# Lesson 8 Complex Computations Practice Understanding

## Jump Start

### 1.

Which vector is the additive inverse of

### 2.

Which vector has a magnitude of

### 3.

Which vector represents the sum of

### 4.

Draw the vector that represents

### 5.

Draw the vector that represents

## Learning Focus

Graph complex numbers in the complex plane.

Use vectors to add, subtract, and multiply complex numbers.

Divide complex numbers.

Find the distance and the midpoint between two complex numbers.

How can we model the arithmetic of complex numbers on the complex plane? What do these models reveal?

How can we visualize the size of a complex number?

## Open Up the Math: Launch, Explore, Discuss

It is helpful to illustrate the arithmetic of complex numbers using a visual representation. To do so, we will introduce the complex plane.

As shown in the figure, the complex plane consists of a horizontal axis representing the set of real numbers and a vertical axis representing the set of imaginary numbers. Since a complex number

**The Modulus of a Complex Number**

It is often useful to be able to compare the magnitude of two different numbers. For example, collecting

In a similar way, we can compare the relative magnitudes of complex numbers by determining how far they lie away from the origin, the point

### 1.

Find the modulus of each of the complex numbers shown in the figure above.

### 2.

State a rule, either in words or using algebraic notation, for finding the modulus of any complex number

**Adding and subtracting complex numbers**

### 3.

Experiment with the vector representation of complex numbers to develop and justify an algebraic rule for adding two complex numbers:

### 4.

How would you represent the additive inverse of a complex number on the complex plane? How would you represent the additive inverse algebraically?

### 5.

If we think of subtraction as adding the additive inverse of a number, use the vector representation of complex numbers to develop and justify an algebraic rule for subtracting two complex numbers:

**Multiplying complex numbers**

One way to think about multiplication on the complex plane is to treat the first factor in the multiplication as a scale factor.

### 6.

Provide a few examples of multiplying a complex number by a real number scale factor:

### 7.

Provide a few examples of multiplying a complex number by an imaginary scale factor:

### 8.

Experiment with the vector representation of complex numbers to justify the following rule for multiplying complex numbers:

### 9.

How do the geometric observations you made in problem 6, problem 7, and problem 3 show up in this work?

**The conjugate of a complex number**

The conjugate of a complex number

### 10.

Illustrate an example of a complex number and its conjugate in the complex plane using vector representations.

### 11.

Illustrate finding the sum of a complex number and its conjugate in the complex plane using vector representations.

### 12.

Illustrate finding the product of a complex number and its conjugate in the complex plane using vector representations. (Use the geometric observations you made in problems 6–8 to guide your work.)

### 13.

If

### 14.

Use either a geometric or algebraic argument to complete and justify the following statements for any complex number

The sum of a complex number and its conjugate is always the real number:

The product of a complex number and its conjugate is always the real number:

**The division of complex numbers**

Dividing a complex number by a real number is the same as multiplying the complex number by the multiplicative inverse of the divisor. That is,

We have also observed that multiplying a complex number by its conjugate always gives us a real number result. We make use of this fact to change a problem involving division by a complex number into an equivalent problem in which the divisor is a real number.

### 15.

Explain why

### 16.

Use this idea to find the quotient

We have been using a vector representation of complex numbers in the complex plane in the previous problems. In the following problems, we will represent complex numbers as points in the complex plane.

**Finding the distance between two complex numbers**

To find the distance between two points on a real number line, we find the absolute value of the difference between their coordinates. (Illustrate this idea with a couple of examples.)

In a similar way, we define the distance between two complex numbers in the complex plane as the modulus of the difference between them.

### 17.

Find the distance between the two complex numbers plotted on the complex plane provided.

**Finding the average of two complex numbers**

The average of two real numbers is located at the midpoint of the segment connecting the two real numbers on the real number line. (Illustrate this idea with a couple of examples.)

In a similar way, we define the average of two complex numbers to be the midpoint of the segment connecting the two complex numbers in the complex plane.

### 18.

Find the average of the two complex numbers plotted on the complex plane provided.

## Ready for More?

Here’s a challenge: Write an algebraic formula for dividing complex numbers.

## Takeaways

Adding complex numbers with vectors:

Subtracting complex numbers with vectors:

Multiplying complex numbers with vectors:

Dividing complex numbers:

The distance between complex numbers

## Vocabulary

- complex plane
- inverse: additive, multiplicative
- modulus
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson we wrote formulas and used vectors to justify the basic operations on complex numbers. We represented complex numbers as vectors and points on the complex plane. Vector representation provided a way to examine the size of a complex number, called the modulus. We learned to divide complex numbers, to find the distance between two complex numbers, and the average of two complex numbers.

Use the graph to write the equation of the line.

### 1.

### 2.

### 3.

Identify the best strategy for solving each equation. Then solve it.