## 21. The Square Root Family

0. General statements about the structure of numbers

0. Introduction
A. Numbers: Counting & Magnitude
B. The Ideal Number

1. The Infinite Continued Fraction and Denominator Series for √2

A. 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 √a

A. 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/b

A. 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 Series

A. 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 Series

A. 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 Series

A. 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 World

A. 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