A Special Case (2)

In the last entry, I started with the sample identity i.e. x = 1, to generate the corresponding polynomial equation (degree 1) i.e. x – 1 = 0.

Now in conventional mathematical terns both of these, i.e. x = 1 and

x – 1 = 0 would be treated as identical.

However something strange occurs when we square both expressions.

Therefore in the former case, we obtain x² = 1 so that x² – 1 = 0, whereas in the latter we obtain (x – 1)² = 0, so that x² – 2x + 1 = 0.

So two different equations are now derived from squaring the two - apparently - similar expressions.

As we know the two roots of (x – 1)²  = x² – 2x + 1 = 0, are + 1 and + 1 respectively and the last blog entry was designed to explain how, from using the unique number sequence associated with this equation, we can in fact properly distinguish these two linear - and indeed in more general terms - n repeated examples of 1.

However, the corresponding roots of x² – 1 = 0 are + 1 and – 1 (which lie as two equidistant points on the unit circle (in the complex plane).

And in more general terms the n roots of xⁿ – 1 = 0, will lie as n equidistant points on the same unit circle.

So we need to explain how we have thereby moved from a linear based notion of dimension (as points on the real number line), to a corresponding circular based notion of dimension (as points on the unit circle).   

Now just as there is a unique number sequence associated with (x – 1)ⁿ = 0, equally there is a unique number sequence associated with xⁿ – 1 = 0 , for every integer value of n.

So in the case of x² – 1, where the coefficient of the x term is 0, again we start with 0,

1. Then we obtain (0 * 1) + (1 * 0) to get 0 as the next term.

So now we have 0, 1, 0.

Then for the next term we obtain (0 * 0) + (1 * 1) = 1.

So the sequence is now 0, 1, 0, 1.

And it continues in this fashion continually alternating between 0 and 1.

In the case of x³ – 1, the coefficients of both the x² and x terms is 0 and we start in this case with 0, 0, 1.

Then to generate the next term we obtain (0 * 1) + (0 * 0) + (1 * 0) = 0.

So we now have 0, 0, 1, 0.

And for the next term we obtain (0 * 0) + (0 * 1) + (1 * 0) = 0. The sequence is now

0, 0, 1, 0, 0.

Then the next term = (0 * 0) + (0 * 0) + (1 * 1) = 1.

Thus the sequence is now 0, 0, 1, 0, 0, 1.

It then continues on indefinitely in this manner with a 1 always followed by two 0’s.

And for the general case, where xⁿ – 1 = 0, in the unique number sequence associated, each 1 is followed by (n – 1) 0’s.

So, as in the previous case for (x – 1)ⁿ = 0, I gave the first 9 digits of the unique digit sequence, for n = 1 to 9, likewise I will now do the same with respect to xⁿ – 1 = 0 (again from n = 1 to 9).

1	1	1	1	1	1	1	1	1
1	0	1	0	1	0	0	0	0
1	0	0	1	0	0	1	0	0
1	0	0	0	1	0	0	0	1
1	0	0	0	0	1	0	0	0
1	0	0	0	0	0	1	0	0
1	0	0	0	0	0	0	1	0
1	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0

  

In the previous case for (x – 1)ⁿ = 0, we were able to successfully use the unique number sequence associated with the polynomial equation in question (for any required value of n) to approximate the required roots (as solutions to the equation).

So again in the case of (x – 1)² = 0, where the unique digit sequence is 1, 2, 3, 4, 5, …the ratio of n/(n – 1) approximates to 1 thereby giving the first root of the equation.

However in the case where n = 2, x² – 1 = 0 is associated with the unique digit sequence 1, 0, 1, 0, 1, …. . Thus when we attempt to calculate a root we face an - apparently - meaningless situation. For the ratio of the n_th/(n – 1)_th term gives either 1/0 = ∞ or 0/1 = 0.

Thus we generate two conflicting answers ∞ and 0 which are at opposite extremes of each other. Likewise in neither case do we obtain the true roots + 1 and – 1, which in conventional mathematical terms are understood to satisfy the equation.

Now the reason for this problem is that we have in fact now moved from a linear interpretation, which applies in the case of (x – 1)² = 0, to a circular interpretation of number, which applies in the case of  x² – 1 = 0.

The implications of this could not be more important for Mathematics as it implies that two distinctive logical systems are in fact required.

The linear (Type 1) system is based on the standard either/or logic whereby opposite signs are clearly separated in an absolute manner.

However the circular (Type 2) system is based on a paradoxical both/and logic whereby opposite signs are viewed as complementary with each other in a merely relative manner.

From the standard linear perspective, a cardinal number is defined as composed of independent homogeneous units (that lack qualitative distinction).

So 2 = 1 + 1.

However from the paradoxical circular perspective, 2 is now seen as composed of two units that are fully interdependent with each other in a qualitative manner.

This in fact corresponds with the ordinal definition of number of 1^st and 2^nd units.

Thus 2 = 1^st + 2^nd

However in conventional mathematical interpretation standard, these units are given a fixed identity, whereby they can be reduced in cardinal terms.

So again 2 = 1 + 1 (in cardinal terms). 2 = 1^st + 2^nd (in ordinal terms).

So by identifying 1^st as the last unit of 1, and 2^nd then (with the 1st unit fixed) as the last unit of 2, 1^st and 2^nd are thereby reduced in an analytic independent manner to cardinal interpretation.

However when we view the ordinal units as truly interdependent in a relative manner, this requires a new holistic interpretation of 2, whereby 1^st and 2^nd units are understood as fully interactive with each other.

And this requires moving from a linear either/or logic (where opposite signs are viewed as absolutely distinct) to a circular both/and logic (where opposite signs are viewed as fully complementary in a relative manner).

As I have illustrated repeatedly in previous blog entries, we can illustrate the two types of logic very well in the case of the definition of left and right turns at a crossroads.

From the linear logical perspective, a left or right turn would be absolutely designated as + 1 (a left turn) and – 1 (a right turn i.e. not a left  turn).

So when one approaches the crossroads from just one direction left and right turns can be unambiguously identified in a linear logical manner.

So for example when approaching the crossroads while heading N, left and right turns can be unambiguously identified.

Equally when approaching the crossroads while heading S, left and right turns can be unambiguously identified.

So the understanding in both cases concurs with linear logic (where opposite turns are absolutely separated). Quite literally this represents 1-dimensional logic, where just one point of reference is required (e.g. heading N) to unambiguously make an identification.

However if one now envisages the approach to the crossroads from both N and S directions simultaneously - in what now represents 2-dimensional logic - what is a left turn from one direction is right from the other and vice versa. So we have now moved to a new holistic circular interpretation entailing the complementarity of opposite signs. So what is + 1 from one direction (N), is equally – 1 from the opposite direction (S) and what is + 1 from this other direction (S), is equally – 1 from the opposite direction.

Thus positive and negative signs have an absolute fixed interpretation in terms of standard linear (1-dimensional) interpretation.

However positive and negative have a relative interchangeable interpretation in terms of  holistic interpretation (with the example here relating to the 2-dimensional case).

In more general terms, holistic interpretation in this context applies where n (as the number of dimensions involved) > 1.

However what is truly remarkable is that there is no formal mathematical recognition whatsoever for this latter type of circular understanding (where the interdependence of the various sub-units of each cardinal number is recognised).

Rather, in every context, the notion of holistic interdependence is formally reduced in an analytic independent manner.

In other words the Type 2 qualitative notion of number is invariably reduced in a Type 1 quantitative  manner.

However, the unique number sequences associated with xⁿ – 1 = 0, (where n > 1) require true holistic understanding to properly interpret the ratios (as corresponding roots of the equations).

Thus in the case of x² – 1 = 0, the respective ratios of 1/0 and 0/1 i.e. ∞ and 0 lie at opposite extremes of each other as complementary opposites reflecting the holistic interdependence of the roots involved.

Therefore the fact that both + 1 and – 1 are the respective roots in this case indicates that their relationship should be understood in a relative holistic manner (as interchangeable) in the context of the unique number sequence involved.

However once again, though these circular roots are of course recognised in conventional terms as valid solutions to the equation, they are interpreted merely in an analytic manner (where positive and negative signs remain fixed).