Understanding negative frequencies: A step-by-step translation of a mathematical manner of speaking

Early in the study of electrical engineering, students are introduced to the powerful concepts of Fourier series and Fourier transforms. In dealing with the complex exponential forms of these transforms, the notion of negative frequencies is introduced. Many students find negative frequencies counter-intuitive and confusing. The article presents the reasoning behind the engineering term `negative frequency´, and how negative frequencies are perceived and used by electrical engineers.