Home Science Page Data Stream Momentum Directionals Root Beings The Experiment
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.
0. General statements about the structure of numbersA. Numbers: Counting & Magnitude
B. The Ideal Number
1. The Infinite Continued Fraction and Denominator Series for √2A. Fraction Series for √2
B. Denominator Series for √2
C. The inverse F Series for √2
2. The Infinite Continued Fraction and Denominator Series for √aA. F&D Series for integer square roots, √a
B. Basic Iterative Theorems for integer square roots, √a
C. Generating the D Series for √a
D. The Inverse F Series for √a
3. The Infinite Continued Fraction and Denominator Series for √a/bA. F&D Series for rational square roots, √a/b
B. Square Root Theorems for √a/b
C. Summarizing the discoveries
4. More on the Denominator SeriesA. D Series foundational to the F Series
B. The Geometric Nature of the Denominator Series
C. Every D Series has a Limit
D. A More General Expression for the D Series
5. More on the Infinite Continued Fraction SeriesA. The Difference between consecutive members of the F Series
B. The Raveling of the F&D Series
6. The Slope of the implied Line of the Difference SeriesA. Difference Graphs for √2
B. The conjugates, x and x'
C. Product of Conjugates: x' * x = c = a/b - 1
D. ‘d’, Ratio of conjugates: x'/x = d or the Divine Ratio
E. The Implications of the Divine Ratio and its Graph
7. The Complex Spiral of the Imaginary WorldA. A Computer Experiment
B. Generating the Equation for the Line
C. Equation for a Complex Spiral
D. What does our equation look like?
E. Philosophy: Perspective is everything..
F. This Complex Spiral only a metaphor
G. Review: Special Effects of the Infinite Continued Fraction Family