audio read-through Introduction to Complex Numbers

Complex numbers were introduced in a prior lesson:  Complex Numbers. Quickly read through this earlier lesson and do plenty of the online exercises to make sure that you understand all the concepts there.

Knowledge of the complex numbers is needed to fully understand the Fundamental Theorem of Algebra in an upcoming lesson.

For your convenience, this current lesson gives an in-a-nutshell review of complex numbers and related concepts.

Review: Complex Numbers and Related Concepts

By definition, the complex numbers are numbers of the form $\,a + bi\,,$ where $\,a\,$ and $\,b\,$ are real numbers, and $\,i = \sqrt{-1}\,.$ The number $\,a\,$ is called the real part of the complex number. The number $\,b\,$ is called the imaginary part of the complex number.

Note that $\,i^2 = -1\,.$ The number $\,i\,$ is not a real number, since there is no real number which, when squared, equals $\,-1\,.$

The solutions to $\,x^2 = -1\,$ are $\,\pm i\,$:

A (pure) imaginary number has real part equal to $\,0\,$: e.g., $\,0 + 7i = 7i\,.$

A real number has imaginary part equal to $\,0\,$: e.g., $\,5 + 0i = 5\,.$

The Complex Plane

The complex numbers are graphed in a coordinate plane where:

In this scheme, the complex number $\,a + bi\,$ is found as follows:

This plane is a perfect representation of the complex numbers, and is called the complex plane:

Equality of complex numbers

For all real numbers $\,a\,,$ $\,b\,,$ $\,c\,$ and $\,d\,,$

$$ \begin{gather} \cssId{s37}{a+bi=c+di}\cr \cssId{s38}{\text{if and only if}}\cr \cssId{s39}{(a=c\ \text{and}\ b = d)} \end{gather} $$ locating numbers in the complex plane

A piece of the complex plane, showing the numbers $\,2 + i\,$ and $\,-1 + 2i$

Arithmetic with complex numbers is covered in the next section.

Concept Practice