Simple harmonic motion (SHM) refers to a type of motion which repeats cyclically, or periodically, in time. Our motivation for study of such motion is threefold: (1) it is an important component of wave motion, (2) in reviewing the analysis of SHM we can develop a general physical plan of attack to address the more complex types of motion associated with waves (Section 4, (3) we will use the familiar results from the analysis of SHM to introduce the powerful technique of analysis using complex numbers (Section 5).