Teaching Partial Fractions

Students commonly encounter the method of partial fractions for the first time (without proofs) in Calculus II, as a method to aid in integrating rational functions. These days, partial fractions are sometimes not taught at all, since students can determine most any common indefinite integral by using a CAS. Without taking sides in the debate over how much methods of integration should be taught, I would like to make a case that partial fractions should be taught in high school or below.

Of course, partial fractions are a technique that comes up when discussing the algebra of rational functions. However, they also come up very naturally in arithmetic. I propose introducing them in the context of solving a problem that students might find interesting. I call this the problem of the base-p rulers.

The smallest distance measurable by an ordinary English-units ruler is 1/2^n inch, where n is typically 5 (32nds) or 6 (64ths). Define a base-2 ruler to be an idealized version of this ruler, where all coordinates of the form a/2^n are marked, where a and n are non-negative integers. It's clear that not all rational distances are measurable with such a ruler, for example 1/3 is not. To measure all rational distances, we can create an infinite number of base-p rulers, where p varies over the prime numbers. A base-p ruler has all co-ordinates of the form a/p^n, where a and n are non-negative integers. A length of length a/b can be laid with base-p rulers, provided a/b can be expressed as a sum of signed base-p numbers a/p^n. For example, the length 1/6 can be laid out by measuring 1/2, and then backing up 1/3: 1/6 = 1/2 - 1/3.

We want to have students discover that every rational number length a/b (in lowest terms) can be expressed using base-p rulers, where p varies over the primes that divide b.

Providing a proof requires some number theory. Clearly, it is necessary and sufficient to show that every number of the form 1/b can be represented in the required form, and the number theory involves finding a generalization of the fact that if (a, b) = 1 there is a solution in integers to ax + by = 1.