Monday, August 22, 2011

Natural Logs

If the natural log of a number n is x, this implies that e^x = n.

So the important point to grasp here therefore is that the log always points to a number that represents a dimension (i.e. power or exponent).

And as I have previously related numbers that represent quantities (raised to a particular power) and numbers that directly represent dimensions (to which base quantities are raised) are properly quantitative and qualitative with respect to each other.

Once again this can be illustrated through the simple example of a number expression such as 1^(1/3). Now 1 and 1/3 are here rational numbers that in isolation can be considered as linear quantities (that are represented by discrete intervals on the straight line). 1 represents the base quantity and 1/3 the dimensional power or exponent.

However though each number can be considered as a linear quantity in isolation, in relation to each other they are as quantitative to qualitative (and qualitative to quantitative).

This in this interdependent fashion if 1 represents the quantitative aspect of number, then 1/3 represents the corresponding qualitative aspect.

And the truth of this observation is then - indirectly - demonstrated by the result of the expression which lies on the circle (of unit radius).

Though this of course is well recognised in conventional (Type 1) Mathematics, because of a merely quantitative bias, its significance is greatly overlooked.

In other words the true (root) reason why this shift from linear to circular interpretation takes place is because the interaction is of a base number quantity with a dimensional number that - relatively - is of a qualitative nature.

Once again when we shift the frame from reference from up to down on a street the turns at a crossroads may both appear similar. So one one of these turns for example moving up the road may appear as left and then coming down the road in the opposite direction the other turn also appear as left.

However in relation the each other they are clearly left and right (and right and left).

So the position in Type 1 Mathematics of recognising a merely quantitative basis for both linear and circular type number quantities is exactly analogous to one who labels in our example the two turns at a crossroads as both left!. Again this arises through attempting to treat numbers (representing both quantities and dimensions) in an isolated fashion.

So once again in a combined expression the number representing the base and dimensional aspects are properly quantitative and qualitative (and qualitative and quantitative) with respect to each other.

Now we have already seen (in the last contribution) that inherent in the very definition of a prime number are both quantitative (analytic) and qualitative (holistic) aspects of equal importance and which in the dynamic generation of prime number behaviour (both with respect to the individual primes and their general distribution) simultaneously arise.

And the simplest expression expressing the general distribution of the primes is given as n/log n (where we are using the natural log base of e).

Now n properly relates to a number representing a base quantity whereas log n properly relates to a dimensional number (that relatively is of a qualitative nature).

So once again - when appropriately interpreted - this simple expression (for the general distribution of primes) demonstrates the inherent connection (in the very nature of primes) as between its quantitative and qualitative aspects.

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