# Chapter 10.1: Complex Numbers

### Learning Objectives

In this section you will:

- Add and subtract complex numbers.
- Multiply and divide complex numbers.
- Simplify powers of .

Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.

In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.

### Expressing Square Roots of Negative Numbers as Multiples of

We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number**. **The imaginary number is defined as the square root of

So, using properties of radicals,

We can write the square root of any negative number as a multiple of Consider the square root of

We use and not because the principal root of is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written where is the real part and is the imaginary part. For example, is a complex number. So, too, is

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.

### Imaginary and Complex Numbers

A complex number is a number of the form where

- is the real part of the complex number.
- is the imaginary part of the complex number.

If then is a real number. If and is not equal to 0, the complex number is called a pure imaginary number. An imaginary number is an even root of a negative number.

### How To

**Given an imaginary number, express it in the standard form of a complex number.**

- Write as
- Express as
- Write in simplest form.

### Expressing an Imaginary Number in Standard Form

Express in standard form.

## Show Solution

In standard form, this is

### Try It

Express in standard form.

## Show Solution

### Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs where represents the coordinate for the horizontal axis and represents the coordinate for the vertical axis.

Let’s consider the number The real part of the complex number is and the imaginary part is 3. We plot the ordered pair to represent the complex number as shown in (Figure)**.**

### Complex Plane

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in (Figure).

### How To

**Given a complex number, represent its components on the complex plane.**

- Determine the real part and the imaginary part of the complex number.
- Move along the horizontal axis to show the real part of the number.
- Move parallel to the vertical axis to show the imaginary part of the number.
- Plot the point.

### Plotting a Complex Number on the Complex Plane

Plot the complex number on the complex plane.

## Show Solution

The real part of the complex number is and the imaginary part is –4. We plot the ordered pair as shown in (Figure)**.**

### Try It

Plot the complex number on the complex plane.