As a prerequisite to appreciating that domain of mathematics referred to as
“calculus”, this chapter re-examines important ideas supposedly (or maybe one should say
“hopefully”), learned in previous studies. The author’s objective in including this chapter is to
emphasize (and thus help to understand) WHY, in contradistinction to merely HOW,
algebraic operations are performed. Notwithstanding that this set of topics had been
developed in previous encounters with mathematics, they are now viewed from an advanced
viewpoint. One begins by reiterating that over a millennium ago arithmetic was simplified
by assigning a number (zero) to “nothing”; thereby causing a paradigm shift that brought
mathematics into the mainstream. A similar new paradigm shift, focused on a number that
represents the concept of “all” (in that philosophical trichotomy of none, some and all) is
herewith proposed. This role will be filled by a new number, denoted as “infinity”, which
includes the infinitely large, the infinitely small, and the infinitely dense. Having made such
an introduction, the rest of this treatise, starting with Chapter 3, examines the relationship
between the set of three “foundational” numbers (zero, one, and infinity) upon which, we
assert, the development of calculus should be formulated, and the familiar arithmetic
operations of compounding and undoing previous operations.
Keywords: Coordinate Systems (Postulates, Cartesian, Polar), Division Involving
Zero, Equations of a Straight Line (5 Common Forms), Functions, Graphing:
Plotting Points vs. Characteristic Curves, Logarithms, Measurement, Memorized
Rules: Why Selected Ones Work, Plane (Circular) Trigonometry, Spherical
Trigonometry, Symmetry.
