Introduction

We have studied numbers from the real number system in math classes up until now. We will now be introduced to a new type of numbers called imaginary numbers. Let's start by solving the following problem.

x2 + 1 = 0 Subtract 1 from each side

x2 = -1 Take the square root of each side

x = ± √(-1)

In the real number system, we cannot take the square root of a negative number. In the real number system, there is not any number that can be multiplied by itself and the product equals a negative number. If you multiply two positive numbers, the product is positive. If you multiply two negative numbers, the product is positive. Imaginary numbers allow us to take the square root of a negative number.

In imaginary numbers, √(-1) = i.

Complex numbers include all numbers. Complex numbers are written in the following form:

a + bi, where a is the real part and bi is the imaginary part.

DES login information. 

Watch the following video to learn more about complex numbers.

Lesson Objectives

Following successful completion of this lesson, students will be able to...

  • Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
  • Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
  • Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Enduring Understandings

  • A basis for the complex numbers is a number whose square is -1. Every quadratic equation has complex number solutions.
  • Real and complex numbers are important in solving and understanding polynomial equations.

The above objectives correspond with the Alabama Course of Study Algebra II and Algebra II with Trig standards: 1, 2, & 3

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