The extent to which complex numbers arise in physics and engineering is almost beyond belief. This chapter starts by defining the complex number field ℂ. It focuses on the geometric and algebraic properties of the complex numbers themselves. One of the main points is the relationship between the complex exponential function and the trig functions. A complex number is a pair (x, y) of real numbers, or simply the point (x, y) in ℝ. By regarding this pair as a single object and introducing appropriate definitions of addition and multiplication, remarkable simplifications arise. The distance function and the triangle inequality enable us to define limits and establish their usual properties. The composition of linear fractional transformations is also a linear fractional transformation. The chapter utilize this observation by reinterpreting linear fractional transformations as matrices.