Partial Fractions in Calculus, Number Theory, and Algebra

Abstract

In second-semester calculus, most students view partial fractions as either a miracle or a torture, sometimes both. Yet, students rarely understand why rational functions can always be written in terms of partial fractions. The rationale comes from abstract algebra, but a simple analogue in the natural numbers reveals many of the intricacies. If you have an interest in examining partial fractions beyond accepting their existence for use in a technique of integration, read on. We begin by recalling the general theorem on the existence of partial fractions in calculus, after which we investigate the corresponding version in the natural numbers. Then we move to a discussion of the calculus version: rational functions over the reals. Finally, we generalize the theorem to rational functions in one variable over an arbitrary field. (For a discussion of the general case of Euclidean domains, see Packard and Wilson [4].)