This book is a mathematical study of Iterative Equations based in the infinite continued fraction. 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.