Hi everyone! Read through the material below, watch the videos.

Lesson 21: Complex Numbers

Topic: This lesson covers Chapter 21: Complex numbers.

WeBWorK: There are four WeBWorK assignments on today’s material:Complex Numbers – Operations

Complex Numbers – Magnitude

Complex Numbers – Direction

Complex Numbers – Polar Form

**Question of the Day: **What is the square root of ?

#### Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## Review of Complex Numbers

How do we get the complex numbers? We start with the real numbers, and we throw in something that’s missing: the square root of .

**Definition 21.1**. We define the **imaginary unit** or **complex unit** to be:

The most important property of is:

**Definition 21.2**. A complex number is a number of the form .

and are allowed to be any real numbers. is called the **real part** of , and is called the **imaginary part** of . The complex numbers are referred to as (just as the real numbers are .

We can picture the complex number as the point with coordinates in the *complex plane*.

**Example 21.3**. Perform the operation.

a)

b)

c)

*VIDEO: Review of Complex Numbers* – Example 21.3

## Polar form

Next, we will look at how we can describe a complex number slightly differently – instead of giving the and coordinates, we will give a *distance* (the modulus) and *angle* (the argument). We call this the **polar form** of a complex number.

Many amazing properties of complex numbers are revealed by looking at them in **polar form**! Let’s learn how to convert a complex number into polar form, and back again.

**Definition 21.4**. Let be a complex number. The **absolute value** of , denoted by , is the distance between the point in the complex plane and the origin . By the Pythagorean Theorem, we can calculate the absolute value of as follows:

**Definition 21.6**. Let be a complex number. The coordinates in the plane can be expressed in terms of the absolute value, or **modulus**, and the angle, or **argument**, formed with the positive real axis (the -axis) as shown in the diagram:

As shown in the diagram, the coordinates and are given by:

Substituting and factoring out , we can use these to express in **polar form:**

**Polar form**:

How do we find the modulus and the argument ?

Note that is given by the absolute value. For , we note that . This leads to the following:

**Formulas for converting to polar form (finding the modulus and argument ):** ,

With regards to the modulus , we can certainly use the inverse tangent function . However, we have to be a little careful: since the arctangent only gives angles in Quadrants I and II, we need to doublecheck the quadrant of .

If is in the correct quadrant then . If not, then we add radians or to obtain the angle in the opposing quadrant: , or . You’ll see this in action in the following example.

**Example 21.7**. Convert the complex number to polar form.

a)

b)

c)

d)

*VIDEO: Converting complex numbers to polar form – Example 21.7*

**Example 21.8**. Convert the number from polar form into the standard form

a)

b)

*VIDEO: Converting complex numbers from polar form into standard form – Example 21.8*

## Multiplication and division of complex numbers in polar form

Why is polar form useful? The primary reason is that it gives us a simple way to picture how multiplication and division work in the plane. The proposition below gives the formulas, which may look complicated – but the *idea* behind them is simple, and is captured in these two slogans:

**When we multiply complex numbers:** we multiply the s and add the s.**When we divide complex numbers:** we divide the s and subtract the s

**Proposition 21.9**. Let and be two complex numbers in polar form. Then, the product and quotient of these are given by

**Example 21.10**. Multiply or divide the complex numbers, and write your answer in polar and standard form.

a)

b)

c)

d)

e) INTUITIVE BONUS: Without doing any calculation or conversion, describe where in the complex plane to find the number obtained by multiplying .

*VIDEO: Multiplication and division of complex numbers in polar form – Example 21.10 *

That’s it for today! Give the WeBWorK a try.

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