This treatise is a mathematical study of the Iterative Equations derived from infinite continued fractions. The 1st Notebook derives some general iterative equations which compute square roots; the 2nd Notebook derives the general iterative equations which compute any root. Then something unusual happened. We, me and all my selves, discovered that these Root Beings exist as a mathematical model in the Quest for Mastery. This is the topic of the 3rd Notebook. Under the same spotlight the 4th Notebook examines the fractal nature of π and other patterned transcendentals, bringing all of these diverse numbers into the same family, hence the name of the book, **The Internal Structure of Numbers & Root Beings**.

### 21. Square Root Family

0. General statements about the structure of numbers

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

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

3. The Infinite Continued Fraction and Denominator Series for √a/b

4. More on the Denominator Series

5. More on the Infinite Continued Fraction Series

6. The Slope of the implied Line of the Difference Series

7. The Complex Spiral of the Imaginary World

### 22. Higher Roots

0. Context for Exploration

1. Cube Root Algebra

2. Fourth Root Algebra

3. General Root Algebra

4. Binomializing the F&D Series

Summary & Conclusions

### 23. Root Beings

1. Laying Foundations

2. Stressed out equations

3. Pictures of Root Beings

4. The Quest Metaphor

### 24. Leftovers

1. The General Inverse Function

2. Dead Ends

3. Organization of the Mandelbrot Set

4. The Inbetween Fractional Root

5. Properties of Binomial Root System

6. Fractal π and other Patterned Transcendentals

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