex numbers'.
Furthermore, we can argue that all complex numbers that can be graphed on the Argand Scale have an 'implied rank' of zero (0). Meaning y= 0
If the above problem interests you, please indicate and I will make follow comments in terms of how this can also relate to 'quadratics'.
Compliments of Col. Rbtx, the Barnyard Physicist of Texas **************