So these represent the positive integers.

However, we can equally show how the Type 1 and Type 2 systems for negative integers are related to subtraction and division with respect to 1.

So in Type 1 terms, 1

^{1 }– 1

^{1 }= 0; 0

^{ }– 1

^{1 }= – 1

^{1}; – 1

^{1}

^{ }– 1

^{1 }= – 2

^{1}; – 2

^{1}

^{ }– 1

^{1 }= – 3

^{1 }and so on!

Then in Type 2 terms 1

^{1 }/ 1

^{1 }= 1

^{0}; 1

^{0 }/ 1

^{1 }= 1

^{– 1}; 1

^{– 1}

^{ }/ 1

^{1}= 1

^{– 2}; 1

^{– 2}

^{ }/ 1

^{1 }= 1

^{– 3 }and so on!

^{}

^{}

^{}Therefore all the integers (positive, negative and 0) are defined in Type 1 terms through addition and subtraction respectively; equally all the integers are then defined in Type 2 terms through multiplication and division respectively.

So returning to the positive integers, once again in conventional mathematical terms, the Type 1 quantitative aspect of multiplication is solely recognised (whereby in effect multiplication is reduced to addition).

In particular in conventional mathematical terms, the primes are considered as the quantitative "building blocks" of the natural number system. So every natural number thereby represents a unique combination of primes.

For example in conventional terms 6 = 2 * 3 (which represents the unique combination of primes for this number).

However, when one carefully reflects on the matter, this notion of multiplication represents but a shorthand reduced form of addition.

So the operator 3 here serves to indicate that the number "2" be added to itself 3 times.

Therefore the multiplication operation 2 * 3 in effect is reduced to the addition operation of 2 + 2 + 2.

The "proof" that this reductionism in fact occurs is that in conventional mathematical terms, the result 6 (from multiplying 2 * 3) is treated as just another number (on the 1-dimensional line).

So therefore in effect, in Type 1 terms, the multiplication operation (2 * 3) = 6

^{1 }is inseparable from the addition operation 2 + 2 + 2 i.e. 2

^{1}

^{ }+ 2

^{1}+ 2

^{1}= 6

^{1}

^{}.

However if we think of 2 * 3 in geometrical terms - say as a table top with length 3 units and width 2 units (representing metres) - clearly the result will now be expressed in 2-dimensional terms (as square units).

So therefore through multiplication, a transformation has thereby taken place in the qualitative nature of the units involved!

This point is in fact central to the appreciation of what the famed Riemann zeros in fact represent!

When I started to seriously investigate the frequency of occurrence (on the imaginary number line) of these zeros, I began to realise that they bear a remarkably close relationship to the proper factors of the natural numbers.

Now the proper factors include all divisors of a number (other than the number itself)!

So for example with respect to the number 12, its proper factors (i.e. natural numbers which divide evenly into this number) are 1, 2, 3, 4 and 6 (So 12 is here excluded as a proper factor).

Basically if one accumulates the total of all proper factors numbers up to n, they will match to ever higher degrees of percentage accuracy the corresponding frequency of the Riemann zeros up to t, where n = t/2π.

Now the reason why n = t/2π is that frequency of factors is measured on a linear scale (the number line) whereas the Riemann zeros are measured on a circular scale (based on the unit circle). And of course where the radius of the unit circle = 1, therefore its corresponding circumference = 2π. Therefore to convert the "circular" units to which the frequency of Riemann zeros relate to the "linear" units (representing the accumulated total of factors) we must divide by 2π.

One might now perhaps ask why there should be such a close relationship as between the accumulated factors of the natural numbers and the Riemann zeros!

And the answer gets to the very heart of this key distinction that I was making (in my previous blog entry) as between the quantitative notion of number independence and the qualitative notion of number interdependence respectively.

So we can indeed - from one valid perspective - start off by viewing the primes as the "independent" building blocks of the natural number system.

However as I was at pains to point out in my previous entry, the very process of multiplication causes a switch from the quantitative notion of number "independence" to the qualitative notion of number "interdependence ".

Therefore though we can initially look - with reference to our illustration - at the two primes "2" and "3" as "independent" numbers in quantitative terms, through the very process of multiplication, a new relationship of number interdependence is created (in a qualitative manner). In other words, in the context of "6", both "2" and "3" (as proper divisors of this number) acquire a new qualitative resonance (which is unique for each number).

So putting it simply, once we uncover a (proper) factor of a number, we thereby identify the qualitative identity of that factor (through its unique relationship with that number).

So again to illustrate with reference to "12" which we have already mentioned, the (proper) factors 1, 2, 3, 4 and 6 all acquire a distinctive qualitative identity (through their unique relationship with the number 12).

Now these factors include composite numbers (viz. 4 and 6) as well as primes.

However, as we know the composites - though depending on the primes - can then be given a Type 1 quantitative identity (as numbers on the 1-dimensional number line). However when considered in relation to other numbers (as factors) they too now acquire a qualitative relationship (through their unique interdependence as factors with those numbers).

However, it is equally true that all such composite numbers already express a relationship with respect to a unique combination of primes. So these unique combinations of primes themselves find a new qualitative significance through their subsequent relationship to other natural numbers (as factors).

Thus it is utterly fallacious - as in conventional mathematical terms - to attempt to view the number system in a merely reduced (Type 1) quantitative manner composed of independent building blocks represented by the primes.

Properly understood, the number system represents a dynamic interactive relationship of both quantitative "independence" and qualitative "interdependence" respectively.

Therefore from one valid perspective, the quantitative ordering of the primes (with respect to the natural numbers) has no strict meaning apart from the corresponding qualitative nature of the natural numbers (through the combined relationship of the primes as factors).

Equally from the opposite perspective, the qualitative nature of the natural numbers (in their unique combinations of factors) has no strict meaning apart from the "independent" nature of the primes (as quantitative building blocks).

Therefore from this dynamic perspective, the ultimate nature of the number system approaches pure synchronicity in an ineffable manner (where the primes and natural numbers mutually reflect each other in an identical fashion).

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