Skip to Content
OpenStax Logo
Precalculus

3.1 Complex Numbers

Precalculus3.1 Complex Numbers
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Express square roots of negative numbers as multiples of  ii.
  • Plot complex numbers on the complex plane.
  • Add and subtract complex numbers.
  • Multiply and divide complex numbers.

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. For example, we still have no solution to equations such as

x 2 +4=0 x 2 +4=0

Our best guesses might be +2 or –2. But if we test +2 in this equation, it does not work. If we test –2, it does not work. If we want to have a solution for this equation, we will have to go farther than we have so far. After all, to this point we have described the square root of a negative number as undefined. Fortunately, there is another system of numbers that provides solutions to problems such as these. In this section, we will explore this number system and how to work within it.

Expressing Square Roots of Negative Numbers as Multiples of i

We know how to find the square root of any positive real number. In a similar way, we can find the square root of a 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 i i is defined as the square root of negative 1.

1 =i 1 =i

So, using properties of radicals,

i 2 = ( 1 ) 2 =1 i 2 = ( 1 ) 2 =1

We can write the square root of any negative number as a multiple of i. i. Consider the square root of –25.

25 = 25(1)          = 25 1          =5i 25 = 25(1)          = 25 1          =5i

We use 5i 5i and not 5i 5i because the principal root of 25 25 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 a+bi a+bi where a a is the real part and bi bi is the imaginary part. For example, 5+2i 5+2i is a complex number. So, too, is 3+4 3 i . 3+4 3 i .

Showing the real and imaginary parts of 5 + 2i. In this complex number, 5 is the real part and 2i is the complex part.

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

Imaginary and Complex Numbers

A complex number is a number of the form a+bi a+bi where

  • a a is the real part of the complex number.
  • bi bi is the imaginary part of the complex number.

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

How To

Given an imaginary number, express it in standard form.

  1. Write a a as a 1 . a 1 .
  2. Express 1 1 as i. i.
  3. Write a i a i in simplest form.

Example 1

Expressing an Imaginary Number in Standard Form

Express 9 9 in standard form.

Try It #1

Express 24 24 in standard form.

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 (a,b), (a,b), where a a represents the coordinate for the horizontal axis and b b represents the coordinate for the vertical axis.

Let’s consider the number −2+3i. −2+3i. The real part of the complex number is −2 −2 and the imaginary part is 3i. 3i. We plot the ordered pair (−2,3) (−2,3) to represent the complex number −2+3i −2+3i as shown in Figure 1.

Plot of a complex number, -2 + 3i. Note that the real part (-2) is plotted on the x-axis and the imaginary part (3i) is plotted on the y-axis.
Figure 1

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 2.

The complex plane showing that the horizontal axis (in the real plane, the x-axis) is known as the real axis and the vertical axis (in the real plane, the y-axis) is known as the imaginary axis.
Figure 2

How To

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

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

Example 2

Plotting a Complex Number on the Complex Plane

Plot the complex number 34i 34i on the complex plane.

Try It #2

Plot the complex number −4i −4i on the complex plane.

Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and combine the imaginary parts.

Complex Numbers: Addition and Subtraction

Adding complex numbers:

( a+bi )+( c+di )=( a+c )+( b+d )i ( a+bi )+( c+di )=( a+c )+( b+d )i

Subtracting complex numbers:

( a+bi )( c+di )=( ac )+( bd )i ( a+bi )( c+di )=( ac )+( bd )i

How To

Given two complex numbers, find the sum or difference.

  1. Identify the real and imaginary parts of each number.
  2. Add or subtract the real parts.
  3. Add or subtract the imaginary parts.

Example 3

Adding Complex Numbers

Add 34i 34i and 2+5i. 2+5i.

Try It #3

Subtract 2+5i 2+5i from 34i. 34i.

Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

Multiplying a Complex Number by a Real Number

Let’s begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. So, for example,

Showing how distribution works for complex numbers. For 3(6+2i), 3 is multiplied to both the real and imaginary parts. So we have (3)(6)+(3)(2i) = 18 + 6i.

How To

Given a complex number and a real number, multiply to find the product.

  1. Use the distributive property.
  2. Simplify.

Example 4

Multiplying a Complex Number by a Real Number

Find the product 4(2+5i). 4(2+5i).

Try It #4

Find the product 4(2+6i). 4(2+6i).

Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or the FOIL method. Recall that FOIL is an acronym for multiplying First, Outer, Inner, and Last terms together. Using either the distributive property or the FOIL method, we get

( a+bi )( c+di )=ac+adi+bci+bd i 2 ( a+bi )( c+di )=ac+adi+bci+bd i 2

Because i 2 =1, i 2 =1, we have

( a+bi )( c+di )=ac+adi+bcibd ( a+bi )( c+di )=ac+adi+bcibd

To simplify, we combine the real parts, and we combine the imaginary parts.

( a+bi )( c+di )=( acbd )+( ad+bc )i ( a+bi )( c+di )=( acbd )+( ad+bc )i

How To

Given two complex numbers, multiply to find the product.

  1. Use the distributive property or the FOIL method.
  2. Simplify.

Example 5

Multiplying a Complex Number by a Complex Number

Multiply ( 4+3i )(25i). ( 4+3i )(25i).

Try It #5

Multiply (34i)(2+3i). (34i)(2+3i).

Dividing Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator. We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of a+bi a+bi is abi. abi.

Note that complex conjugates have a reciprocal relationship: The complex conjugate of a+bi a+bi is abi, abi, and the complex conjugate of abi abi is a+bi. a+bi. Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.

Suppose we want to divide c+di c+di by a+bi, a+bi, where neither a a nor b b equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.

c+di a+bi  where a0 and b0 c+di a+bi  where a0 and b0

Multiply the numerator and denominator by the complex conjugate of the denominator.

( c+di ) ( a+bi ) ( abi ) ( abi ) = ( c+di )( abi ) ( a+bi )( abi ) ( c+di ) ( a+bi ) ( abi ) ( abi ) = ( c+di )( abi ) ( a+bi )( abi )

Apply the distributive property.

= cacbi+adibd i 2 a 2 abi+abi b 2 i 2 = cacbi+adibd i 2 a 2 abi+abi b 2 i 2

Simplify, remembering that i 2 =−1. i 2 =−1.

= cacbi+adibd(1) a 2 abi+abi b 2 (1) = (ca+bd)+(adcb)i a 2 + b 2 = cacbi+adibd(1) a 2 abi+abi b 2 (1) = (ca+bd)+(adcb)i a 2 + b 2

The Complex Conjugate

The complex conjugate of a complex number a+bi a+bi is abi. abi. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

  • When a complex number is multiplied by its complex conjugate, the result is a real number.
  • When a complex number is added to its complex conjugate, the result is a real number.

Example 6

Finding Complex Conjugates

Find the complex conjugate of each number.

  1. 2+i 5 2+i 5
  2. 1 2 i 1 2 i

Analysis

Although we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find the complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real number from an imaginary number, we can simply multiply by i. i.

How To

Given two complex numbers, divide one by the other.

  1. Write the division problem as a fraction.
  2. Determine the complex conjugate of the denominator.
  3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
  4. Simplify.

Example 7

Dividing Complex Numbers

Divide ( 2+5i ) ( 2+5i ) by ( 4i ). ( 4i ).

Example 8

Substituting a Complex Number into a Polynomial Function

Let f(x)= x 2 5x+2. f(x)= x 2 5x+2. Evaluate f( 3+i ). f( 3+i ).

Analysis

We write f(3+i)=−5+i. f(3+i)=−5+i. Notice that the input is 3+i 3+i and the output is −5+i. −5+i.

Try It #6

Let f(x)=2 x 2 3x. f(x)=2 x 2 3x. Evaluate f( 8i ). f( 8i ).

Example 9

Substituting an Imaginary Number in a Rational Function

Let f( x )= 2+x x+3 . f( x )= 2+x x+3 . Evaluate f( 10i ). f( 10i ).

Try It #7

Let f(x)= x+1 x4 . f(x)= x+1 x4 . Evaluate f( i ). f( i ).

Simplifying Powers of i

The powers of i i are cyclic. Let’s look at what happens when we raise i i to increasing powers.

i 1 =i i 2 =1 i 3 = i 2 i=1i=i i 4 = i 3 i=ii= i 2 =(1)=1 i 5 = i 4 i=1i=i i 1 =i i 2 =1 i 3 = i 2 i=1i=i i 4 = i 3 i=ii= i 2 =(1)=1 i 5 = i 4 i=1i=i

We can see that when we get to the fifth power of i, i, it is equal to the first power. As we continue to multiply i i by itself for increasing powers, we will see a cycle of 4. Let’s examine the next 4 powers of i. i.

i 6 = i 5 i=ii= i 2 =1 i 7 = i 6 i= i 2 i= i 3 =i i 8 = i 7 i= i 3 i= i 4 =1 i 9 = i 8 i= i 4 i= i 5 =i i 6 = i 5 i=ii= i 2 =1 i 7 = i 6 i= i 2 i= i 3 =i i 8 = i 7 i= i 3 i= i 4 =1 i 9 = i 8 i= i 4 i= i 5 =i

Example 10

Simplifying Powers of i i

Evaluate i 35 . i 35 .

Q&A

Can we write i 35 i 35 in other helpful ways?

As we saw in Example 10, we reduced i 35 i 35 to i 3 i 3 by dividing the exponent by 4 and using the remainder to find the simplified form. But perhaps another factorization of i 35 i 35 may be more useful. Table 1 shows some other possible factorizations.

Factorization of i 35 i 35 i 34 i i 34 i i 33 i 2 i 33 i 2 i 31 i 4 i 31 i 4 i 19 i 16 i 19 i 16
Reduced form ( i 2 ) 17 i ( i 2 ) 17 i i 33 ( 1 ) i 33 ( 1 ) i 31 1 i 31 1 i 19 ( i 4 ) 4 i 19 ( i 4 ) 4
Simplified form ( 1 ) 17 i ( 1 ) 17 i i 33 i 33 i 31 i 31 i 19 i 19
Table 1

Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method.

Media

Access these online resources for additional instruction and practice with complex numbers.

3.1 Section Exercises

Verbal

1.

Explain how to add complex numbers.

2.

What is the basic principle in multiplication of complex numbers?

3.

Give an example to show the product of two imaginary numbers is not always imaginary.

4.

What is a characteristic of the plot of a real number in the complex plane?

Algebraic

For the following exercises, evaluate the algebraic expressions.

5.

If f(x)= x 2 +x4, If f(x)= x 2 +x4, evaluate f(2i). f(2i).

6.

If f(x)= x 3 2, If f(x)= x 3 2, evaluate f(i). f(i).

7.

If f(x)= x 2 +3x+5, If f(x)= x 2 +3x+5, evaluate f(2+i). f(2+i).

8.

If f(x)=2 x 2 +x3, If f(x)=2 x 2 +x3, evaluate f(23i). f(23i).

9.

If f(x)= x+1 2x , If f(x)= x+1 2x , evaluate f(5i). f(5i).

10.

If f(x)= 1+2x x+3 , If f(x)= 1+2x x+3 , evaluate f(4i). f(4i).

Graphical

For the following exercises, determine the number of real and nonreal solutions for each quadratic function shown.

11.
Graph of a parabola intersecting the real axis.
12.
Graph of a parabola not intersecting the real axis.

For the following exercises, plot the complex numbers on the complex plane.

13.

12i 12i

14.

2+3i 2+3i

15.

i i

16.

34i 34i

Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

17.

( 3+2i )+(53i) ( 3+2i )+(53i)

18.

( 24i )+( 1+6i ) ( 24i )+( 1+6i )

19.

( 5+3i )(6i) ( 5+3i )(6i)

20.

( 23i )(3+2i) ( 23i )(3+2i)

21.

(4+4i)(6+9i) (4+4i)(6+9i)

22.

( 2+3i )(4i) ( 2+3i )(4i)

23.

( 52i )(3i) ( 52i )(3i)

24.

( 62i )(5) ( 62i )(5)

25.

( 2+4i )( 8 ) ( 2+4i )( 8 )

26.

( 2+3i )(4i) ( 2+3i )(4i)

27.

( 1+2i )(2+3i) ( 1+2i )(2+3i)

28.

( 42i )(4+2i) ( 42i )(4+2i)

29.

( 3+4i )( 34i ) ( 3+4i )( 34i )

30.

3+4i 2 3+4i 2

31.

62i 3 62i 3

32.

5+3i 2i 5+3i 2i

33.

6+4i i 6+4i i

34.

23i 4+3i 23i 4+3i

35.

3+4i 2i 3+4i 2i

36.

2+3i 23i 2+3i 23i

37.

9 +3 16 9 +3 16

38.

4 4 25 4 4 25

39.

2+ 12 2 2+ 12 2

40.

4+ 20 2 4+ 20 2

41.

i 8 i 8

42.

i 15 i 15

43.

i 22 i 22

Technology

For the following exercises, use a calculator to help answer the questions.

44.

Evaluate (1+i) k (1+i) k for k=4, 8, and 12. k=4, 8, and 12. Predict the value if k=16. k=16.

45.

Evaluate (1i) k (1i) k for k=2, 6, and 10. k=2, 6, and 10. Predict the value if k=14. k=14.

46.

Evaluate (1+i)k (1i) k (1+i)k (1i) k for k=4, 8, and 12 k=4, 8, and 12 . Predict the value for k=16. k=16.

47.

Show that a solution of x 6 +1=0 x 6 +1=0 is 3 2 + 1 2 i. 3 2 + 1 2 i.

48.

Show that a solution of x 8 1=0 x 8 1=0 is 2 2 + 2 2 i. 2 2 + 2 2 i.

Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.

49.

1 i + 4 i 3 1 i + 4 i 3

50.

1 i 11 1 i 21 1 i 11 1 i 21

51.

i 7 ( 1+ i 2 ) i 7 ( 1+ i 2 )

52.

i −3 +5 i 7 i −3 +5 i 7

53.

( 2+i )( 42i ) (1+i) ( 2+i )( 42i ) (1+i)

54.

( 1+3i )( 24i ) (1+2i) ( 1+3i )( 24i ) (1+2i)

55.

( 3+i ) 2 ( 1+2i ) 2 ( 3+i ) 2 ( 1+2i ) 2

56.

3+2i 2+i +( 4+3i ) 3+2i 2+i +( 4+3i )

57.

4+i i + 34i 1i 4+i i + 34i 1i

58.

3+2i 1+2i 23i 3+i 3+2i 1+2i 23i 3+i

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions
Citation information

© Feb 10, 2020 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.