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This Notebook explores the family of numbers from the perspective of the infinite continued fraction. The infinite continued fraction allows us to compute square roots.
In the first section we derive the Infinite Continued Fraction Series, i.e. the Fraction Series, i.e. the F Series, whose limit is all positive square roots, (depending upon parameters of course). The F Series is based upon a series of fractions. We find that the numerator of these fractions is a simple function of the denominator. With this fact we also derive another series, which consists entirely of whole numbers. The ratio of the consecutive members of this series, with specified scaling, also yields the F Series whose Limits are the square roots. We call this the Denominator Series, i.e. D Series. Because whole numbers, rational numbers, and square roots can be expressed in the same way, we call this a family of numbers.
Both the F&D Series are based upon iterative equations, i.e. they are based upon continual feedback. Because of this they have an unusual symmetry. In the second section we dig deeper into the D Series finding that the ratio of its members approximates roots by going a little over and then a little under. We also find that the Denominator Series has an amazing crystalline structure. Further we see that the structure of the D Series is raveled in a power series, which illustrates the fractal nature of our family of numbers.
In the graphing of the approach of our F&D Series to their square root limit from a logarithmic perspective, we find that the approach is linear. We call the series generated by difference between the F Series and the Root, the Difference Series. Section 3 derives the slope of the approach. In so doing we discover an inverse F Series which reveals the polar properties of the infinite continued fraction family of numbers. In Section 4 we derive a Complex Spiral that imitates the Difference Series. We end this first Notebook by examining how oneÕs level of perception determines the ÔtruthÕ that one perceives.
Before looking at the new, let us ground ourselves in the old. Let us first examine numbers from the perspective of the decimal system in order to understand traditional perceptions of number.
On the surface, numbers seem quite simple. The first contact with numbers is counting. We use numbers to count things. He has five apples. This is exact. Counting is used to count unitary elements. Each unitary element always equals one. Hence counting unitary elements is exact even in the real world. When we speak about the average family containing 3.4 people, we find it amusing because we know that there is no such thing as a partial person.
After counting comes magnitude. First we count how many hands high a horse is. Or we count how many feet long the plot of ground is. We quickly learn that while the notions of magnitude and counting are related, that they are distinctly different. While counting is exact, practically speaking, magnitude is always approximate. No matter what measuring stick is used to determine magnitude there is a margin of error that is insurmountable in the real world.
When talking about ideal magnitude, each distance has a specific ideal length. If two sides of a right triangle are equal and designated as 1, then the hypotenuse has a magnitude of the Ã2. Of course we are talking in ideal terms. Our triangle is ideal. Our right angle is ideal. Our measurements are ideal. In any real world measurement there is a margin of error. There is a margin of error in the right angle. There is a margin of error in the magnitudes of the sides. There is a margin of error in the connections of the endpoints of the two sides of the right angle. Thus although we can easily imagine an ideal magnitude of the Ã2, from an ideal right triangle, from ideal sides of 1, practically speaking our magnitude of Ã2 has a great deal of error included in its measurement. So although our ideal measurement is exactly Ã2, not even plus or minus an infinitesimal point, our practical measurement is plus or minus an infinitude of points. So while ideal magnitudes of precise lengths, to the exact point, can exist in the imagination, practically speaking magnitudes are pretty fuzzy because of imprecisions in the measuring and constructing devices.
Because of this inherent imprecision in practical magnitude, we are going to shift to the concept of an ideal number disassociated from magnitude. In other words we are going to shift from a geometrical approach to number to an analytical algebraic approach. The geometric approach works well with spatial structures but falls apart when speaking of the real number line. However when talking about ideal numbers another type of imprecision emerges, which we shall soon see.
Our global culture now employs the decimal system to achieve any level of precision desired. For practical purposes the decimal system works perfectly well. However problems arise immediately when expressing the most simple of fractions, 1/3. This number cannot be represented precisely in decimal form. It must be represented as the repeating decimal 0.3333É. . While this repeating decimal will yield any desired level of practical accuracy, it will never reach 1/3. While the fractional representation is the most accurate way of representing 1/3, at least the pattern for generating 1/3 in decimal form is easily recognizable and duplicable. Every rational number, the ratio of one integer to another, can be represented in decimal form by a repeating fraction. Hence fractions based upon integers are dealt with adequately by the decimal system. Hence 0.3333É is said to equal 1/3.
However, while rational fractions are dealt with adequately by the decimal system, roots are not. Even the simplest root, the square root, yields a non-repeating decimal. It has been proved that while rational numbers yield repeating decimals, irrational numbers have non-repeating decimals. Hence Ã2 Å 1.414, ¹ Å 3.14159 are only approximations. There is no exact representation of irrational numbers in decimal form.
While the Ã2 is an exact representation of itself, it is only a symbol and reveals nothing about the internal structure of the Ã2. In a similar way ¹ is exact but only a symbol. We, of course, have complicated algorithms that will yield an approximate representation of Ã2 or ¹ in decimal form. Because the decimals are non-repeating, the algorithm must be continued forever and still no pattern emerges, no matter how long the algorithm is continued. The decimal system is an abysmal flop when it comes to irrationals. While the decimal system is a great system practically, on ideal levels it is inadequate for expressing the pattern of irrationals. Similarly a geometric analysis of irrationals yields some exact ideal representations of square roots but again reveals little of the underlying structure of irrationals.
In this paper we are going to derive a way of simply expressing both rationals and irrationals. Because of the simplicity of expression we are going to say that they belong to the same family. On the contrary the decimal system inherently divides numbers into two categories, repeating and non-repeating decimals. Rational numbers, which are repeating decimals, are divorced from irrationals and transcendental numbers, which are relegated to the family of non-repeating decimals. This paper puts these numbers in the same family by marrying repeating and non-repeating decimals into the same family. While the decimal system puts them in different families, this system joins them again.
The system that we refer to is based upon infinite continued fractions, discovered over 4 centuries ago. While the decimal system deals adequately with finite fractions, here we are speaking of infinite continued fractions. These are virtually impossible to compute in a straightforward fashion into the decimal system. Instead we derived an iterative expression, whose limit is the rational roots of any positive number, which can be easily computed. These iterative expressions become the true foundation of our system.
In Part 4, called Leftovers including ¹, we will also see that ¹, the transcendental number, can also be represented exactly in infinite continued fraction form. Infinite continued fractions can be broken further into patterned and non-patterned transcendentals. We will see that our family of infinite continued fractions easily reveals the differences between square roots, patterned transcendentals and unpatterned transcendentals. Rational numbers in this system become part of the Root family.
The infinite continued fraction and the iterative series at its foundation reveal the underlying structure of different types of numbers. The family of decimals only distinguishes between repeating and non-repeating decimals, while revealing nothing about the underlying structure of any irrational number except that it has no pattern in decimal land.
While acknowledging the possibility of many different systems of number structure we are going to focus our attentions upon the number system based upon infinite continued fractions. This system is inordinately difficult computationally; hence we are not offering it as a practical substitute for the decimal system.
One last thing. The infinite continued fraction system that we are introducing, based upon interactive feedback, is a context-based system. The traditional way of describing numbers is content-based. Dead matter is described incredibly well by content-based equations with their content-based numbers. Because life, however, responds to context, it is not very well described by content-based analysis. These contextual equations that we develop model life in fascinating ways. Part 3 of this paper, i.e. Root Beings, will focus heavily upon the metaphorical significance of these iterative expressions. Thus another justification for the development of this iterative perspective is that it models life processes in a unique way.