Tuesday, September 30, 2014

Holistic Synchronous Nature of Number System (1)

The Erdős-Kac Theorem relates to the distribution of the (distinct) prime factors of a number demonstrating that it approaches a perfect normal distribution when the relevant numbers (to which the prime factors relate) are sufficiently large.

So for example if we were in the region of the number system where n > 1020 and were to take in this vicinity a significant sampling of numbers we would find that the number of (distinct) prime factors would vary considerably with the majority close to a central value with higher and lower recorded values symmetrically distributed around the central value in accordance with the normal distribution.

We would equally expect the distribution of H’s with respect to similar sized samples, relating to the tossing of an unbiased coin, to be likewise normally distributed thereby indicating the random nature of the trials involved.

So likewise the Erdős-Kac Theorem demonstrates the random nature with respect to the  occurrence of the distinct prime factors of a number.

However there is also the more familiar way in which we speak about the randomness of the primes which relates to the distribution of the (individual) primes with respect to the overall natural number system.

Thus when we look at these two aspects of prime randomness in the appropriate dynamic interactive manner, we realise that they represent complementary aspects of prime behaviour (with respect to the natural numbers).

So the familiar Type 1 notion entails the (individual) primes in the context of the (collective) natural number system.

However the alternative Type 2 notion entails the complementary notion of a (collective) group of prime factors in the context of a related (individual) natural number.

Also the Type 1 notion entails viewing the primes as base quantities. In relative terms the Type 2 notion relates to the dimensional qualitative aspect of these numbers (as factors).

Notice also how the relationship as between the primes and natural numbers is inverted as we switch from the Type 1 to Type 2 definition! In Type 1 terms the (collective) natural number system is understood in terms of its (individual) prime constituents. In Type 2 terms the primes as a (collective) group of factors are understood in terms of each (individual) natural number!

Thus in truth the relationship the two-way relationship as between the primes and the natural numbers is of a dynamic interactive nature, entailing the complementarity of these two opposite polarities.

Now we have that the notion of randomness itself can only be properly appreciated in a dynamic interactive context where it always implies the opposite notion of order.

Therefore the Type 1 and Type 2 aspects of the primes intimately depend on each other in a holistic synchronous manner.

Now whereas the (Type 1) individual primes are random, we also have the opposite demonstrated characteristic of their overall collective order with respect to the natural numbers (when taken as an entire group) as for example expressed through the prime number theorem.

Likewise whereas the (Type 2) collection of primes is random, we also have the opposite characteristic of individual order (indirectly expressed through the prime roots of 1, where all roots other than 1 are uniquely determined).

Therefore we can validly say that the random nature of the primes in Type 1 terms is intimately related to their corresponding ordered nature in a Type 2 manner; likewise the random nature of the primes in Type 2 terms is intimately related to their corresponding ordered nature in Type 1 fashion.

Thus we can now state this remarkable overriding fact regarding the nature of the number system, which contains the seeds to not only dramatically revolutionise our very appreciation of the nature of Mathematics but likewise all of its related sciences.

And this fact simply relates to the clear recognition of the inherently dynamic nature of the number system with its underlying holistic synchronous basis.