For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry. In this article, the author draws back the curtain to expose some of the mathematics behind computational wizardry and introduces some fundamental ideas that are accessible to precalculus students and that inspire many of them to pursue mathematics. In the process, he shows how these ideas provide a wonderful opportunity to make connections between seemingly disparate mathematical threads--such as use of data analysis and trigonometric identities and the behavior of polynomials and sines and cosines--to link them and provide teachers with new ways to reinforce important concepts and methods. All that a calculator can do is addition and subtraction; thus, "every" other operation--multiplication, taking the square root, and so on--necessarily involves some very clever numerical scheme. The author considers the problem of how one approximates a function, typically a transcendental function such as the sine or the cosine, with a simpler function, usually a polynomial, to calculate values of the function. (Contains 14 figures and 3 tables.)