Algebra 2 Pre Calculus
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In logic, the terminology seems to have been influenced by two factors. The very early development of various deductive systems was done by people who were more philosophers than mathematicians and who seem to have used "calculus" to refer to anything that looked mathematical. Also, that development took place before "algebra" had acquired all of its current meanings.
My impression is that the use of "calculus" in logic is restricted to the meaning of "formal deductive system" --- and usually rather old systems. As for the SKI system of combinators, I would call it a calculus if you're talking about rules of inference. But if you mean the system of all combinators, with the operation of application, generated by S and K (I is redundant), then this is an algebra.
This agrees very much with the definitions I have encountered within mathematics. Free variables are not a requirement; indeed, even variables are not strictly required as objects within a calculus. (I am aware that there exist proof calculi that don't deal with the concept of variables; perhaps someone could give the name of such a one.)
Mathematics is an activity of investigation and exploration. Informally, both calculi and an algebras are tools which consist of sets of symbols and systems of rules (usually called axioms) for manipulating those symbols.
Calculi tend to be specified/defined/explored/used to answer questions of "calculation" or reckoning, in some very general sense. Calculi tend to be used to investigate properties of objects (i.e "What is the area under the curve?")
Algebras tend to be specified/defined/explored/used to answer questions about how different "things" are related, in some very general sense. Algebras tend to be used to study the relationship between objects. (i.e. "Is this equation 'the same' as that equation?")
The Calculus (as taught in high-school or undergraduate university), also known as "infinitesimal calculus", is a calculus focused on limits, functions, derivatives, integrals, and infinite series. It is chiefly concerned with calculations or answering questions about change. The Calculus uses the complex numbers (chiefly) as a foundation for this investigation.
Opening a book on computer science, you might find a "calculus of computation" which might involve symbols and rules which let one "calculate" or "discover" behavioral properties of a computer program. As a foundation, such a calculus might use "states" and "transitions", instead of the complex numbers, to ground the investigation.
Elementary Algebra (ie. high-school algebra) is, informally, the study of relationships of variables and structures (e.g. equations) arising from combining variables according to certain rules (i.e. performing "operations"). It uses the complex numbers as the basic foundation in which one could "check" or "verify" statements, but quickly one finds that "calculating with numbers" is not that useful (or practical) in investigating relationships between equations.
In that sense, Elementary Algebra is more "abstract" than arithmetic, and is often the subject where schools (specifically bad teachers) lose a student's interest and attention in mathematics. It is a tragedy, since it is exactly at Elementary Algebra that things get interesting.
In computer science or other engineering disciplines, you might find a "process algebra" when reasoning about how various states of a computer program relate to each other. We can ask questions like "is a specification of a collection of processes 'functionally equivalent' to another specification (i.e do they do the same thing? as in the case of a particular hardware design versus a software program)? The same "process algebra" could possibly be used to reason about how the various "states" of a garage door opener relate to each. Such an algebra might use states, transitions, and time as a foundation.
a calculus is a symbolic system for computation where computation can most generally be seen to be a spatial reorganization of symbols. any kind of numerical computation can be described in terms of recursion which can fundamentally be seen as a symbolic manipulation process. logic calculi and the calculus that set theory uses is also describable as a symbolic computation system.
an algebra is a mathematical structure in the informal sense which turns out to be a vector space with the added ability to multiply the actual vectors together. the complex numbers with addition and multiplication is an algebra over the complex numbers; _over_a_field
In an algebra we would work within a known system and the results tend to be within those domains of interest and nothing new is found. In calculus the situation is different and we would encounter new things other than those we have been dealing. For example if we are working with one kind of curves and think about an operation like an integration or a differentiation we would be taken up/down another system. There is always a scope to see a new thing in a calculus and never a new thing in algebra. Whether it is a conventional mathematical system or relational algebra/calculus or logic (first order and more) we can observe this tendency.
Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.[1]
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.[2][3] Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.[4]
In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, sequent calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus.
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India.
The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle.[10][11] In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method[12][13] that would later be called Cavalieri's principle to find the volume of a sphere.
Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus.[17] Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.[18]
Significant work was a treatise, the origin being Kepler's methods,[18] written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[19] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving predecessors to the second fundamental theorem of calculus around 1670.[20][21]
The product rule and chain rule,[22] the notions of higher derivatives and Taylor series,[23] and of analytic functions[24] were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.[25]
These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton.[26] He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.[27]
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