In mathematics, as in baseball, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, avoiding errors is not always a good idea. Sometimes an analysis of errors provides much deeper insights into mathematical ideas. Certain types of errors, rather than something to be eschewed, can be extremely valuable. One of the most important branches of mathematics today deals with numerical methods for rapidly approximating quantities that need to be calculated to high levels of accuracy. By definition, any approximation involves some error; otherwise, it would not be an approximation. In this article, the author looks at a situation at the precalculus level that demonstrates how an understanding of errors leads to much better approximation methods. Other articles consider a number of other instances at the calculus level--involving errors in numerical differentiation and in numerical integration--that similarly demonstrate the insights gained from an analysis of errors. (Contains 9 figures and 2 tables.)