A Special Case (5)

Again I wish to return to the second grid of the unique number sequences (first 9) for the simple polynomial equation xⁿ – 1 = 0 (for 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

This provides greater clarity on the true nature of - what I refer to as - the Zeta 2 function.

Now clearly the linear base of this equation occurs for x – 1 = 0.

Thus when we divide xⁿ – 1 = 0 by x – 1 = 0 we obtain

x^{n – 1} + x^{n – 2}  + … + x¹  + 1 = 0.

Or to put this in the form that is generally presented by reversing the direction of terms, we have

1 + x¹ + x² + x³ + … + x^{n – 1} = 0.

Then when the additional restriction is placed, that n is prime, we then have the Zeta 2 equation.

And the zeros to this equation then provide the holistic interpretation of the notions of

1^st, 2^nd, 3^rd, ….(n – 1)^th members of a prime number group of n.

In other words they indirectly provide a numerical expression of the holistic interdependence of the various members of the group (where ordinal positions are interchangeable). Of course even here, as interdependence must necessarily start from independence, one member of the group i.e. the n^th member is necessarily excluded. So for example, we can only recognise the interdependence of two turns at a crossroads, if initially we can view each one separately in an independent manner.

And because, such holistic interdependence properly belongs to the Type 2 aspect, numerical estimates in a Type 1 quantitative independent manner are thereby non-intuitive from this perspective. So the use of successive ratios to approximate solutions of x to the respective equations appears meaningless, with interdependence in numerical terms being represented as 0.

Now again with reference to the equations for xⁿ – 1 = 0 above, when we now include entries solely for prime values of n from 1 to 9, i.e. 2, 3, 5 and 7 we obtain the following:

x = 2	1	0	1	0	1	0	1	0	1
x = 3	1	0	0	1	0	0	1	0	0
x = 5	1	0	0	0	0	1	0	0	0
x = 7	1	0	0	0	0	0	0	1	0

If we then go back to our original grid for (x – 1)ⁿ = 0, again providing the first nine numbers of the unique sequences, where n takes on the prime values (from 1 – 9) of 2, 3, 5 and 7. we obtain the following:

x = 2	1	2	3	4	5	6	7	8	9
x = 3	1	3	6	10	15	21	28	36	45
x = 5	1	5	15	35	70	126	210	330	495
x = 7	1	7	28	84	210	462	924	1716	3003

When we now express the above table in modular (clock) arithmetic using a modulus of 2, 3, 5 and 7 respectively we then obtain:

x = mod 2	1	0	1	0	1	0	1	0	1
x = mod 3	1	0	0	1	0	0	1	0	0
x = mod 5	1	0	0	0	0	1	0	0	0
x = mod 7	1	0	0	0	0	0	0	1	0

We can see now that entries are identical in this Mod (circular) form for (x – 1)ⁿ = 0 with the corresponding (linear) table earlier for xⁿ – 1 = 0.

This in fact illustrates well the key (unrecognised) feature of prime numbers, which however can only be adequately understood in a dynamic interactive context, where both quantitative (linear) and qualitative (circular) features are understood as complementary.

Thus from the quantitative perspective, each prime is indivisible as a unique building block of the cardinal number system i.e. with no (non-trivial) constituent factors .

However from the corresponding qualitative perspective, each prime group is already composed in natural number terms of a unique set of ordinal members.

Now both of these aspects are connected through the number 1.

So from the cardinal perspective, each prime is always divisible by 1 (as factor).

Then from the ordinal perspective one member, when indirectly represented in quantitative terms through the n roots of 1, = 1.  

When one then properly appreciates the dynamic nature of prime number behaviour, it becomes apparent that both the prime and natural numbers ultimately serve as perfect mirrors of each other in a fully synchronistic manner.

The implications therefore for the true nature of number - and indeed the true nature of Mathematics - couldn’t be more fundamental, with nothing less that the most radical possible change in perspective now of the greatest urgency.        

A Special Case (4)

I am going back again to that grid table that lists the first 9 numbers in the unique sequences associated with the equation (x – 1)ⁿ = 0 from n = 1 to 9.

1	1	1	1	1	1	1	1	1
1	2	3	4	5	6	7	8	9
1	3	6	10	15	21	28	36	45
1	4	10	20	35	56	84	120	165
1	5	15	35	70	126	210	330	495
1	6	21	56	126	252	462	792	1287
1	7	28	84	210	462	924	1716	3003
1	8	36	120	330	792	1716	3432	6435
1	9	45	165	495	1287	3003	6435	12870

Now an especially interesting feature with respect to these numbers can be made where n is a prime number. .

If we look at the entry with respect to n = 2, a unique pattern unfolds, whereby numbers follow a cyclical pattern of 2 members, where the 1^st when divided by 2 leaves a remainder of 1, and where 2 is then a factor of the 2^nd number.

Then when we look at the entry with respect to n = 3, another unique pattern unfolds, whereby numbers now follow a cyclical pattern of 3 members, whereby the 1^st when divided by 3 leaves a remainder of 1, with 3 then a factor of the other two numbers.

Finally to again illustrate, with respect to n = 5, another unique pattern unfolds, whereby numbers follow a cyclical pattern of 5 members, whereby the 1st when divided by 5 leaves a remainder of 1, with 5 a factor of the other 4 members of the cyclical group.

Then when n is not prime, no such unique cyclical pattern unfolds.

For example when n = 4, the 1st number (in the first group of 4) leaves a remainder of 1 (when divided by 4). However the 1^st number (in the next group of 4 i.e. 35) leaves a remainder of 3 (when divided by 4).

Also, though 4 for example is a factor of the second member (of the first group of 4) it is not a factor of the 3^rd member.

So again in an “Alice in Wonderland” type manner this provides in principle a valid manner for testing whether any number is prime, which in general terms can be expressed as follows.

With respect to the unique number sequences associated with the polynomial equation,

 (x – 1)ⁿ = 0, for n = 1, 2, 3, 4,…,

if n is prime then a unique cyclical sequence of n individual members will unfold, whereby the 1st member of each cyclical group will always leave a remainder of 1 (when divided by n) with n then being a factor of the other (n – 1) members of the group.

And if this is not the case, then n is not a prime number.

As the unique number sequences involved are potentially infinite, this would then strictly prove a never-ending process.

However if the required pattern for prime numbers applies to the 1st cyclical group of members involved, then it will apply to all further groups.

Therefore we can more simply restate the position as follows.

With respect to the unique number sequences associated with the polynomial equation,

 (x – 1)ⁿ = 0, for n = 1, 2, 3, 4,…,

if n is prime, then a unique cyclical sequence of n individual members will unfold, whereby the 1st member of the first cyclical group will leave a remainder of 1 (when divided by n) with n then a factor of the other (n – 1) members of the group.

For example if 7 is prime, the unique cyclical sequence of 7 individual members will unfold, whereby the 1st member (i.e. 1) of the first cyclical group leaves a remainder of 1 (when divided by 7) with 7 then being a factor of the other 6 members of the group.

And as we can see from the table, these six members are 7, 28, 84, 210, 464 and 924 respectively with 7 a constituent factor of all these numbers.

So 7 therefore is a prime number. (Incidentally the next number, which is then the 1^st member of the next cyclical group i.e. 1716 as expected, then leaves a remainder of 1, when divided by 7).

However 8 clearly is not prime. Whereas the 1^st member - as is always the case - leaves a remainder of 1 (when divided by 8), 8 is not a factor for example of the second i.e. 36 of the remaining seven members of the group.

As I have mentioned on many occasions in these blog entries, there are really two complementary ways of defining a prime number.

The standard (Type 1) approach, based on cardinal number identity, defines a prime as an indivisible “building block” of the natural number system, so that it has no constituent factors (other than itself and 1). This corresponds directly with the quantitative aspect of number.

However the alternative (Type 2) approach based on ordinal number identity defines a prime in reverse as a group that is already composed of natural number members.

So for example 5 as a prime number is thereby necessarily composed of its 1^st, 2^nd, 3^rd, 4^th, and 5^th members.

This directly corresponds with the qualitative aspect of number, which then indirectly can be expressed in quantitative fashion through obtaining the 5 roots of 1.

Now interestingly as one of the n roots of 1 is always 1, this thereby is not unique.

So for a prime number n, the remaining (n – 1) roots are unique.

So what we have uncovered therefore through these unique number sequences of a simple type of polynomial equation, is an approach to the primes that reflects the ordinal (qualitative) rather than cardinal (quantitative) definition of a prime.

So once again from the standard cardinal perspective, a number n is prime if it has no constituent prime factors.

However, from this alternative ordinal perspective, a number n is prime if it is a constituent factor of a unique group of n – 1 members (with the remaining member = 1).  

A Special Case (3)

In the last blog entry, specifically with reference to the number “2”, I showed how there are two equally important ways in which the number can be defined.

Thus in standard linear terms, where its sub-units are considered in an absolute independent manner, 2 is given an analytic quantitative identity (in cardinal terms).

However in paradoxical circular terms, where its sub-units are now considered as relatively interdependent - and thereby fully interchangeable with each other - 2 is given a holistic qualitative identity (in an ordinal manner).

Though the ordinal nature of number is of course recognised in conventional mathematical terms, it is invariably reduced in a merely quantitative manner (whereby each position is given a fixed identity).

Thus using a physical analogy from quantum physics, every number can manifest itself in both a particle and wave-like manner, and these two aspects necessarily keep switching with each other in the dynamics of understanding.

However the conventional mathematical approach grossly misrepresents the true nature of number by attempting to (formally) interpret it in a static fixed manner.

Now, I have frequently referred to the two aspect of number as Type 1 and Type 2 respectively!

Type 1 corresponds with the linear conventional interpretation of number in analytic quantitative terms.

The natural numbers from this perspective (1, 2, 3, 4, 5, …) are more fully represented as   

1¹, 2¹, 3¹, 4¹, 5¹, …

Type 2 corresponds (indirectly) with the circular paradoxical interpretation of numbers in a holistic qualitative manner

The natural numbers from this perspective are more fully represented as

1¹, 1², 1³, 1⁴, 1⁵, …

Now these have no distinctive meaning in quantitative terms. However from a holistic qualitative perspective 1² represents the interdependence of two related units.

Then indirectly this interdependence can be expressed in a holistic circular manner as + 1 and  – 1, where both units are interchangeable.

(Of course + 1 and – 1 can equally be expressed - as in conventional mathematical terms - in an absolute manner where they are clearly separated in an analytic manner).

However it is the continued failure to recognise the true holistic aspect of number, which I am addressing in these blogs.

So mathematical understanding necessarily contains both analytic and holistic aspects, which in direct terms are represented by reason and intuition respectively.

For example I have just read this quote from Poincare.

“It is by logic, we prove, by intuition we invent”

Logic - in the sense that Poincare intends - represents the analytic aspect of understanding which is of an unambiguous rational nature.

However intuition represents the corresponding holistic aspect, which inherently is of a paradoxical circular nature.

Therefore though analytic reason and holistic intuition are clearly distinct from each other, in conventional mathematical terms the holistic aspect is inevitably reduced in formal terms to the analytic.

In this sense formal mathematical interpretation is of a grossly reduced nature (and ultimately not fit for purpose).

But before this crucial point can be properly grasped, the distinctive holistic aspect of all mathematical understanding must be properly recognised.

Now if we go back to the original simple expression i.e. x = 1, we can perhaps now better appreciate what happens when both sides are raised to the power of n.

So xⁿ = 1ⁿ.

Therefore when n = 2, x² = 1².

Therefore, the number on the right side of the equation properly belong to the Type 2 aspect of the number system.

However in conventional mathematical terms, as 1² has no distinctive quantitative value, it is inevitably reduced to 1 (i.e. 1¹) where it is interpreted as the first natural number in the Type 1 system.

Thus a number that should be treated in a holistic manner - and indirectly expressed in a circular interdependent fashion - is now treated in analytic terms.

And this is what then creates the conflict with earlier analytic type interpretation.

So once again when x = 1, x – 1 = 0.

Therefore squaring both sides (x – 1)² = 0.

And this equation in Type 1 terms has two “linear” solutions i.e. + 1 and + 1 (as independent).

However equally when x = 1, x² = 1² .

And this equation in Type 2 terms has two “circular” solutions i.e. + 1 and – 1 (as interdependent).

Thus we obtain two “different” answers because both correspond respectively to different notions of dimensions (that are analytic and holistic with respect to each other).

However again, these two sets of answers cannot be reconciled satisfactorily through conventional mathematical interpretation (that solely recognises the analytic aspect of number as quantitative).  

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).