CALCULUS
We are taught that the mathematical process referred to as integral ‘calculus’ involves sub-division of the graphical area defined by an analog mathematical function into an infinite number of sub-divisions as illustrated here. The sub-divisions are then added together in order to derive a resultant value representative of the value of the area between two pre-defined mathematical limits (indicated here as X1 and X2). We are advised that the subdivision into an infinite number of sub-divisions eliminates potential errors in the resultant value because the potential error inherent in each individual sub-division must be infinitely small, or zero.
In essence, the 'integral value is represented by the combined area of an infinite number of adjacent vertical lines, each of which is said to have a width of 'dX' and a height equal to the distance between the horizontal axis and the line representing f(X) at the specific horizontal location which corresponds to that line (dX) of current interest.
That which we are taught may be true in a world of pure imaginary mathematics, but it is not true in the universe of realities. The reason that it is not true in the universe of reality is that the number of the sub-divisional parts is not infinite, but is totally dependent on the arbitrary fixed magnitude of the units of measure which have been defined for real factors such as distance, time lapse, and force.
In order to derive a practical solution to the integral process, the number of vertical lines must be defined in terms of numbers representative of units of pre-defined 'units' of measure for real parameters, and the actual number of sub-divisions will be simply that number which corresponds to the difference between the pre-specified limits which were placed on the integration equation.
Before the mathematical process of integral calculus can begin, it is necessary to determine the relationship of the function of current interest in terms of the same type of unit of measure used along the horizontal coordinate. In graphical form, the vertical axis of the graph is then displayed in terms of the number of that same unit of measure. In mathematical terms the horizontal axis is referred to as ‘X’, and the vertical axis is referred to as a ‘function of‘ X, and symbolized as f(X).
The total area included in the graphical area of interest it simply the sum of the product of the height of each column of units times the number of those units included between the pre-defined horizontal limits of the graph. For example, if the resultant value of interest can be portrayed by a graph extending between the limits of 1 and 5 units of measure displayed along the horizontal axis, then there will be only four (5-1) horizontal subdivisions involved in the integration process. And the number 4 is very different from the concept of the infinite number of subdivisions that we were told that would be used to define the integrated value.
The number displayed as the solution to an integral will be simply the number of squares of measure displayed between the limits of the integral function times a ‘sort of averaged’ number of those same type of units indicated by the vertical coordinate of the graphical image between the limits as indicated along the horizontal axis. It is necessary to explain that ‘sort of averaged’ number because it has not been otherwise defined by a specifically recognized word.
By ‘sort of averaged’ I refer to division of the difference in the values of the f(X) function at the upper and lower limits of the integral divided by a number which is one level greater than the exponent which may have applied to the overall function defined as f(X). if the exponent of f(X) is one, then the resultant integral value will have an exponent corresponding to 1 plus 1 (or 2), and that resultant integral value will be divided by 1 plus 1 (or 2) . That results in simply the true average value between the values for f(X) existent at the two limits specified by the limits of the integral.
If however, the exponent of the entire f(X) function is more than one, then the resultant value for the integral will be equal to the difference between those two limit values of f(X) to the exponent plus one, divided by that same factor of the exponent plus one.
Hence, the ‘sort of averaged ’ term refers to the difference between the two limiting values of f(X) divided by the same number that appears as the final exponent of the f(X) factor.
A simple example: The derivation of an ‘area’ of interest based on the use of an X axis factor defined as a linear distance, and a Y axis factor expressed as a mathematical function of linear distance. The resultant area is defined as a linear distance raised by one order of magnitude. That raised order of magnitude of the distance creates the concept referred to as ‘area’, having an exponent of 2 (rather than 1). And the difference between the mathematical values of areas defined at the limits of the integral is then divided by the same exponent (2) to obtain a final answer for the integral.
However, the derivation of a ‘volume’ of interest based on the use of an X axis factor defined as a linear distance, and a Y axis factor expressed as a mathematical function of linear distance raised to the power of two (referred to as an ‘area’). The resultant values of the integral will involve exponents of two plus one (3), and the mathematical value corresponding to the difference between the f(X) values existent at the limits of the integral must now be divided by two plus one (3) in order to obtain the ‘sort of averaged’ final value between the two extreme vales. That raised order of magnitude of distance creates the concept referred by the word ‘volume’.
There are no words corresponding to dimensional units of distance raised to powers greater than 3 (a volume), but the imaginary mathematical process is not limited to creation of concepts having exponents greater than 3. The mathematical integration process for these unnamed concepts would simply continue to involve exponents and divisional factors having one additional level of value above the imaginary f(X) mathematical exponent.
Another interesting way to look at the reality of the integration process is to think of the resultant value (ie, the 'integral value')
as the averaged value of the difference between simply two different areas of the graph. The first such area is defined as all the area in a rectangle of space between the two coordinate lines and the values corresponding to the f(X) function at the maximum limit of the integral. The second area is defined as all the area in a rectangle of space between the two coordinate lines and the values corresponding to the f(X) function at the lower limit of the integral. That difference between the two areas is then 'sort of averaged' just as described above.
The process of integral ‘calculus’ is one example, out of many, wherein current science which is based on imaginary mathematics does not necessarily agree with the imaginary ground rules which may apply to mathematics. In cases where there is obvious differences between the reality of nature, and the imagination of mathematics, the reality is usually by far the simpler of the two concepts. Perhaps this was best phrased by Einstein when he advised that:
"As far as the Laws of Mathematics refer to Reality, they are not certain, and as far as they are certain, they do not refer to Reality." ....Einstein
Another strange anomaly in the imaginary world of pure mathematics is the practice that defines the function of X^N when the value of N is less than zero. According to the established rules, when N is less than zero, then the function of X^N is interpreted as if the function were displayed as the inversion form with the negative sign deleted. Which is to say that the function X^(-N) is replaced by the function 1 / X^(+N).
I have not researched the actual history for that seemingly strange rule, but one possible reason could be that the rules of calculus simply fail when N is less than zero. For according to the rules of calculus discussed above, the integral of the function X^(-1) would resolve to a value of 1 divided by zero. And division by zero produces an answer of infinity - which can only be explained as imaginary nonsense. The mathematicians must have been faced with a dilemma when they realized that the integral value of X^(-1) resulted in nonsense.
And at that point the mathematicians were forced into creating a new mathematical rule. It was declared that the integral of X^(-1) would be defined by a new word. And the word selected was 'logarithm'. At which point it was also declared that when the function is a logarithm, the use of exponential values less than zero is simply mathematically 'illegal', and when the exponential value is exactly zero, then the integral shall not be infinity, but rather it shall be, by definition equal to 1.0.
And so it was that the logarithmic value of every number which had an exponent of zero became 1.0. Perhaps we are not supposed to ask why the value defined by the integral of X^(-1) is not equal to infinity as the rules of calculus would indicate - we are simply advised that in this case, the value of infinity has simply been changed to 1.0. And we must accept that this is so because the integral of X^(-1) is now referred to as an 'logarithm' rather than another imaginary mathematical value named 'infinity'.
This redefined form of imaginary mathematics called 'logarithms' is discussed in the next section. A similar problem was encountered when the general rules of calculus were applied to geometric and trigonometric functions. This relationship is discussed in a thrid section of this document.
SKIP to Geometric Functions
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