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Let us set up a context for this Notebook before we get too far into the abstract numbers and symbols.
Basically the algebra of this paper accomplishes a few tasks. One of those tasks is to create a family of numbers that includes the rational numbers as a subset of the irrationals.
Bear in mind that the decimal system does not include any irrational or transcendental numbers in its family. Thus while the decimal system contains the necessary approximations for any number, it is unable to represent irrationals or transcendentals. Even fractions are tackled uncomfortably in the decimal system with the simple 1/3 written as an infinite string of 3s after the decimal. However what the decimal system loses in inclusiveness it makes up for in practicality. Although the system that we will discover is more inclusive than the decimal system in that it includes all rational roots in its family (and some transcendentals, too), we will regularly refer back to the decimal system to check our work.
One of the features of our system of numbers is that of pattern.
An example of a patterned number in the decimal system is the repeating decimal to indicate a fraction or rational number. All the patterned numbers in the decimal system are based upon addition. In our example: 3/10ths + 3/100ths + 3/1000ths + (as far as you want to go) approximately equals 1/3. If any of these numerators of this series is different than 3, then it is a different number. One of the beauties of the decimal system is that each number has one and only one representation. Fractions are not like that, as any school child, who has had to reduce fractions, will tell you. Because each decimal number is unique they are compared and combined easily in a number of operations, including addition and multiplication.
Further because the series is added onto, all of the non-zero patterned numbers approaches their fractional limit from below. In other words, the approximation based upon the pattern will always be less than the goal, never above it. (Rounding up is a different process.) Most of the common systems of numbers are based upon the approach to a limit from above or below.
In the number system developed in this paper each number is a pattern, rational and irrational. While the patterns of the decimal system are based upon adding a series of numbers together, the patterns of this system are iterative, i.e. based upon feedback. These iterative numbers bounce around the desired limit, sometimes above and sometimes below, closer and closer. Within an appropriate range the number of iterations necessary to reach maximum computer accuracy is between 10 and 20.
There are two different types of iterative series that we will develop, the Fraction Series and the Denominator Series, i.e. the F & D Series. These series are used to generate our family of numbers. While the Denominator Series is derived from the Fraction Series, both series stand alone in their ability to generate iterative patterns that will yield all the rational roots as a limit.
In the terminology of iterative functions, the ‘seed’ is the number that is fed into the iterative function to get it going, while the ‘orbit’ is where the function is going after repeated iterations. In terms of our iterative functions, the ‘orbits’ of the F Series always approach a limit, while the ‘orbits’ of the D Series always approach infinity, positive or negative. Further and more remarkably, to our way of thinking, the ‘seed’ of these iterative functions has virtually no effect upon the limits that are approached, with only one notable exception. The seed of the D series can’t be zero, or it kills the series. Basically the initial point is insignificant in determining the limits or ratios of our series while the pattern is everything.
For comparison let us look at two famous sets based upon iterative functions, the Mandelbrot and Julia sets.
Both sets are based upon the same iterative function, which we call the Julia function so that it has a name. Following is the verbal description of the iterative Julia function. The new element of each iterative series is generated by squaring the last element and then adding a constant number, c, to it.
The initial element of the series is the seed. The Mandelbrot set is based upon keeping the seed constant and varying the constant, c. The Julia set is based upon keeping the constant fixed and then varying the seed. From these statements alone it is seen how important the ‘seed’ is to these iterative equations. But let us be a little more specific.
For the Mandelbrot set the ‘seed’ of the Julia function is not nearly as important as the constant that is added to it. But the seed does have an effect, acting somewhat as a translation factor. This is seen visually in the Julia set. While the seed of the Julia function is not very interesting in terms of the Mandelbrot set it does have an effect upon the result, i.e. whether the ‘orbit’ goes to infinity or not. The Mandelbrot set contains the set of constants, c, that don’t drive the ‘orbit’ of the Julia function to infinity. If the ‘orbit’ goes to infinity the number is not in the set, while if it doesn’t go to infinity then it is in the set. Remember that the number in the set refers to the constant added on, not the seed. Because the effect of the ‘seed’ is unimportant, the ‘seed’ of the Julia function for the Mandelbrot set is given as zero, for simplicity. Let us restate that while the ‘seed’ is unimportant it does have an effect. With the F&D Series the ‘seed’ is unimportant and also has no effect.
Although there is only one Mandelbrot set based upon the behavior of all the constants, when the seed is given as zero, there is an infinity of Julia sets. There is a different Julia set for each constant. The Julia set, when the constant c equals two, contains all the seeds that don’t drive the ‘orbit’ of the Julia function to infinity. Thus the Julia sets are a set of ‘seeds’ each of which does something different to the iteration. Each of the Julia sets graphs into a unique fractal pattern. The existence of Julia sets and the interest in them illustrates how important the ‘seed’ is to the iterative equation of this system. Reiterating (excuse the pun) the iterative pattern for the F&D Series is so dominant that the ‘seed’ has virtually no influence on the outcome, or ‘orbit’, of the iteration.
Further each of infinite F&D Series approaches a distinct limit, the Limit. The iterative function that drives the Mandelbrot and Julia sets, depending upon seed and constant, may or may not have an infinite limit, may or may not have multiple limits, may cycle or may not, may be random or may be ordered. In short nothing definitive can be said about the orbits of the Julia function. Further all we can say about the members of the Mandelbrot and Julia sets is that their orbits don’t go to infinity. These orbits can be indeterminate, fixed, with a cycle, or with a limit. In contrast every F&D series has a distinct and namable limit.
In summary within this Notebook we develop an iterative system where each rational root is characterized by an iterative pattern that generates the number independent of initial starting point or ‘seed’. Thus in this more inclusive system the numbers exist as patterns rather than as independent numbers. They are more process and less fixed element.
Before developing our system let us summarize the results of our previous paper on the same topics.
In our last ‘Notebook’ we examined the square roots written as infinite continued fractions. While successfully representing square roots the infinite continued fraction was next to impossible to compute. Thus we first explored an iterative function, which expressed the same idea. As the number of iterations of this function approached infinity, the value approached our square roots. We put the elements of this iterative function into a series called the Infinite Continued Fraction Series or F Series. From the F Series we generated a Denominator series, where the ratio of consecutive members of the set approached our square roots. Both the Fraction Series and the D Series included all square roots and all rational numbers within their family.
While rational numbers were included with the square roots in a common system, it was found that all the numbers of this family have a built-in imprecision which was irresolvable. This contrasts with the decimal system, where an infinity of numbers are represented exactly, for instance all of the whole numbers. With the fractional number system all of the rational numbers are represented exactly as a ratio. While fractional numbers are all exact, it is impossible to represent any number but rational numbers as a fraction. While some decimal numbers are exact, no irrationals are included in its family. Another problem with the fractional representation is that it is hard to organize the fractions into larger and smaller, while the beauty of the decimal system is that it organizes so easily from small to large. In summary, fractional numbers have the capacity to precisely represent all of the rational numbers and none of the irrational numbers, hence their name. The decimal system is able to represent many numbers exactly and is also able to represent every rational number as a patterned number. Further the decimal system is able to represent all irrationals and even the transcendentals to the level of precision required, even though these numbers are unpatterned. The number system developed in this paper is able to represent all irrationals and rationals and an infinite number of transcendentals as patterned numbers, while none of the numbers is ever precise, even the whole numbers.
While the imprecision was irresolvable, the patterns were simple. Hence the non-repeating decimal of the irrational square roots breaks into a simple pattern, similar to the infinite string of threes after the zero that signifies one third. Just as the repeating decimals allows all of the fractions to be included in the decimal family, similarly so do these patterned fractions allow all of the square roots to be included with the rational numbers in a common family.
Indeed this family of numbers is characterized by ultimate imprecision for all numbers except zero, the point of origin. Even one is only approximate. In the last paper we discussed the difference between the ultimate precision of counting numbers versus fractional numbers. We pointed out that while the rational number line is founded on the ultimately precise counting numbers, that the reality of measurement assured relative imprecision and approximations with all but the unitary counting numbers. Thus the ‘one’ of ‘one apple’ is much different than the ‘one’ of ‘one mile’. While the 'one apple' is exactly precise, the ‘one-mile' is simply an idealized measurement with external standards.
While our Denominator Series and Fraction Series, the F & D Series, included all square roots and rational numbers, they did not include the higher roots. Hence this Notebook is oriented to including the higher roots into its family. The quest in this Notebook is to include all irrational and rational numbers into a common family. At the end, we will even include transcendental numbers, such as Ļ into our family. At this point we will see a differentiation between patterned and unpatterned transcendentals. However just as the decimal system couldn’t include the irrational numbers into its scheme, we will find that unpatterned transcendentals, by definition, aren’t included in this system. Thus the ultimate thrust of this paper was to include more numbers into a common family. Purpose stated let the algebra begin.