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MATHEMATICA: A Treatise Written by: Marilynn Lea Stark
Preface: The study of mathematics originally had taken precedence over all other subjects in my education, and although I did not major in it, it remained as a first love, a supreme conceptual plane and driving interest for me as I grew as a scientist. Due to the rigorous application of metaphysics on the precipitous political path upon which I landed during the Cold War from 1980 onward was I also one day gifted with a breakthrough in my understanding of mathematics. This breakthrough in my understanding of mathematics resolved the quest which had remained with me throughout all of my studies in the scientific disciplines which had become my occupation and contemplative endeavor. In July of 2001 I had derived e. This inspired me further to develop "A Treatise on the Chemistry of Living Things" at that time. Once the 9/11 disaster struck our nation shortly thereafter the continuity of the work I had begun with the derivation of e was placed on the back burner; I began to develop the Web site Starkmusic.com with the hopes of ever disseminating my classical music compositions to the wider people. Starkmusic.com was offered as an indirect answer to the plight of our nation in the wake of 9/11 in the way of guidance culturally. Following that, the political way for me became most tenuous as to residence, and I was uprooted from all I had lived and worked for in the way of recovery from the results of the loss of freedom I had known for so many years as I led in the Cold War. Bereft of religious freedom, my impending marriage and financial stability, once again, the strife for survival took over my life. Still today, the socio-political forum in my locale unjustly fights my return to academia on a day-to-day basis, and so I am offering this work through the media of the Internet for proper dissemination in the name if not spirit of freedom and democracy. May this work be well appreciated, and may justice be won. May God bless America. // Marilynn Stark September 5, 2005
The logarithmic function is defined: y=logbx Equation 1. This function sets forth a variable x which changes in regard to a constant b, or base, thus relating by value to the function y discriminately. What is the shape or behavior of x as it varies? The independent variable x can simply increase as it will, and its logarithm can be found as the exponent, by=x Equation 2. E.g., log10100=2 and log1010,000=4 102=100 104=10,000. However, in order to study in the abstract how x might vary we can view the expansion of x by looking at a given variable 't' found in this expression: (1+t)1/t Then, this expression can be substituted in for x in the function y=logbx Intuitively, x would start by adding to the first natural number, the unit value, 1. Everything starts from 1. Therefore, (1+t) is established as the natural sequence of integers starting at 1. This is fundamental; however, since this natural sequence will be related to y, the logarithmic function, then numbers which are described in sequence from (1+t) can be viewed as possible multiplicands of exponential powers, or as bases, also of increasing, natural whole numbers. This (1+t) is how numbers change in sequence, i.e., (1+t), most fundamentally, and then, since a log is an exponent, the same discrete series must be applied logarithmically to this (1+t) sequence. In order to see how the logarithmic function works, we could say (1+t)t , using t twice for symmetry. However, this expression (1+t)t would approach infinity upwardly or outwardly, which is partial symmetry. It would be more useful, therefore, to explore (1+t) as a possible base to an exponential power series by inverting the exponent to which it is held, thus as (1+t)1/t For (1+t)1/t neatly approaches 1, since 1/t approaches 0. This approach to 1 gives a perfect symmetry; that is:
(1+t)1/t=(1+t)
~0É(1+t)/(1+t)
É
1 Equation 3. Notice that the logarithmic function overall greatly compacts numbers. Great changes are reduced to simple, or smaller changes. Thus, instead of moving conceptually from, say, 102 to 109, which written out is 100 to 1,000,000,000, the course is shortened to the span from 2 to 9. Now comes the idea of understanding the function y=logbx by its derivative, which particularizes it to the infinitesimal. That is to say, the logarithmic function has its own mode of change, and to study how that change occurs we can divide into it by a limit taken by that divisor, which is a form of arbitrary unification, but not to 1, which is why it is only arbitrary.
Therefore, the derivative dy/dx of the function
of x is the
ratio of Äy
to Äx
seen at the limit as Äx
approaches zero for any function y=f(x).
(Eq.2)
dy
Äy Equation 4.
The ratio Äy/
Äx
is made minute unto change and which changes unify the ratio by the
approach of Äx
The limit as Äx¾
x2
Ê
x1
= Äx Equations 5 & 6. So now, it is stated that the limit of the expression (1+t)1/t = e, as t approaches infinity, Equation
7. and this limit is given by the infinite series that sums the reciprocals of factorials: e
= 1 + 1 + 1 + 1 +...+ 1 +... Equation 8. e1=
(1+0)0=1 (This method of computation is indirect.) Observation: the first term in the infinite series which is given by e is 1 because at infinite t, which is never reached, the exponent 1/t approaches zero, so that the entire expression approaches 1. Since 1 is never realized, since infinity cannot be quantized by increasing natural numbers, we know that 1 is thus always modified, as if 1 would "found" its own modification in an ever-changing fashion. The way lim
Äy is used in
the Fundamental Theorem of calculus causes us to compare this expression
and its limit as
t¾0
,since
in this case, there is not a ratio of change; rather, there is an
exponent, 1/t,
to consider also as ratio. So the change is exponential, and does
not linearize a curve by diminishing intervals to the limit of zero,
which says that the curvature of a function is less real or prominent
dimension Thus, knowing that the approach is infinite unto 1, how can this approach be characterized? Just as the natural numbers proceed one-by-one in a sequence, then they may contribute to the lim(1+t)1/t in a sequential fashion, yet, if one calculates e by taking discrete values of t, one at a time to form a simple summation, one has erred. Why? This is because these values are discrete -- it would be like calling the interval xn+1 Ê xn = Äx a point: the point is only approached by infinitesimally small intervals. Similarly here, to write in a value for t defies the nature of the expression, which though composite through the character of t, only particularizes to 1 in a dynamic sense, never reaching 1, just as a curved line is never really a straight one, no matter how small the interval. This is further complicated by the fact that due to the inversion of the exponent to 1/t from what could have been t itself, explained previously, discrete values for t particularize into roots, such as the cube root of 4. This leads to a second particularization of the expression, since taking the varied roots of given values compresses the expression. This stands in contrast to the expression (1+t)t as t¾º whose limit would be true infinity. With compression occurring by root formation using discrete term-taking, there is the setting up of internal limits, which defies the nature of the expression (1+t)1/t as an ever-expanding expression if t¾º. Knowing this, that we must imitate the conceptual awareness offered by the power of the exponent yet inverted, 1/t , to cause the limit of the overall expression (1+t)1/t to approach unity, or 1: we must realize that there exists here not a value-by-value relationship with the variable at hand, but rather a series, and an infinite one, where each member is as important as the next. Therefore, we must seek to characterize that infinite series. Any discrete values ordinated from (1+t)1/t and listed comprise a sequence of given numbers. If we wish to characterize that sequence, then we may apply a limit concept to the varying factor, so as to see unto what value the sequence changes. Here we have already seen that the expression (1+t)1/t , if seen as the limit of t as t approaches infinity: then the sequence born out of that increment upon 1 (one) of an ultimately varying t actually approaches 1 then again, since 1/t in such a case, the exponent, approaches zero. The discrete values of the series of reciprocal factorials added to 1 which is e, or the limit as t approaches infinity of (1+t)1/t can be calculated discretely: e
= 1 + 1 + 1 + 1 + 1 + 1 +...+ 1 +... = 1 + 1 + 1 + 1
+ 1 +
1 This sum arithmetically amounts to 2.00000 + 0.50000 + 0.16667 + 0.4167 + 0.00833 = 2.71667. The normative value of e to five decimal places is given as 2.71828 for the first six terms of the series. By calculating e only with the first five factorials a nominal value of 2.71667 is obtained. Thus, we can reasonably conclude that since e is the characterization of the expression (1+t)1/t as t approaches infinity limit-wise, that values can be taken for the actual natural numbers whose comprise define e, and even that they are in a series, and even that this series is infinite. This proof of the series given in classical mathematics causes the observer to want to further understand that series. Why this series? In the strict expression (1+t)1/t, the 1/t exponent causes the base (1+t) to be figured as roots, such that even as t increases, it compresses into an overall value which, when multiplied by the given power of the root at hand, equals the nth root. This forms a certain sequence of roots.
To take a limit on this expression of sequential roots as t approaches
infinity is to perform an operation upon that sequence, since this limit
is a determinative factor, and the factor is infinity
This series is said to be:
When 1/t is introduced as an exponent to (1+t) remember that then an internal symmetry is introduced due to the ratio which is 1/t. This causes the sequence formed of (1+t)1/t to be compressed. When the operation of limit-taking is performed upon (1+t)1/t , then the exponent 1/t is seen as zero, and (1+t) is perceived as 1, no matter how t had varied in its approach to infinity. This root-taking at the sequence level must be modified through the limit-taking operation cited heretofore, since all of the operations are subsumed under the leverage of the limit. There are three such operations at hand which are operated upon by the fourth, or the limit-taking operation in the expression lim(1+t)1/t
First, there is addition, (1+t), t being added to 1 variably and in the natural sequence of integers. Then, there is multiplication, which could be more simply, to demonstrate by example, to the tth power at which the base (1+t) is used to multiply itself against itself. Next, there is division, as the exponent is actually inverted to the ratio of 1/t . When the fourth operation, that of taking the limit of this overall expression, is applied to the expression as t approaches infinity:
lim(1+t)1/t
the internal symmetry which had been introduced by the division in 1/t as an exponent is neatly reduced to zero. This reduction also affects the base (1+t), by symmetrizing it to 1 by identity, thus:
ê(1+t)ï
(1+t) Equation 10. When this symmetrization occurs, what happens to the operations of multiplication and addition which were cited earlier, and are composite to the overall expression (1+t)1/t under consideration now as the power of the operation of the limit-taking which governs the symmetrization is applied at the given limit? These two operations must also accomplish each a step in symmetry, so that the base (1+t), which had been compressed into root-level products, will realize an expansion meet with the approach of t to infinity. Rather than to be held to the constraint of an internal sum, which had been imposed by the internal symmetry of the multiplier, the exponent 1/t, which is also itself a dividing factor, such that nth roots of (n+1) are taken; the way to symmetrize those multiplying bases now freed from root-taking, would be for straight digit multiplying as the digits increase by 1. The founding term of this series of increasing groups of multiplicands should be 1! itself, since even in a series of whole numbers starting as internally multiplying and increasingly multiplying groups, the number 1 is that to which all other groups symmetrize, thus, 1! or 1 x 1, would be included by definition of the unitary principle now extended in this case. The unitary principle can be understood as the truth that quantification starts with one, and relates back to one therefore, whether one is seen as the digit 1 such as in the natural numbers, or as a unit value or entity. The unitary principle allows the derivation of symmetrization, wherein its truth is applied specifically to mathematical operations and the concepts which underlie those operations as simplification to one (1) is elicited accordingly. Now again, how would the multiplication change once the symmetrization to 1 of the overall expression occurs dynamically as t¾º, |
Chapter
1 
Derivation of e