Learning Resources
Overview
Throughout the history of mathematics problems have arisen that could not be solved using the available number systems. This often led to the development of new numbers and number systems. First was the development of the number zero. Later, negative numbers were developed once a need for them arose. Finally, the real number system was devised and, as a result, many problems became solvable.
As you are aware, however, there are still many problems that cannot be solved using the real number system. For example, consider the equation x2 = -1. The goal here is to determine which number squared will produce an answer of -1. You're probably thinking by now that this is an impossible task. And it is, of course, under the restraints of the real number system. The arising of problems such as this led scientists to develop the complex number system.
This final chapter of study is devoted to the use of such numbers. In Mathematical Modeling, Book 3 you were introduced to the concept of imaginary numbers as you solved equations such as x2 = -4. Perhaps, though, you are wondering why they are called imaginary numbers and how can anything imaginary exist? What you have to keep in mind however, is that whether or not a concept makes sense depends on the context in which it is used.
This unit of study contains three sections as outlined below.
Section 5.1: examines the relationships between various number systems. The imaginary unit is introduced, followed by the definition of a complex number. You will learn to perform arithmetic operations on both imaginary and complex numbers. As well, you will graph these numbers on the Argand diagram and examine arithmetic operations from a graphical point of view. The associative, commutative, and distributive properties will be explored, as you make connections between them, real numbers and complex numbers. In addition, you will solve equations that have complex roots, and develop and evaluate mathematical arguments and proofs relating to complex numbers.
Section 5.2: explores polar coordinates and their use. You will realize the need for polar coordinates and see that using them is often a simpler way to represent equations. You will learn to convert equations in rectangular form into polar form and vice versa, as well as visualize polar equations.
Section 5.3: explores patterns involving relationships in operations with polar coordinates. You will investigate patterns that eventually lead to De Moivre's Theorem for powers and a quick method for multiplying polar coordinates, or complex numbers. As well, you will graph polar coordinates.
This unit of study takes approximately 15 hours to complete.