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.
It is a tragedy, since it is exactly at Elementary Algebra that things get interesting. 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. "The general theory of arithmetic operations is algebra: so we can also develop an algebra of set theory." - Concepts of Modern Mathematics, Ian Stewart 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. equations) arising from combining variables according to certain rules (i.e. high-school algebra) is, informally, the study of relationships of variables and structures (e.g. As a foundation, such a calculus might use "states" and "transitions", instead of the complex numbers, to ground the investigation.Įlementary Algebra (ie. 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. The Calculus uses the complex numbers (chiefly) as a foundation for this investigation. It is chiefly concerned with calculations or answering questions about change. 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. I think it is safe to say that the term "algebra" today, carries a bit more meaning to most mathematicians than the general teram "calculus". "Is this equation 'the same' as that equation?") Algebras tend to be used to study the relationship between objects. Calculi tend to be used to investigate properties of objects (i.e "What is the area under the curve?")Īlgebras tend to be specified/defined/explored/used to answer questions about how different "things" are related, in some very general sense. 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.Ĭalculi tend to be specified/defined/explored/used to answer questions of "calculation" or reckoning, in some very general sense.
Mathematics is an activity of investigation and exploration.
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