Again, as we have seen associated with x2 – 1 = 0 is the infinite number sequence 1, 0, 1, 0, …
So we have a repeating cycle of 2 digits (1, 0) in this case,
Now remember that the ratio of the nth/(n – 1)th term is used to approximate the value of x. In like manner, the ratio of the nth/(n – 2)th can be used to approximate the value of x2.In this context, two answers thereby are suggested by the number sequence.
So from one perspective, if the nth term is 1, then the (n – 2)th term is also 1.
Therefore the ratio of nth/(n – 2)th terms = 1.
Thus x2 = 1, as we would expect from the initial equation.
However, when we recognise both the Type 1 and Type 2 aspects of the number system, this can be expressed more fully as x2 = 1 (where 1 is interpreted in Type 1 linear terms as a point on the real number line where x = 1).
However equally from an alternative perspective if the nth term is 0, then the (n – 2)th term is also 0.
Therefore the ratio of nth/(n – 2)th terms = 0/0 (which has no meaning from the Type 1 perspective).
What this in fact entails is that an intuitive - rather than rational - type recognition is now required in interpreting x. In other words, it is now understood as a number on the unit circle (in the complex plane) where it is necessarily interdependent with its related unit.
So quite literally in such number recognition, – 1 is understood as interdependent – and ultimately identical, with + 1 (as complementary opposites of each other).
Now this intuitive recognition, which literally is of a circular logical nature entailing paradox, is inherently dynamic in nature entailing - like the fusion matter and anti-matter particles in physics - the psychological interaction of opposite polarities.
And this annihilation of mental form leads to the psycho-spiritual energy (which in fact represents intuitive recognition) = 0.
However it is important to bear in mind that 0 in this context relates to a holistic rather than analytic type meaning.
So the ratio of 0/0 can be looked on as the complementary fusion of – 1 and + 1 (= 0) in relation to the reverse fusion of + 1 and – 1 (= 0), where both are viewed, like the two turns at a crossroads, in purely relative terms.
However when we attempt to approximate the value of x through the nth/(n – 1)th term, we get the apparently meaningless choice as between 1/0 and 0/1 respectively.
This is because we are attempting to express, in mere Type 1 terms, a relationship that properly entails both the Type 1 (linear) and Type 2 (circular) aspects of number interpretation.
Again in conventional Type 1 terms, when x2 = 1, we would give x two separate answers in linear terms, i.e. + 1 and – 1 which appear valid in an isolated independent context.
However properly from a holistic Type 2 perspective + 1 and – 1 are seen as interdependent in a purely relative manner.
Therefore we cannot properly express such paradoxical interdependence, of a holistic nature, in a reduced linear manner (that is unambiguous in nature).
Once more it would be helpful to envisage the scenario of interpreting left and right turns at a crossroads.
Now the holistic (Type 2 circular) interpretation of this scenario, entails the approach to the crossroads simultaneously from two opposite directions (both N and S).
Through such intuitive recognition, we realise that left and right turns are merely relative, so that what is left (+ 1) from one direction is right (– 1) from the other and what is left from the opposite direction (– 1) is right from the other (+ 1).
However the analytic (Type 1 linear) interpretation entails treating the approach to the crossroads from just one independent direction (either N or S).
Then what is left (+ 1) is unambiguously separated in absolute fashion from what is right (– 1) as the other direction.
So this concurs with linear rational interpretation.
And in the understanding of the crossroads, we can now perhaps see more clearly how both types of interpretation (analytic and holistic) are inevitably involved.
However in conventional mathematics, though Type 1 (rational) and Type 2 (intuitive) recognition are necessarily involved in the interpretation of symbols, formal interpretation is always inevitably reduced in a merely Type 1 (rational) manner.
And though readily admitting the massive developments in specialised mathematical reasoning of an analytic variety, when one properly appreciates the true interactive nature of mathematical relationships (entailing both quantitative analytic and qualitative holistic aspects in dynamic relationship with each other), one must accept that our present understanding - most fundamentally in terms of number - is ultimately unfit for purpose!
Though it is easiest to illustrate these general points with respect to the relatively simple case where x2 – 1 = 0, these can be readily extended in understanding to xn – 1 = 0, where n is a positive integer > 1.
For example in the case where x3 – 1 = 0, the unique number sequence associated with this equation is 1,0,0,1,0,0, …
So the 3 numbers 1, 0, 0 repeat here in a regular cyclical manner. And in more general terms, where
xn – 1 = 0, 1 followed by (n – 1) 0's will occur in a regular cyclical sequence.
The introduction of the extra 0 indicates an extra degree (or dimension) of interdependence, which in turn requires a more highly refined intuitive ability for proper appreciation.
The 2-dimensional case i.e. x2 – 1 = 0, implies a situation involving two unitary objects where both are considered as interdependent - and thereby interchangeable - with each other. However in order to recognise the interdependence of 2 objects, each must initially be understood in an independent (1-dimensional manner). So there is only one degree of freedom here, relating to the interdependent aspect.
So again with respect to our left and right turns at the crossroads, we must initially be able to understand each in an independent analytic manner (i.e. when approached from just one direction).
This means that we holistically consider the interdependence of these turns, only one other interchangeable option is available. Thus what was initially considered for example as a left turn, can now equally be given a right location.
So the 1 in the unique number sequence for x2 – 1 = 0, can be directly identified with the initial independent identification of the unit (in this case a left turn).
The 0 then by contrast is identified with the other interchangeable option (i.e. right turn) reflecting in this context the pure relativity of both units.
The 3-dimensional case i.e. x3 – 1 = 0, implies an extension of the previous example, where now 3 unitary objects are considered as interdependent and interchangeable with each other.
However once again, this involves initial analytic identification of the 3 units in an independent (1-dimensional) manner. Thus the "1" in each number sequence of 3, relates directly to the independent unit. This then leaves - with respect to recognition of their holistic interdependence - two other options for interchange. So these two “higher” dimensions, represented by the two successive 0’s in the number sequence, relates directly to the increased holistic interdependence of these objects.
So the very notion of mathematical dimensions (as numbers representing powers or exponents) can carry both (linear) analytic and (circular) holistic interpretations.
So the notion of a cube as a 3-dimensional object through 3 independent sides represents the (linear) analytic interpretation.
However the corresponding notion of the interdependence of 3 related units represents the (circular) holistic interpretation.
And just as mathematicians accept that the ability to interpret “higher” dimensional objects e.g. 4-dimensional, in (linear) analytic terms requires an increased specialisation in abstract rational ability, likewise the ability to interpret “higher” dimensional interdependent objects requires a corresponding increase in (circular) holistic intuitive ability.