Expressed in an equivalent manner, through the importance of imaginary numbers is now well established in quantitative terms, their corresponding qualitative significance is not yet formally recognised.
However we will now indirectly probe the true nature of imaginary numbers through the use of the unique digit sequence associated with the polynomial equation x2 = – 1, i.e
x2 + 1 = 0.
Now the unique digit sequence associated with this equation is 0, 1, 0, – 1, 0, 1, 0, – 1, …
Now when we take the ratio of nth/(n – 2)th terms, from one perspective we obtain – 1/1 or alternatively 1/– 1 = – 1.
This therefore suggests indeed that x2 = – 1 (which was our starting point).
However from an alternative perspective, the ratio of the nth/(n – 2)th terms = 0/0.
Again, as we saw in the last entry, this result is somewhat meaningless from a Type 1 linear perspective (where number is viewed in an independent manner).
So the result of 0/0 suggests a relative relationship involving interdependent - rather than independent - units.
In this respect, x2 + 1 = 0 with the two imaginary roots of x, is similar to the the two real roots of x2 – 1 = 0 (which we dealt with in the last entry) where interpretation properly entails both linear notions of independence and circular notions of interdependence respectively.
The difference however with respect to the imaginary roots is that the sign of 1 keeps switching alternatively as between positive and negative values, and herein lies the true clue as to the nature of imaginary numbers.
Once again in conventional mathematical terms, based on linear rational notions of quantitative independence, positive (+) and negative (–) signs are considered as absolutely independent. So as we have seen from this perspective, a left turn at a crossroads (+ 1) is thereby clearly separated from a right turn (+ 1).
However there is equally - a largely unrecognised - qualitative interdependent interpretation of number, of a purely relative nature, where each ordinal position is fully interchangeable with each other ordinal position.
So in this case of two items, (+) and negative (–) signs become fully interchangeable.
Thus as we have seen in our crossroads example, when one envisages the approach to the crossroads from two opposite directions simultaneously, a left turn (+ 1) cannot be distinguished from a right turn (– 1) and vice versa.
Therefore left and right turns are now purely relative.
So the former understanding of absolute independence corresponds to linear rational type appreciation; however the latter understanding of relative interdependence corresponds to circular intuitive type appreciation.
And we can refer to the former as analytic and the latter as holistic type understanding respectively.
Now, both of these two types of understanding (analytic and holistic) are necessarily involved in the interpretation of left and right turns at a crossroads!
However in conventional terms - consistent with the absolute quantitative interpretation of number as points on a real number line - the analytic is solely recognised in formal terms.
However, the latter holistic aspect is then recognised in conventional mathematical terms, through “conversion” into analytic meaning of an “imaginary” rather than “real” nature.
Therefore though the imaginary points directly to the holistic meaning of interdependence of a qualitative nature, through being then indirectly converted in an analytic independent manner, this enables the new imaginary aspect of number to be treated in a quantitative manner.
So in analytic terms i (as the imaginary unit) = √– 1 i.e. (– 1)1/2.
This therefore provides the indirect quantitative means of expressing a 2-dimensional (circular) notion as a point on the unit circle (in the complex plane) in a 1-dimensional i.e. linear manner.
Now if we look at 2 holistically i.e. through the Type 2 aspect of the number system, it is represented as 12. So 2 here represents the 1st and 2nd dimensions respectively that are + 1 and – 1 respectively.
Now + 1 as the 1st dimension is independent (like the unambiguous identification of a left or right turn at a crossroads). However the 2nd dimension then properly relates to the interdependence of two units (as in the case of the crossroads the interdependence of both left and right turns).
Properly understood when we use – 1, in this holistic interdependent context, it becomes dynamically related to + 1. And the interaction of both + 1 and – 1, just like matter and anti-matter particles in physics, leads to a fusion in energy.
And we refer to this psycho-spiritual fusion in understanding (where opposite polarities are recognised as complementary) as intuition.
From an equivalent perspective, in holistic terms + 1 (where literally a phenomenon is posited in understanding) represents the conscious direction of understanding.
– 1, then as the corresponding (dynamic) negation of the posited unit, represents the unconscious direction of understanding.
So both conscious and unconscious aspects interact in mathematical understanding through the corresponding interaction of rational and intuitive understanding.
However, though the importance of intuition - especially for creative work - is recognised in conventional terms, invariably it is reduced in a merely conscious rational manner. And strictly this greatly distorts the true nature of mathematical understanding!
So the key function of the imaginary notion in Mathematics is to enable the unconscious aspect of understanding with respect to number (which is of a holistic intuitive nature) be indirectly assimilated in a conscious rational manner.
Put yet again in an equivalent manner, the imaginary notion serves to convert the qualitative aspect of number indirectly in a quantitative manner.
So for example 1 (one) represents the real conscious understanding of the quantitative notion of a unit in a rational manner.
Then 1 (oneness) represents the corresponding unconscious appreciation of the qualitative notion of a unit in an intuitive manner.
And in the dynamics of understanding quantitative and qualitative notions are positive (+) and negative (–) with respect to each other.
Then √– 1 (= + i or – i) indirectly represents in a quantitative rational manner the corresponding qualitative aspect of the unit (which is inherently of a holistic intuitive nature). And this indirect understanding represents the imaginary - as opposed to the real - aspect of number.
However just as all analytic quantitative notions have holistic qualitative counterparts in Mathematics, likewise this also implies to the imaginary notion.
Thus when appropriately understood i.e. in a dynamic interactive manner, the Riemann Hypothesis can be given a simple - yet compelling - interpretation.
As is well known this postulates that all the non-trivial zeros of the Riemann zeta function lie on an imaginary line (through ½).
Now in conscious rational terms it is assumed that all the real numbers lie on a straight line.
However this begs the question as to whether the assumption is also valid from an unconscious holistic perspective.
Again in conventional mathematical terms - because of conscious reductionism - it is just blindly assumed that the qualitative aspect corresponds with the quantitative.
So no distinction is made as between numbers as independent quantitative entities and the interdependent relationship as between these numbers (which is strictly of a qualitative nature).
However by resorting to the imaginary notion, it is thereby possible to isolate both quantitative and qualitative aspects (by indirectly representing the qualitative aspect in an imaginary quantitative manner).
Thus the requirement that all the (non-trivial) zeros lie on an imaginary line, in effect amounts to the requirement that the unconscious qualitative aspect of mathematical understanding - which is inherent in all interdependent relationships as between numbers - corresponds in a consistent manner with the conscious quantitative aspect (where numbers lie on a real straight line).
Therefore if all the zeros do not lie on an imaginary line, we can no longer have faith in the very consistency of the number system, with the entire mathematical edifice thereby built on faulty foundations.
However, clearly this most fundamental question of all cannot be proven within the accepted mathematical axioms (as they already blindly assume the truth of the Riemann hypothesis).