The definition of calculus is often confusing. Like many terms used over time in mathematics (and the sciences, for that matter), there is often an overlapping of names and terms. Thus, the term “calculus” is often a generic name for any area of mathematics dealing with calculation; arithmetic could be called the “calculus of numbers.” It is also why there are such terms as imaginary calculus (a method of looking at the relationships between real or imaginary quantities using imaginary symbols and quantities in algebra) that do not mean the type of calculus discussed in this chapter.

Most mathematicians say that, in general, “a” calculus is an abstract theory developed in a purely formal way. “The” calculus is different, as it is a branch of mathematics that deals with functions; another name for the calculus is real analysis (a more archaic term is infinitesimal analysis). This type of calculus evaluates constantly changing quantities, such as velocity and acceleration; values interpreted as slopes of curves; and the area, volume, and length objects bounded by curves (remember, curves can also mean straight lines). It involves infinite processes that lead to passage to a limit, or the approaching of an ultimate, usually desired value. The tools of the calculus include differentiation (differential calculus, or finding a derivative) and integration (integral calculus, or finding the indefinite integral), both of which are foundations for mathematical analysis.