Saturday, August 20, 2011

The Number e

The number e which is 2.718281828... approx. is certainly one of the most important constants in Mathematics.

One issue that has long fascinated me relates to its use with respect to prime numbers as for example the general distribution of the primes which is given in its simplest form as approximating n/log n. So we would expect to find roughly n/log n primes in the first n (natural numbers)!

Once again I am mainly concerned here with the qualitative holistic approach to mathematical symbols and in this regard e is especially interesting.

In psychological terms - as we have seen in the last contribution - development is characterised by both (conscious) differentiation and (unconscious) integration. These two aspects likewise correspond with the linear (1) and circular (0) use of logic respectively.

Successful development entails both differentiation and integration. In the mystical contemplative literature as the spiritual aspirant approaches union (discrete) phenomena of form are so fleeting and short-lived that they no longer even appear to arise in experience but rather seem to have finally merged with the continual present moment. So here both conscious and unconscious are so closely related that it is no longer possible to distinguish differentiated rational form from integral i.e. holistic intuition.

Put another way it is no longer possible to separate the quantitative from the qualitative aspect of experience.

Remarkably such experience corresponds to the holistic mathematical interpretation of e (where extremely refined discrete phenomena (of a differentiated nature) can no longer be distinguished from a continual spiritual intuitive awareness (of a corresponding holistic integrated nature).

Now we can approach the analytic interpretation of e in a similar fashion.

Imagine I invest €1 for a year at 100% rate of interest. So my investment will be worth €2 at the end of the year.

Now say I am allowed to compound interest at shorter time intervals. So instead of waiting a year I can invest for six months getting a 50% return (for the 1/2 year involved) and then reinvest. Well! I will make more money this way for at the end of the year the investment will be €(1 + 1/2)^2 = €2.25. In other words I will get 75 cent additional interest on the €1.50 invested for the 2nd six months bringing the total for the year to €2.25.

If the time periods were reduced to 3 months (with 25% return over that period the investment would be worth (1 + 1/4)^4 = $2.44.

So the general formula here is (1 + 1/n)^n.

Now if we keep shortening the time periods (1/n) so that ultimately n is infinite as conventionally understood we then obtain the value of e. So our investment would be then worth at the end of the year €2.78 (to the nearest cent).

What in effect happens here is that the discrete time intervals (over which interest is calculated) eventually become so short that we can no longer distinguish them so that interest now appears to accumulate on a continual basis.
So differentiation (with respect to discrete time intervals) can no longer be distinguished from integration (of infinitesimal intervals on a continual basis).

Now there is an especially remarkable feature about e (from a Type 1 mathematical perspective) that is worth commenting on:

if y = e^x, then dy/dx = e^x.

So with respect to this simple function integration is indistinguishable from differentiation.

As we have seen in corresponding holistic (Type 2) mathematical terms integration (approaching contemplative union) cannot be distinguished from differentiation.

Or as we have seen - put another way - the quantitative aspect cannot be distinguished from the qualitative aspect.

In Type 1 terms unfortunately the qualitative aspect is inevitably reduced to quantitative interpretation. So e is merely understood in rational terms as a quantity thus obscuring its true nature. In other words e is properly a transcendental number that entails both linear (finite) and circular (infinite) aspects. So e can be approximated in value in reduced quantitative terms. However the true value remains elusive due to its infinite qualitative aspect whereby the decimal sequence continues indefinitely with no discernible pattern.

As is well known e has a vital role to play with respect to understanding the general distribution of the primes.

This immediately suggests - or at least should suggest - that inherent in the very nature of primes is the key feature that they contain quantitative and qualitative aspects that ultimately are indivisible. We will return shortly to this vital point!

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