Learning Resources
Outcomes
In this lesson you will learn
- the imaginary unit
- the concept of complex numbers
- to perform operations with complex numbers
By the end of this section students will be able to:
- explain the connection between real and complex numbers
- represent complex numbers in a variety of ways
- apply operations on complex numbers both in rectangular and polar form
- construct and examine graphs in the complex and polar planes
- develop and evaluate mathematical arguments and proofs
- analyze and solve polynomial, rational, irrational, absolute value and trigonometric equations
Introduction
You will, first of all, examine the relationships between various number systems. The introduction of the imaginary unit will lead to the definition of a complex number. You will then discover how to perform arithmetic operations on both imaginary and complex numbers and solve equations that have complex roots. The stage will be set for working with polar coordinates.
The main application of complex numbers is that of electricity. For example, in direct current (DC) circuits, the basic relationship between voltage and current is given by Ohm's Law. If V is the voltage across a resistance R, and 
is the current flowing through the resistor, then Ohm's Law states that V =R.
There is a similar equation for alternating currents (AC). If V is the voltage across an impedance Z, and I is the current flowing through the impedance, then V =Z.
The main difference in these equations is that the DC circuits are expressed as real numbers. Using complex numbers with the AC circuits and subsequent equations allows you to take the same simple form as the DC circuits and equations, except that all quantities are complex numbers.
Some of these relationships will be explored in this section, which should take 6 to 7 hours to complete.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- imaginary numbers
- properties of numbers
- number systems (rational, irrational, integers, whole numbers, natural numbers, real numbers)
- quadratic functions
- quadratic formula
- algebraic operations
- how to simplify radicals