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 now square both
expressions.
Therefore in the former case, we now obtain x^{2 }= 1
so that x^{2 }– 1 = 0, whereas in the latter we obtain (x – 1)^{2 }= 0, so
that x^{2 }– 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 }= x^{2 }– 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 x^{2 }– 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 x^{n }– 1 =
0, will lie as n equidistant points on the same unit circle.
So we have 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 x^{n }– 1 = 0 , for every integer value of n.
So in the case of x^{2 }– 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 x^{3 }– 1, the coefficients of both
the x^{2} 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 x^{n }– 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 x^{n }– 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

Now 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, x^{2 }– 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 x^{2 }– 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.
Now, 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}
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 1dimensional 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
2dimensional 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 (1dimensional) interpretation.
However positive and negative have a relative
interchangeable interpretation in terms of
holistic interpretation (with the example here relating to the 2dimensional 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 subunits 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 x^{n }– 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 x^{2 }– 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).
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