Home Science Page Behavioral Dynamics Directionals Root Beings The Experiment
In the last part of this book we will explore a few related areas of mathematics; tie up a few loose ends; reach some dead-ends; and include some research upon ¹. Because of this hodgepodge of topics we call this Notebook Leftovers. The following topics will be dealt with herein:
1. First we extend the research of our square root paper looking at the Inverse Function of our Root Equations. 2. In searching for the slope of the Line of the Higher Roots, we reach some Dead Ends, which are exhibited and commented upon here. 3. Then we explore parallels between the Mandelbrot set and our Root Beings. 4. This exploration leads to an examination of the In-between Fractional Root and its significance. 5. This leads us into the unique properties of the Binomial Root System developed in these pages. 6. Finally we connect ¹ and other ÔpatternedÕ transcendentals with the infinite continued fraction system. If the Reader has any curiosity left, read on. ItÕs worth it.
1. The General Inverse FunctionA. Proof: The Slope of the Difference Series for Square Roots
B. Iterative functions are weird
C. The Inverse Limit, x’
D. What does x’ equal?
E. The Inverse Fraction Series, F’
2. Dead EndsA. The Correlation & Slope Table for 3rd and 4th Roots
B. The difference between x’ and x
C. The difference between ƒ’(N) and ƒ(N)
D. Last correspondences
E. Universal Theorem for Cube Roots?
3. Organization of the Mandelbrot SetA. The Algebra of the Mandelbrot set
B. The Geometry
C. Generating Inbetween Fractions: The Process
4. The Inbetween Fractional RootA. Inbetween Theorems
B. The F Series product, neither Inbetween Root, nor Root product
C. Reexamining the Process, its Parallels
D. Inbetween Root Theorems 1 & 2
E. Other Relations, Inbetween Root Theorems 3 & 4
5. Properties of Binomial Root SystemA. Exponentiation and Multiplication
B. Implications of these properties or Vanishing Ones
C. How the F Series fits into the concept of Number
D. Cracking the iron ball to look inside
E. Primacy of ‘one’ in the exponential system
F. Parallels and Connections with the Mandelbrot Set
G. Neutralization and Transcendence
6. π and other Patterned TranscendentalsA. Micro-History
B. A Fractal Function Set for π, or another iterative series
C. π as an infinite continued fraction, a Patterned Transcendental
D. π a Ratio
E. Summary