Calculus: The Logical Extension of Arithmetic re-examines the calculus paradigm by expanding the set of 'indeterminate forms' espoused by l'HĂŽpital 320 years ago. Starting from the 54 possible binary combinations of the foundational numbers (zero, one and infinity), a replacement for the function theory formulated earlier by Newton and Leibniz, is presented. A logical extension of the three concepts of differentiation, integration and the Naperian base number e follows this introduction, which is interpreted as 'zero divided by zero', 'infinity times zero' and 'one raised to the infinite power', respectively. The concept that a number postulated as representing 'nothing' is reinforced a useful connection to calculus theory. This treatise proposes a 'similar' number to denote and quantify 'all', in order to understand the concept of infinity. Other topics covered in this text include analytical geometry, infinite sequences and infinite series. Calculus: The Logical Extension of Arithmetic is a useful reference on advanced calculus theory for mathematics students and researchers.
Zero Divided by Zero â and the Concept of Differentiation
Abstract
This chapter begins with a brief historical introduction, wherein the seeming paradox of Achilles and the turtle is examined. Although this treatise does follow part of the tradition and progresses to the concepts of limits and continuity, including the more formal perspective of using epsilon and delta type proofs that have been the touchstone of calculusâs foundation for over three centuries, no further development of such a protocol is undertaken. In its place a very different âWeltanschauungâ (world philosophy) that focuses on what this author asserts is the appropriate underlying foundation of calculus is promulgated; namely, the relation of the concepts of none, some and all (algebraically expressed as 0, 1 and â) to the six fundamental operations of numbers (addition, subtraction, multiplication, division, raising to a power and extracting a root). From the 54 potential binary combinations of these sets, the seven traditional indeterminate lâHĂŽpital forms, as well as three additional related forms that mathematicians have missed for over three centuries are distilled. In the process, attention is focused on combinations deliberately disallowed in previous mathematics courses; especially those that arise with respect to infinity and division by zero. One particular combination, which has as its objective the determination of those extreme values that the given function can reach both globally (over all of space), and locally (in a given interval), is postulated to be the foundation upon which, provided the appropriate constraints are included, the first of the major techniques of calculus is to be built. The philosophy espoused herein views a specific related function, derived from the given function and thus named as âthe derivative of that functionâ, as the division of two, considered to be even more elementary, functions, called âdifferentialsâ. Each of these differentials, which are primarily algebraic constructs, is equivalent to having a limit value of 0. Consequently, the derivative may be viewed as giving meaning to the indeterminate form
, under a set of constraints to be designated at a later time. Meanwhile, selected other entities, which had been historically defined, such as the concept traditionally expressed as âconcavityâ, are viewed as having been relegated to the status of insignificance. This is, in contradistinction to many traditional calculus textbooks which belabor concavity as being nearly equal in importance with the extreme values of maxima and minima. The topological subtleties, often forming the basis of theoretically biased courses, are included only when they add to an intuitive understanding of the subject matter, and thus become of interest to applied scientists and engineers. Two other lâHĂŽpitalian combinations, which are similarly depicted as forming the foundation for the other two significant terms that comprise the principal domain associated with calculus will be introduced and developed in Chapters 4 and 6 respectively.
Keywords: : Arithmetic Operations Involving Infinity, Continuity, Curve Sketching, Derivatives (Definition, Poly- vs. Multi-nomials, Trig Functions), Differential Calculus, Epsilon-Delta Processes, Extrema (Maximum, Minimum, Point of Inflection), Implicit Differentiation, âLastâ Number and Interpreting Infinity: lâHĂŽpital Indeterminate Forms vs. lâHĂŽpital-Elk, Indeterminate Forms, Limits, Related Rat...
