Continued Fractions, An Introduction


If r is a real number, there are many sequences of rational numbers that approach r. One sequence is defined implicitly by the decimal expansion of r. For example, π is approached by the sequence {3, 3.1, 3.14, 3.141, 3.1415, …}. But other sequences will do just as well. Students of calculus might multiply the following series by 4 to get π.

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + …

The "continued fractions" form a particular sequence of rational numbers that approach r. The terms of the sequence are constructed recursively; then one must prove that the sequence converges to r.

Continued fractions have some nice properties that prove useful in other areas of mathematics. For example, continued fractions can be used to describe the fundamental unit in a quadratic number field. Perhaps we can explore some of these relationships, but first, let's define the continued fractions for r.