Introduction to Proofs and Proof Strategies

In this chapter, we introduce the complex numbers system – an extension of the well-known real numbers. Complex numbers arise naturally in many problems in mathematics and science and allow us to study polynomial equations that may not have real solutions (such as x2 + 5 = 0). As we will see, many familiar algebraic properties remain valid in the complex number system. In particular, we show that the complex numbers form a field and that the quadratic formula and the triangle inequality can still be used in this new number system.