To say that the understanding and teaching of calculus has not changed much since its formulation by Liebnitz and Newton, over three centuries ago, may be overkill. Then, again, it may not be! Upon reviewing all of the calculus books on the shelf in any large university library, the astute observer will become aware just how staid the associated pedagogy actually is! Not only will they conclude that, the subject matter which was developed in the late seventeenth/early eighteenth centuries still loudly resonates today, but also that the focus of the respective past and present authors has always been related to HOW, in contradistinction to WHY. Except for the inclusion of computer graphics, which assists in visualizing selected applications, there has been a scarcity of new ideas that would assist comprehending the foundation upon which the underlying mathematics has been built. IFF1 the subject matter was readily understood, this would not be a liability. TO THE CONTRARY, most otherwise educated persons consider the very word “calculus” as being synonymous with “way beyond comprehension”.
The perspective that is herewith being promoted is predicated on showing that the understanding and use of calculus should be no more difficult than was the mastering of the earlier studied subsets of mathematics, namely arithmetic and high school level introductions to algebra and geometry. In this treatise it shall be demonstrated that, when properly presented, the fundamentals of calculus are the very same ideas that were presented in those earlier more intuitive, and consequently deemed to be more elementary, studies. The only major difference is that they are now being adjoined with a single, supposedly already familiar, concept: namely, that there is no last number in the counting scheme ─ an idea that will be re-enforced in the succeeding pages. In the same manner as the recognition that a number called “zero” was a useful concept when it was postulated a little over a millennium ago, a similar useful number called “infinity” underlies the foundation for the development of a comparable extension of the number system. In fact, these two numbers plus the most primitive of all numbers, one, forms a set whose various binary combinations extend arithmetic into the domain of “calculus”.
At this point, in order to counter the prevalent mindset that has been associated with calculus ever since its earliest development, the reader’s attention is directed to one of the many muses who influenced the writing of this monograph: An important fundamental classic of mathematics literature, that nearly all educated mathematicians, along with most physical and biological scientists and engineers of all fields, as well as economists and social historians, and anyone interested in understanding either the evolution or the implementation of mathematical thought is a monograph by Richard Courant and Herbert Robbins entitled What is Mathematics? Excerpting from that treatise, which was first published in 1941 and is still in print and in wide circulation nearly three-quarters of a century later, one encounters the following passages which describes precisely what this author hopes to accomplish in his limited approach to the understanding of that very important, but sadly misunderstood, branch of mathematics referred to as “calculus”.
“For more than two thousand years some familiarity with mathematics has been regarded as an indispensible part of the intellectual equipment of every cultured person. . . . Teachers, students, and the educated public demand constructive reform . . . It is possible to proceed on a straight road from the very elements to vantage points from which the substance and driving forces of modern mathematics can be surveyed. . . . The present book is an attempt in this direction. . . . It requires a certain degree of intellectual maturity and a willingness to do some thinking on one’s own. This book is written for beginners and scholars, for students and teachers, for philosophers and engineers, for class rooms and libraries.”
Our approach begins by noting that calculus has been, and continues to be, taught as though it was a whole new subdivision of mathematics, with a unique protocol that required the concomitant “mastering” of rules associated with a new discipline. Instead of pursuing such an ossified path by solving a given set of arcane problems using an even more arcane set of previously developed “algorithms” (i.e., memorization), this author follows a protocol which historically had been deliberately rejected; namely, the domain of calculus is viewed as an extension of what has come to be called “elementary algebra”2,3,4. This is equivalent to asserting that the unthinking application of many of the rules established in traditional arithmetic/algebra courses need to be rejected. Such a proposition is especially true when the limits on these rules are breached. In particular, we focus our attention on the eight possible binary combinations of three “tripartite numbers” (0, 1 and ∞)5 as they impact the six fundamental operations of arithmetic (addition, subtraction, multiplication, division, raising to a power, and extracting a root). In the process we shall discover that our proposed term “tripartite number” bears an important relationship to the philosophical ideal of “none”, “some”, and “all”, along with its place in symbolic logic and in probability/statistics. Furthermore, we shall conclude that the (so named) ‘Fundamental Theorem of Calculus’ is not-so-fundamental, or, more accurately, that it is insignificant; not even worth the appellation of a “corollary”, no less a “theorem”! One would not be amiss in concluding that the philosophy espoused comes directly from the Gilbert and Sullivan operetta H.M.S. Pinafore: “Never mind the Why and Wherefore!”
In the evolution of our perspective, we probe both the denotation and the connotation of that mathematically definable ‘term’ number, along with three related concepts of importance:
- extension from the primitive idea of counting;
- the ideal of repetitive performance of an arithmetic operation
- the ideal of “inverses”; i.e., the undoing of an arithmetic operation;
In practice, one notes that addition is merely a means of counting from a different starting point than from the heuristic concept of “none”. Additionally, in retrospect, one observes that it was not until a little more than a millennium ago, that the foundation of mathematics was shifted from its emphasis on what we today relegate to the subdomain of “geometry” to a new paradigm with the independent postulation of positional notation by the Hindu and the Arabic civilizations, thereby replacing the then prevalent concept of an additive numbering protocol. With the elevation of that amorphous concept of “none” as a “number”, denoted by “zero”, this new paradigm accentuated the formalization of counting in the reversed direction; i.e., “subtraction”. This was then followed by repetitive operations of addition and subtraction, which were designated as “multiplication” and “division” respectively and by a second repetition of these two processes to yield “exponentiation” (alternately referred to as “raising to a power”) and root extraction. This set of six “elementary” operations is the foundation, not only for “numbers”, but also for many of the more sophisticated concepts in mathematics, which comprise advanced domains of both understanding and of manipulation, calculus being one of them.
At this time, one further expands their purview from the intuitive concept of none as the number zero to the intuitive concept of all as another mathematical term, loosely called “infinity” in the common domain, which shall be more accurately defined later in this text; and thirdly to the word “some”, with its implication of “one or more”. To redirect this latter word into being a term we delimit its scope so that it has the denotation of precisely 1 in the following development of the foundations of calculus. That this can be done without loss of generality is illustrated in Section 1.2. where one of the arbitrary choices in the process of establishing a measuring system is the choice of a unit reference. The judicious selection of such a reference simplifies the process of computation..
Our entire new premise (hopefully a new paradigm) as to “WHAT IS CALCULUS?” begins by focusing attention on the fifty-four permutations of the eight possible binary combinations of the above delineated six fundamental operations of arithmetic. This is then followed by the recognition that such a protocol gives rise not only to the seven traditional “l’Hospital indeterminate forms”6, but to a larger set of similarly-related indeterminate forms. Such a development is promoted to center stage early in Chapter 2. Only a selected re-examination7 of pre-calculus material8 in Chapter 1, and on the concepts of limits and continuity in the first section in Chapter 2 has been given priority over what we interpret as the fundamental subject matter of calculus.
We further assert that the concept of infinity is an important number. YES! NUMBER! (even though, to some, it may have “unfavorable” social or political associations). At this point one is reminded of the begrudging acceptance, over a millennium ago, of a mathematical term to denote “nothing”. It was that introduction, and the acceptance, of zero as such a number which caused the paradigm shift which facilitated mathematics entering the public psyche. It is important to be aware that in the previous prevailing counting systems, exemplified by Roman numerals, there was (and remains today) no symbol for nothing. To the contrary, numbers were “naturally” limited to the set of positive integers, which were designated as “natural numbers”. With the establishment of zero as a number, arithmetic entered the domain of the masses, rather than being confined to an esoteric pursuit of a, mostly religious, elite. Additionally, it fostered new developments in algebra and analysis.
We herewith sponsor the introduction of a similar perspective concerning infinity, examining its influence on mathematics in general, and on calculus in particular. This is done by focusing attention on a so-called “Eleventh Commandment”:
Thou shalt not divide by zero!
This, admittedly irreverent, term which had been satirically proposed by some unrenowned anti-clerical mathematician, spawned the even more infamous Twelfth Commandment. A treatise “Hegel, Haeckel, Kossuth and the Twelfth Commandment” satirized one earlier and two contemporary “would be intellectuals”. That treatise10 avowed that ever since Moses brought the Ten Commandments from Mount Sinai, society has sought an eleventh. Whether there is such a commandment is speculation; HOWEVER, just in case there does exist such an eleventh commandment, here is a twelfth.11,12 It should be regarded equally as important as the original ten:
Thou shalt not set thyself up as an expert in a field thou doeth not understand!
Chowolson disdained anyone venturing too far afield without the prescribed, authenticated knowledge background. He avowed a “genius” in one field 13 might well be an “idiot” in another. Moreover, even articles published in highly prestigious journals do not guarantee erudition.14
The problem with such a premise, however, is its unstated assumption: that the previous “geniuses”, those who established the paradigms that we believe today, are infallible. Were this to be true, all of mankind would have reached the end of discovering new knowledge and all progress would be impossible.15 To the contrary, there is no indelible line that should NOT be crossed. We often need to suffer “innovators” (translation = fools) graciously.
In our present treatise, we knowingly accept the accusation of non-conformity. However, just as truth refutes liable, we dismiss the premise that only recognized authorities have the prerogative of advancing new ideas; i.e., that all others stand accused of heresy and of violating the Twelfth Commandment.
To the contrary, we challenge the historical method of teaching calculus. We place our emphasis on a re-examination of what are called “indeterminate” forms. Such forms emanate from performing algebraic manipulations “in the wrong place” or ”at the wrong time”. We further assert NEVER say “Thou shalt NOT .... . “ to anything in mathematics or science. This is in contradistinction to religion — which is based on faith. We espouse probing the consequences that result from violations of traditional protocol. We further champion Lakatos’s technique16 and examine how violations can be evaded. In our approach, meaning is given under the appropriate conditions to otherwise indeterminate forms: For example, division by zero, when performed blindly, leads to ridiculous statements. However, with the appropriate restraints, such division leads to calculus and transfinite numbers. Each of these different protocols adds substantially to both our communal and our individual understanding of logic, mathematics and science.
Similarly, we re-examine the role played by infinity in the developing of mathematical processes. The inverse operation of “anti-differentiation” is envisioned as adding meaning to “infinity multiplied by zero”. A third and fourth of these indeterminate forms (infinity minus infinity and one raised to the infinite power) underlie a system of “natural” logarithms. The base of this logarithm system is a fundamental constant designated by the letter (e).
Moreover, when these conditions are not met, mathematical chaos results. The worst fears, expressed by the computer slang term GIGO (Garbage In yields Garbage Out) are encountered. This includes familiar nonsense such as a “proof” that: 2 = 1. We shall return to this quote-proof-unquote in Chapter 1, Section 3. Were such an equation to be true, all of mathematics would be useless!
Notwithstanding what many traditionalists will undoubtedly view as the admitted bias of this author, we place a high priority on refuting over-simplified explanations that are fundamentally flawed! In order to advantageously use the principles being here espoused as “calculus”, we redevelop, from basic principles, and thus expand, that set of combinations which are known as the “l’Hospital indeterminate forms”. Additionally, in Chapter 2, we shall demonstrate the relationship of these forms to an even more fundamental mathematical foundation.
Reiterating our obsession with over-simplified explanations, we avow that, the desirability of simplicity SHOULD NEVER BE at the expense of accuracy. Knowing that the reader will lack the depth of knowledge to be able to raise an objection is insufficient justification for “taking short-cuts”. To the contrary, an author knowing that his audience is unprepared for a “better” explanation MUST:
- a). supply the missing background whenever it is not too advanced
- b). acknowledge this limitation and advise what future study is desired.
Two important ideas that permeate this book are:
Their’s not to make reply.
- “Half right” answers are much worse than wrong answers! This is notwithstanding that such answers can be found in many, often highly regarded, textbooks, as well as, unfortunately, other mathematics and science publications. Intellectual honesty demands that when better explanations are available they need to be acknowledged. A problem area should never be “swept under the rug”. We disdain placing blinders on the advanced student and feel honor-bound to discuss (a) “what is the source of the problem?’ and (b) ‘what are our limitations?’
- As well as the “algebraic logic” which underlies calculus, this text stresses “geometric logic”. For example, consider the ‘slices’ vs. ‘shells’ method of determining volumes of revolution. We object to the traditional “Charge of the Light Brigade” mentality:
Their’s not to reason why.
Their’s but to do, and to die.
Instead of just dumping the traditional integration formulas onto the student, the discussion of this technique is framed as part of a two parameter system. Now, by differentiating the formula for the volume of a cylinder
V = π r2h
by each of its independent variables r and h, we obtain the resultant formulas referred to as the “method of ‘shells’ vs. ‘slices’”. In other words, this pedagogic “trick” simplifies the remembering and correct usage of the appropriate formulas, which is especially useful when applied to “moving” an axis of rotation.
Of far more geometric importance is a major mistreatment common in vector arithmetic. Our perception “What a Vector is” is CONTRARY TO WHAT MOST CALCULUS TEXTBOOKS TEACH. Because we never want the advanced student to have to “unlearn” mathematically wrong ideas, we vociferously object to regarding the cross product as just another vector. This would be the scenario for those continuing advanced studies in electrodynamics if the traditional description of a vector as “a quantity having magnitude and direction” was employed. Such a misconception in the late 1800s helped to reinforce belief in the ether. We avoid this physics inaccuracy by giving a more accurate definition for that quantity called a “vector”. Because a physically accurate presentation MUST allow for all three metric motions: translation, rotation AND reflection of the coordinate axes, we examine some properties of a more complex form of mathematical structure called a “dyadic” and introduce a more advanced (third) type of vector multiplication. This is then combined with a geometric interpretation of tensor analysis. Only then are we able to fulfill our stated goal which is to present a mathematically correct presentation of the foundation of vector algebra and calculus, while still maintaining our description at the student’s present level of understanding. In Chapter 6, this subject matter is developed along with an examination of the problems associated with attempts at formulating a system for vector division. The background material is a series of published research articles by this author1.
The historical development of another subdomain of calculus, that this author finds undesirable, involves the traditional assigning of names to a set of functions based on circular trigonometry, but then this is inconsistently followed by giving the formal definitions of these functions in terms of exponential functions. Such a protocol to the questioning student will appear to be equivalent to drawing such functions and their associated names out of a hat. This not to say that such a protocol is mathematically wrong, We object precisely because, similar to the above discussion of vectors, there is a familiar, consistent, viable analogy to the trigonometry that was learned earlier. Consequently, we are able to define these new function using a reference hyperbola in the same manner as their similar named functions were defined using a reference circle; i.e., we define the “hyperbolic trigonometric functions” in terms of their lengths in a reference hyperbola. Then, in a later chapter we show that the correlation to the exponential functions is a derived relation, rather than being the basis for their definition.
Meanwhile, although we shall examine all the traditional properties, our presentation is often unorthodox. For example, consider the algebra “rule” for multiplication by a negative number: Reverse the direction of the original inequality. We treat this NOT AS A RULE, but rather as a logical protocol. One merely adds a common term to both sides of an inequality (see Chapter 1). In other words, this result is an application of arithmetic, NOT some memorized rule. Similarly, consider “the chain rule”, which is traditionally taught in calculus courses. Note this is NOT some “erudition” discovered by an astute mathematician. Neither is it the combined genius promulgated by an august assembly of learned mathematicians. TO THE CONTRARY, it is merely simple algebra: Multiplying by one in a convenient form. Likewise, in many instances, differentials supply an intuitively simpler technique than derivatives. We recommend using them when performing implicit differentiation. This is in contradistinction to the perspective espoused in traditional textbooks. There, the focus is almost exclusively on derivatives. We, on the other hand regard a derivative as a secondary concept that is merely the division of two differentials (i.e., 0 divided by 0). This must be done, however, UNDER THE APPROPRIATE CONDITIONS. See Chapter 2. Similarly, an integral is EXISTENTIALLY the summation of an infinite number of differentials (i.e., ∞ multiplied by 0). Again this must be done UNDER THE APPROPRIATE CONDITIONS. See Chapter 3.
A further comment of importance is that, when evaluating the rigorous development of any idea, the bias of the writer should always be taken into consideration. This author regards many developments as being too esoteric for the applied student of science or engineering.19 We often will take a more mundane approach20, when we believe that it is NOT mathematically wrong. For example, in Chapter 2, we focus on that order of infinity associated with “countability”.21 This is unabashedly described as “larger than any number you can count to”. Likewise, in Chapter 7, in the discussion of sequences and series, we introduce that “order of infinity” associated with the continuum22, as well as including a superficial extension to what are referred to as “the higher orders of infinity”. In a similar manner, we include an “applied” definition of “function” in Section 1.1.23. This is in contradistinction to developing one or more of the various “pure” mathematics definitions. Such definitions are more remote from the heuristics that scientists and engineers utilize.
Two personal comments that are herewith interjected are:
- I have never once failed a student in calculus for not knowing calculus. I have, however, failed many students for not knowing algebra (or often arithmetic). Without such background knowledge, the ideas introduced in the calculus class COULD NOT make sense. This is in contradistinction to other subjects in which the student may excel. There are no prerequisite courses in the study of a Shakespearean historical play. No knowledge of English history is required for an essay on Richard III. The author supplied everything you need to know to write your essay. This is NOT to say that such historical knowledge would, or should, go unused. In mathematics, on the other hand, one does not have the ability to do addition without having mastering counting. Likewise, it is extremely unlikely to master subtraction without knowing both addition and counting. This same principle applies to each new level studied. Competence in division requires proficiency in multiplication and subtraction. This cascades to imply the above mentioned competency in addition and counting, etc. Demand for proficiency in algebra plus arithmetic frustrates many calculus students.
- This proposal is especially, but not exclusively, useful to those teachers of calculus who have a progressive student body and an administration willing to allow for experimentation with new ideas. Many years ago, I had just such a combination when I taught geometry to tenth grade students at an elite private school in New York City. My specific “would-be” paradigm shift was to integrate traditional plane and solid geometry into a proposed radical textbook: “Geometry – From ‘Gee’ Through ‘Why’”. For example, after developing the properties of triangles (call them “2-simplexes”; i.e., the simplest figures in two dimensional space), my syllabus continued to the corresponding ideas related to tetrahedra (i.e., “3-simplexes”) and even dared to superficially venture into hyperspace (i.e., “n-space”) BEFORE studying the various important four sided planar figures. This protocol worked magnificently for the top track of an ability segregated student body, especially those who were intellectually mature enough to welcome new ideas; while being a disaster for lower tracks. In retrospect, if my protocol had been applied to the appropriate audience, it most probably would have worked. However, for the environment in which this experiment was done, it was political naivety. The competition not only for the initial assignment to the highest possible track, but more importantly for being allowed to continue in successive semesters to remain in such a track created an unremitting environment that is akin to a caste system. Although such a protocol could be countermanded by a highly flexible mobility between the different tracks (depending on how the individual students perform in each succeeding module), it was rightly regarded as having toxic consequences. Most important of these was the psychological consequence of the need to move those students who were evaluated as being at the bottom of one track down to the next lower track, in order to make room for those at the top of that lower track to move up. Although this can be easily done in an authoritarian system, where the administration has the power to dictate, through the awarding of grades, and of class assignment, it is much more difficult in a democracy where any value judgments may be viewed as subjective, and where “objective values” are a legalistic fiction that is, at best, arbitrary and capricious. The degree of difficulty in achieving “fairness” (whatever that means) is even more difficult in a private high tuition school where the parents, and not the state, are paying the bill.
On the other hand, such a protocol can achieve its educational objective in a democracy when the attendees are self-selected and where the only presiding judge is the marketplace in which all who can hope to more successfully compete are free to do so. This is precisely where Courant and Robbins’ treatise, “What Is Mathematics?”, now regarded as an outstanding classic text, earned its place in mathematical literature. We hope to see a similar paradigm shift in the understanding and teaching of calculus. If NOT by us, then let it fall to others who produce a better product.
1IFF is NOT a typographical error, but rather a common abbreviation in mathematics denoting the two propositions IF and ONLY IF simultaneously.
2The term3 “algebra” refers to either one of two subdivisions of mathematics, depending on the context in which the word appears: Traditionally, and most often colloquially, this term, which often is accompanied by the adjective “elementary”, denotes the mathematical subject matter that follows arithmetic in the curriculum taught in most junior high schools, At higher levels of education (college and graduate school), on the other hand, with or without the adjective “modern”, a domain that includes more advanced topics, such as matrices, tensors, rings, fields, etc. is frequently implied .
3An important distinction between the nouns “term” vs. “word” shall be maintained throughout this treatise: “Term” shall be reserved for a precise denotation by professionals in that field of study. This is in contradistinction to “word” which is in common usage by lay persons and which allows for much less precision in its connotation. Moreover, remember “precision” is a matter of mathematics vs. “accuracy” which is dependent on the science. A further important comment, notwithstanding that many may consider it to be “tangential”, is the role played by “orismology” 4 (the science of defining words) both in the development of calculus as well as in the earlier evolution of our concepts of arithmetic, especially in the computer age. Remember that a digital computer, unlike the human mind, has as its algorithmic base of computation a much more efficient (BUT SIMULTANEOUSLY MIND-NUMBING) numbering system with only two distinct numbers (zero and one), rather than on the biological ‘accident’ that the human species has ten digits to count on.
4The following passage, which this author published in an earlier treatise “A New Unifying Biparametric Nomenclature that Spans all of Chemistry”, Elsevier, 2004, describes both the denotation and the connotation of the term “orismology”:
5Note that the term “tripartite number” was chosen over others who have referred to this same set of three numbers, often in a meta-mathematical context. For example, Michael Van Laanen in his dissertation “Encounters with Infinity”, Trafford Publishing, 2002 introduced this particular set of three numbers with the term “boundary numbers”. We eschew using his term inasmuch as we view the concept of boundary as being associated with other very different denotations, especially in mathematical physics.
6Although until recently most textbooks did not introduce the familiar 7 l’Hospital’s forms until late in the second semester of a traditional three semester calculus sequence, and even those more recent ones that do at the end of the first semester, they are still regarded as not much more than a glorified footnote.
7That should have been mastered in preparatory algebra, geometry and trigonometry courses.
8Some of which is here presented at a more sophisticated level.
9Note the origin of this concept in our everyday language. This is true for “numerous” other mathematics-based terms.
10Originally published in 1906 anonymously. Later acknowledged as authored by Professor O. D. Chowolson, Department of Physics, Kaiserlichen University, Saint Petersburg, Russia.
11One of my mentors12 advised this was required reading for German doctoral students in science and mathematics in pre-World War II Germany.
12Dr. Julius Friedrich Vandrey, who had been one of the German scientists at Peenemunde during World War II was the Director of Research at The Glenn L. Martin Company in the 1950’s. During that time I was a Research Engineer in his department.
13The butt of this satire was the biologist Ernst Heinrich Haeckel and philosopher Georg Wilhelm Friedrich Hegel. Haeckel had extrapolated his biological finding to predict a heat sink at the South Pole. Hegel, upon venturing into science avowed:
- The term “orismology” has been resurrected from being an arcane synonym of “terminology” to denote a study of the entire evolution of ideas inherent in a term, rather than being limited to the specific connotation of the present usage of the term - the usual meaning associated with “terminology”.
This Luddite approach to knowledge in general, and science in particular, caused Karl Marx to comment:
- “The fixed stars are a pimple on the firmament.
There can be only 7 planets.
But that doesn’t jibe with the facts!”
So much the worst for the facts.
“I found Hegel standing on his head and set him upright.!”
14“Some Observations on Haeckel’s Riddle” by H. Kossuth in “Zeitschrift für Philosophie und philosophische Kritik” 1903. This article displayed an appalling lack of understanding of elementary physics. Without Chowolson’s treatise, the only claim to fame of H. Kossuth was popular confusion with another Kossuth. Lajos Kossuth was the Hungarian poet who led the 1848 revolution against the Hapsburg monarchy. Historians blame Lajos for the failure of that uprising, due to his arrogance. It is asserted that any decent general, with a modicum of military knowledge, would have won.
15A common urban legend attributed to Charles H. Duell, Commissione of the U.S. Office of Patents in 1899 was that “everything that can be invented has been invented”. Nowadays, there are many websites about such “bad predictions” such as http//www.rinkworkks.com. These include, among other things, the comment of Thomas Watson, president of IBM, that there would be a market for at most 5 computers in the entire world.
16Imre Lakotas in his 1976 treatise: “Proofs and Refutations - The Logic of Mathematical Discovery” Cambridge University Press
http://en.wikipedia.org/wiki/Johann_Bernouilli. Bernoulli was hired by Guillaume de L'Hôpital for tutoring in mathematics. Bernoulli
and L'Hôpital signed a contract which gave l'Hôpital the right to use Bernoulli’s discoveries as he pleased. L'Hôpital authored the first textbook on
infinitesimal calculus, Analyse des Infiniment
Petits pour l'Intelligence des Lignes Courbes in 1696, which mainly consisted of the work of Bernoulli, including what is now known as L'Hôpital's rule.
International Journal of Mathematics Education in Science and Technology: (a) “Refining the Definition of Indeterminate Forms", 30(6),1999,924-928; (b) "Is Calculus Really that Different from Algebra? A More Logical Way to Understand and Teach Calculus", 29(3), 1998, 351-358; (c) "The Cross Product of Two Vectors is NOT Just Another Vector - A Major Misconception Being Perpetuated in Calculus and Vector Analysis Textbooks", 28(4), 1997, 531-543.
19This decision is purely a heuristic one.
20Some may say naïve.
21Denoted as ℵ0 in the development of transfinite numbers by Georg Cantor
22Cantor’s ℵ1, also often denoted by c
The International Dictionary of Applied Mathematics, Van Nostrand, Princeton, 1960, p.386.