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Before plowing ahead into our new topic, ‘Root Beings’, let us take brief stock of where we’ve been and where we are going. In the last Notebook, we demonstrated or derived a simple universal equation to represent the Fractional Roots of the Rational Numbers. In finding this equation, we also found some other equations that worked under ‘Normal’ Circumstances but failed to work under ‘Exceptional’ Circumstances. This intriguing attribute led us to further explore these equations. The results of these explorations are the topic of this present Notebook.
Before talking about the explorations, we need to talk a little about computer experimentation, which is what these mental adventures are based upon.
Generally speaking in the old days, i.e. before computers, the mathematicians only had the proofs to rely upon to determine the veracity of their equations. A mathematician would throw some derived truths out to the mathematical community, who would examine his derivations with care, trying to ferret out any logical inconsistency. After going through academic scrutiny the proofs would be determined to be logically true. Future mathematicians would then build upon these proofs.
In modern times, since the computer, mathematicians, in certain circumstances, have been able been able to check their work, through high speed computer computations that were previously prohibitive or impossible. For instance iterative equations have been successfully explored with the computer. As mentioned in the previous Notebook, these are the type of equations that we are using in our explorations. Hence the computer was the ideal tool to check our results.
As an aside, we also call these feedback equations, because they rely upon feedback to generate themselves. The square roots take one level of feedback to generate themselves; cube roots need two levels of feedback, fourth roots three levels of feedback, and so forth. Thus the higher the root the higher the number of levels of feedback needed to generate it. Traditional computational mathematics is barely able to compute equations requiring one level of feedback, much less multiple levels. Nearly all of traditional computational mathematics is based upon techniques for dealing with zero levels of feedback. The best of traditional computational mathematics is able to deal with continuous one level feedback with its exponential functions. Discontinuous multiple level feedback is for computers only. Thus from the beginning we have tested our iterative equations with the computer. If you, the Reader, are interested in the results of this computer testing read on.
Because of the inherent precision of our topic, i.e. irrational numbers, the first topic we will tackle is the inherent imprecision of the computer. In the previous Notebook, to describe the Fractional Roots of Rational Numbers, we introduced three series, the X Series, the Fraction Series, i.e. the F Series, and the Denominator Series, i.e. the D Series. Let us take a look at the relation between these series.
The X Series is made up of a set of expanded equations, each of which exactly equals the Root minus one. The F Series is made of a set of fractions that approximate the Root minus one. The D Series is that it is made up purely of whole numbers, the ratio of the consecutive members approaching the Root minus one. This is one of the beauties of D Series. No square roots. Not even fractions. Only whole numbers. With these most elementary of numbers, we can build the entire universe of Rational Roots. We bow before her majestic glory.
The members of the X Series are the expanded versions of the iterated equation that is exactly the square root. While it is exact, it can’t be computed. If it can’t be computed then it is not real. Thus the X series while exact is not real. Alternately the F & D Series are computable and thus real, while always approximate. Hence that which is exact is unreal while that which is approximate is real. As is life.
The binomial expression that we derived for the F Series is definitely the simplest expression of the General Root Equation. While the F Series is probably simpler in expression, the D Series creates the F Series. The D Series is the foundation of the F Series, just as atoms are the foundation of molecules. While the D Series is the foundation of the F Series, the binomialized iterative equations that we derived are at the heart of the structure of both the D and F Series, just as physical dynamics is at the heart of both atomic and molecular structure. In this paper we will attempt to include some analysis of each of these three elements.
There is an inherent imprecision in fractional ratios and their iterative Roots that they approach. If this is not enough, there is also a built-in imprecision in the computer computations. Let us first look at the flaws of the F Series. Most of us think of rational numbers as exact because we can easily get close enough for practical purposes with the decimal system. However even simple fractions are expressed imprecisely with the decimal system. These fractional ratios are expressed a little imprecisely in decimal form because of the limits of the computer. Any rational number is stored in a computer in binary form. Even the elementary fraction, 1/3, one of the first inputs in the cube root series, is only an approximation for the binary computer system, no matter how powerful the computer, IBM or Mac. Any rational number with a non-2 factor in the denominator is ultimately an approximation with the binary computer system.
Thus we are looking at an approximate limit, derived from imprecise data. Further this imprecise data is used to calculate the next numbers of the series. Hence each successive member of the F Series is computed from the inherently imprecise data that preceded it, which was based upon the inherently imprecise data that preceded it. Normally this is only going to lead to disastrously imprecise results. If we start with corrupted data we are certainly going to arrive at corrupted results. One would think that this imprecision would be magnified with each additional iteration – because the new is created from the old. One’s mind would tell them from experience that any system built upon the feedback of this corrupted data is destined for increasing imprecision. With traditional systems the imprecision always multiplies with each iteration.
But, not to worry. We have the inherently precise D Series, based only upon the exact whole numbers. Further these precise whole numbers are used to compute the next whole numbers of the Series. While each of the members is exact, the ratio is inherently inexact. Thus the only thing inexact about the D series is the ratio, which approximates the Root minus one. All of the data and its feedback are based upon the ultimately precise whole numbers. No corrupted data is fed into the system and so it is as pure as possible. While the members of the F Series are only pretending to be exact in terms of the computer, the members of the D Series are really exact, based only upon whole numbers.
Each member of the Denominator Series is generated from the only really precise numbers, the ultimately precise whole numbers. The F&D Series are both based upon feedback. However the inherently imprecise decimalized rational numbers provide the feedback for the F Series, while the inherently exact whole numbers provide the feedback for the D Series. While the members of the F Series provide the feedback material and the approximation, the members of the D Series provide the feedback material and the ratio of consecutive members provides the approximation. Hence the D Series itself is independent of ratio, while the ratio of its consecutive members provides the approximation of the Root. The only human limit for the D Series is how much time they want to devote to calculating a Root, which a simple hand calculator could compute in a few seconds, at most.
In Summary, the members and roots of the X series are exact, while the results are unreal. The members and roots of the F Series are inexact, while the results are real. While the Roots of the D Series are inexact, the members of the series are exact, and the results are real.
However as we are of scientific bent, we need a hypothesis to test our mental conclusions. If the F Series is fed upon imprecise fractional corrupted computer data and the D Series is fed upon exact whole numbers, then one would predict that the D Series would approach the Root more quickly, while the F Series would drift off.
To test our hypothesis we set up a spreadsheet program, which we assumed would establish for once and all the superiority of the whole number D Series over the imprecise fractional F Series in determining our Root. We used the F&D Series for the 3rd, 4th and 5th root to compare their performance with hundreds of iterations. Surprise of surprises. The D Series did not out-perform the F Series. While they both approached the Root at a pretty similar rate, many times the F Series approached the Root more quickly.
What happened to all of our mental logic? The first misconception surrounded the precision of whole numbers on the computer. While the computer can express most whole numbers precisely with its computer chips, it must round off numbers that demand over 16 places of precision. Thus the computer limit for the D Series is how many decimal places it can ‘remember’ before beginning to round off. Although whole numbers are exact and fractions are approximations, practically speaking whole numbers with over 16-digit accuracy are also an approximation. The F Series is based upon 16 digits of accuracy just as is the D Series. Because of the feedback nature of the D Series, it quickly reaches the limits of computer accuracy, forcing rounding off. This is especially true when finding the Roots of larger numbers or when finding the higher Roots of any number. Thus the 'real' accuracy of our beloved whole numbers evaporates under the pressure of actual computation.
While the elements of the D Series are 'really' accurate, they are not accurate 'practically'. Their elements are continually rounded off at a certain point. While the elements of the F Series are not ‘really’ accurate, they are accurate 'practically', in the sense that their elements are the approximate Ratio. While the elements of the X Series are exactly accurate, they only exist as pure forms in the imagination.
Both of our iterated series are based upon corrupted data. Therefore with increasing iterations they should both be drifting ever more away from the True Root. However the reverse was the case. Both the F & D Series, even though based upon the imprecise computer approximations, headed inexorably to the Real Root regardless of the number of iterations based upon this corrupted data. Why?
Both our F&D Series are self-correcting. Just as the starting point has very little effect upon our equation that approaches the Root with increasing iterations, similarly the midpoints also have very little effect. Basically the pattern is so overriding that the closer the answer is approximated, the closer one gets more quickly. Thus even if there are inaccuracies and built-in imprecision, the pattern is never led astray from her ultimate goal. With increasing repetitions the pattern seeks out the Root like a heat seeking missile. Although both the F& D Series must approximate their elements at a certain point because of the limits of any computer, the patterned feedback reaction in the equation always drives us to the Root. Thus another aspect of the F&D Series is the fact that they are self correcting. They are self-correcting iterative equations. Soon we will analyze these iterative equations in more detail to see which factors determine the Pattern that drives our equations inexorably towards the Root. We will see how important the binomial coefficients are in determining this dominant pattern.