Calculus: The Logical Extension of Arithmetic

Calculus: The Logical Extension of Arithmetic

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Understanding Calculus as a Logical Extension of Arithmetic re-examines the calculus paradigm by expanding the set of ‘indeterminate forms’ espoused by l’Hôpital 320 years ago. Starting from ...
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Re-Examination of Basic Algebra and Trigonometry

Pp. 17-108 (92)

Seymour B. Elk

Abstract

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.

Affiliation:

Elk Technical Associate, New Milford, New Jersey 07646 U.S.A.