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Precalculus

Mrs. Snow's Math

McNeil High School

Notes are intended to compliment the current text in use at McNeil High School Precalculus, Enhanced with Graphing Utilities, Texas Edition By Michael Sullivan and Michael Sullivan, III

FALL SEMESTER

Fundamentals

MATHXL FOR SCHOOL – where to find it on line and where are all the assignments!

Algebra II Review Annotated Notes

Chapter 2:  Functions and Their Graphs All assignments are due at the start of the next class period.  The first assignments of this school year will be completed on paper: 

Current assignments are currently not available on Mathxl, the precalculus online homework program.  For full credit, use separate paper:   please write your full name, class period, and identify the assignment with chapter.section number.  Write out each problem, show all work and circle the answer.

Lesson 2.1 and Lesson 2.2 Functions and 2.2 The Graph of a Function Mathxl Homework 2.1 and 2.2 Solutions 2.1 2.2 Mathxl -Textbook Homework 2.3 and 2.4 Solutions Textbook 2.3 and 2.4 Mathxl – Textbook Homework 2.5 Solutions 2.5 Annotated Notes Video Lesson 2.1 and 2.2 Lesson 2.3 and Lesson 2.4 Properties of Functions,   2.4  Piecewise and Greatest Integer Functions Annotated Notes 2.3 and 2.4 Video Lesson 2.3 and 2.4 Lesson 2.5   Graphing Techniques:  Transformations Annotated Notes 2.5 Video Lesson 2.5 In-class Transformation Practice Annotated Transformation Practice mrssnowsmath.com/…/chapter2review.2023r.pdf 2.5 Homework Worksheet Chapter 2 Review Review Solutions  

Chapter 4:  Polynomial and Rational Functions

Lesson 4.1  Polynomial Functions and Model Annotated Notes Video Lesson 4.1 Lesson 4.2 The Real Zeros of a Polynomial Function Lesson 4.3 Complex Zeros; Fundamental Theorem of Algebra Annotated Notes 4.2 and 4.3 Video Lesson 4.2 and 4.3 Lesson 4.4 and Lesson 4.5  4.4 Properties of Rational Functions and.4.5 The Graphs of a Rational Function Annotated Notes 4.4 and 4.5 Video Lesson 4.4 and 4.5 Lesson 4.6 Polynomial and Rational Inequalities Annotated Notes 4.6 Video Lesson 4.6 Chapter 4 Spiral Ch 2 Review    Updated 9/2023 Spiral Review Solutions    

Chapter 6 Trigonometric Functions

Quizlet Links: Sine, Cosine, and Tangent Radian Values Practice with radians from the unit circle; flashcards and more Unit Circle practice page   2 unit circles for drill practice

Lesson 6.1 and 6.2 Part 1 Angles and Their Measure, The Unit Circle Annotated Notes 6.1 And 6.2 Part 1 Video Lesson 6.1 and 6.2 part 1 Lesson 6.1 and 6.2 Part 2 Trigonometric Functions: Unit Circle Approach Annotated Notes 6.1 and 6-2 Part2 Video Lesson 6.1 and 6.2 part 2 2 Unit Circles   Practice Form Lesson 6.3 Properties of the Trigonometric Functions Annotated Notes Video Lesson 6.3 Review 6.1-6.3 Chapter 6.1-6.3 Review Solutions   

Unit Circle Chart – In Order Lesson 6.4 Graphs of the Sine and Cosine Functions Annotated Notes 6.4 Video Lesson 6.4 Worksheet6.4 Lesson 6.5   Graphs of Secant and Cosecant Functions Annotated Notes 6.5 Part 1 Lesson 6.5 Graphs of Tangent and Cotangent Functions Annotated Notes 6.5 Part 2 Worksheet 6.5    Worksheet for: secant, Cosecant, Tangent and Cotangent Functions Video Lesson 6.5    this covers both part 1 and part 2:  Secant, Cosecant, Tangent and Cotangent Functions Lesson 6.6 Phase Shift Annotated Notes 6.6 Video Lesson 6.6 Worksheet 6.6 Lesson 8.5   Simple Harmonic Motion Annotated Notes 8.5 Video Lesson 8.5   “The Ferris Wheel Problem”   Video showing Simple Harmonic Motion Spiral Review 6.1-6.6, 8.5   Spiral Review Solutions

Lesson 5.1 and Lesson 5.2 Composite Functions and One-to-One Functions, Inverse Functions Annotated Notes 5.1 and 5.2 Video Lesson 5.1 and 5.2

Lesson 7.1 and 7.2   The Inverse Sine, Cosine, and Tangent Functions and The Inverse Trigonometric Function, Continued 7.1 and 7.2 Annotated Notes Video Lesson 7.1 Extra Examples Lesson 7.3 Trigonometric Equations Annotated Notes 7.3 Video Lesson 7.3 Review 5.1-5.2  and 7.1-7.3 Review 7.1-7.3 Solutions (revised 11/2/2019)

Unit Circle Sine, Cosine and Tangent values – Quizlet     link to a quizlet that has flash cards for our unit circle values.  The “Learn” link will quiz you with the flash cards out of order.

Trigonometric Identities Reference Sheet The first page will need to be memorized! Lesson 7.4 Trigonometric Identities Annotated Notes 7.4 Video Lesson 7.4 Worksheet Proofs Trig Identities   fall 2022 we are only doing the odd problems.  The even are available for additional practice Lesson 7.5 Sum and Difference Formulas Annotated Notes 7.5 Video Lesson 7.5 Lesson 7.6 Double-angle and Half-angle Formulas Annotated Notes 7.6 Video Lesson 7.6 Review Spiral 5.1-2, 7.1-3 7.4-7.5 Review Answers    Corrected!

 Chapter 10

Lesson 10.1 and 10.2 Conics and the Parabola Annotated Notes 10.1 and 10.2 Video Lesson 10.1 and 10.2

Lesson 10.3 The Ellipse Annota te d Notes 10.3 Lesson 10.4 The Hyperbola Annotated Notes 10.4 Lesson 10.7 Plane Curves and Parametric Equations Annotated Notes 10.7

Fall Final Exam Review Fall Final Exam solutions

GUIDELINE FOR NOTECARD

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK  for Fall and Spring Semesters

SPRING SEMESTER 

Chapter 8 Lesson 8.1 Right Triangle Trigonometry; Applications Annotated Notes 8.1 Video 8.1 Trigonometry Applications Worksheet Lesson 8.2 The Law of Sines Lesson 8.2 The Ambiguous Case Example Annotated Notes 8.2 Video 8.2 Lesson 8.3 The Law of Cosines Lesson 8.4 Area of a Triangle Annotated Notes 8.3 and 8.4 Video 8.3 and 8.4 Review Chapter 8 Review Chapter  Solutions  

Lesson 8.5 Simple Harmonic Motion (presented in the fall with Chapter 6) Annotated Notes 8.5 are located with Chapter 6

Chapter 9 Lesson 9.1 Polar Coordinates Annotated Notes 9.1 Video Lesson 9.1 Lesson 9.2 Polar Equations and Graphs Annotated Notes .9.2 Video Lesson 9.2 Limacon Examples Polar Graph Paper Lesson 9.4 Vectors Video Lesson 9.4 Annotated Notes 9.4 9.4 Static Equilibrium Problem #19 example Lesson 9.5 The Dot Product Annotated Notes 9.5 Video Lesson 9.5 Spiral Ch 8 and 9 Re view Spiral Review Solutions

Chapter 5 and 11.5 Lesson5.3 Exponential Functions Annotated Notes 5.3  and corrected Video Lesson 5.3 Lesson 5.4 Logarithmic Functions  and Lesson 5.5 Properties of Logarithms Annotated Notes 5.4 and 5.5 V ideo Lesson 5.4 Video Lesson 5.5

Lesson 5.6 Logarithmic and Exponential Equations Annotated Notes 5.6 Video Lesson 5.6 Lesson 5.7 and Lesson 5.8 Financial Models   and  Exponential Growth and Decay Models Annotated Notes 5.7 and 5.8 Video Lesson 5.7  Video Lesson 5.8 Chapter 5 Review  Review Solutions Chapter 5

Extra Lessons 11.5 and Rational Expressions Addition and Subtraction of Rational Expressions Annotated Notes Video Lesson Addition and Subtraction of Rational Expressions Multiplication and Division of Rational Expressions  Annotated Notes Video Lesson Multiplication and Division of Rational Expressions

Lesson 11.5 Partial Fraction Decomposition Annotated Notes 11.5 Video Lesson 11.5 https://www.youtube.com/watch?v=8lSCAY15BdQ

Chapter  12 Lesson 12.1 Sequences Annotated Notes 12.1 Video Lesson Lesson 12.2 Arithmetic Sequences Annotated Notes 12.2 Video Lesson Lesson 12.3   Geometric Sequences Lesson 12.3 Lecture Notes modified for voice over lecture Annotated Notes 12.3 Video Lesson Lesson 12.4 Mathematical Induction Lesson 12.4 Lecture Notes modified for voice over lecture Annotated Notes 12.4 Video Lesson 12.4 Homework Worksheet Lesson 12.5  The Binomial Theorem Lesson 12.5 Lecture Notes modified for voice over lecture Annotated Notes 12.5 Video Lesson Review Chapter 12    

Solutions Review Ch 12 and 11.5   if you want credit for the review, make sure you show your work,  copying these answers will result in a 0%

Chapter 14 –  A Preview of Calculus  

Lesson 14.1 Finding Limits Using Tables and Graphs Annotated Notes 14.1 *    Video Lecture 14.1 14.1 Worksheet

Lesson 14.2 Algebra Techniques for Finding Limits *    Video Lecture 14.2 Annotated Notes 14.2 14.2 Worksheet

Lesson 14.3  The Tangent Problem:  The Derivative Annotated Notes 14.3 *     Video Lecture 14.3 14.3 Worksheet Helpful videos for visualizing secant lines morphing into a tangent line: The Tangent Line and the Derivative   an 11 minute video, all is very good, the definition by derivative starts at 4 minutes. Applet:  Ordinary Derivative by Limit Definition   as an interactive applet to let the student see as the distance between points goes to zero,  the secant line becomes the tangent line.

Lesson 14.4  Limits at Infinity Annotated Notes 14.4 14.4 Worksheet *    Video Lecture 14.4

Lesson 14.5 The Area Problem:  The Integral Annotated Notes 14.5 14.5 Worksheet *    Video Lecture 14.5

Solutions to Chapter 14 practice problems

14.3, problem #7: correct answer is f'(x)=-5

Spiral Review Chapter 14 and Chapter 12 -remember, no work no credit!! Review Ch 12 and 14 Solutions

Note Card For Precalculus Final All students, whether exempting the final exam or not,  are required to complete the Final Exam Review.  Work must be shown for credit.   Spring Final Exam Review Spring Final Review Solutions

Other Notes

Intro to Calculus Lesson on Derivatives and Integration Annotated Notes Derivatives Worksheet #1 – Product and Quotient Rules Worksheet #2 – Chain Rule Chain Rule Notes Worksheet #3 – Integrals Integral Notes

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK for 2020-2021 School Year

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Precalculus

Unit 1: composite and inverse functions, unit 2: trigonometry, unit 3: complex numbers, unit 4: rational functions, unit 5: conic sections, unit 6: vectors, unit 7: matrices, unit 8: probability and combinatorics, unit 9: series, unit 10: limits and continuity.

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About This Course

Welcome to the Math Medic Precalculus course! The lessons and activities in this course are designed to thoughtfully prepare students for AP Calculus or other college math courses. Each lesson is designed to be taught in an Experience First, Formalize Later (EFFL) approach, in which students work in small groups on an engaging activity before the teacher formalizes the learning. Alignment to the CCSS can be found here .

The course begins with an in-depth study of function families, with a focus on connecting multiple representations, followed by an advanced investigation of trigonometry and its applications, including polar graphs, parametric equations, and vectors. The course continues with a unit on sequences and series followed by an accessible and engaging introduction to the big ideas of Calculus. The unit overviews and learning targets for the Math Medic Precalculus course can be found here .

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Activities – Chapter 3

Activity 3a – inverse functions.

This activity explores the details of finding an inverse to a function and the implications on the domain and range of the function and its inverse. It is related to the discussion of the inverse trig functions in Topics 3.1, 3.2 and 3.3.

It is a very visual lesson, and students should have some piece of technology to graph several functions at once.

Inverse Functions

Activity 3b – Angle Sum Formulas Part I

This activity is the first of three that generates formulas useful to simplify trig expressions. These formulas are used in Topics 3.5, 3.6 and 3.7.

There are several issues that students will have to sort out, including interpreting function notation, some geometry, symmetry, and some pretty intense algebra.

Angle Sum Worksheet – Part I

Activity 3c – Angle Sum Formulas Part II

This activity builds upon Activity 3b and continues to generate formulas used in Topics 3.5, 3.6 and 3.7.

Again, students will have to sort out function notation, geometry, symmetry, and algebra.

Angle Sum Worksheet – Part II

Activity 3d – Double Angle Formulas

This activity is directly related to the formulas in Topic 3.7. It finishes the work that began in Activity 3b and 3c.

Double Angle Formulas

Precalculus Copyright © by Mike Weimerskirch and the University of Minnesota Board of Regents is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

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

Learning Objectives

In this section, you will:

  • Express square roots of negative numbers as multiples of   i i .
  • 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

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.

So, using properties of radicals,

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

We use 5 i 5 i and not − 5 i − 5 i 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 + b i a + b i where a a is the real part and b i b i is the imaginary part. For example, 5 + 2 i 5 + 2 i is a complex number. So, too, is 3 + 4 3 i . 3 + 4 3 i .

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 + b i a + b i where

  • a a is the real part of the complex number.
  • b i b i is the imaginary part of the complex number.

If b = 0 , b = 0 , then a + b i a + b i 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.

Given an imaginary number, express it in standard form.

  • Write − a − a as a − 1 . a − 1 .
  • Express − 1 − 1 as i . i .
  • Write a ⋅ i a ⋅ i in simplest form.

Expressing an Imaginary Number in Standard Form

Express − 9 − 9 in standard form.

− 9 = 9 − 1 = 3 i − 9 = 9 − 1 = 3 i

In standard form, this is 0 + 3 i . 0 + 3 i .

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

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.

Plot the complex number 3 − 4 i 3 − 4 i on the complex plane.

The real part of the complex number is 3 , 3 , and the imaginary part is −4 i . −4 i . We plot the ordered pair ( 3 , −4 ) ( 3 , −4 ) as shown in Figure 3 .

Plot the complex number −4 − i −4 − i 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:

Subtracting complex numbers:

Given two complex numbers, find the sum or difference.

  • Identify the real and imaginary parts of each number.
  • Add or subtract the real parts.
  • Add or subtract the imaginary parts.

Adding Complex Numbers

Add 3 − 4 i 3 − 4 i and 2 + 5 i . 2 + 5 i .

We add the real parts and add the imaginary parts.

Subtract 2 + 5 i 2 + 5 i from 3 – 4 i . 3 – 4 i .

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,

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

  • Use the distributive property.

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

Distribute the 4.

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

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

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

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

Given two complex numbers, multiply to find the product.

  • Use the distributive property or the FOIL method.

Multiplying a Complex Number by a Complex Number

Multiply ( 4 + 3 i ) ( 2 − 5 i ) . ( 4 + 3 i ) ( 2 − 5 i ) .

Use ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i ( a + b i ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i

Multiply ( 3 − 4 i ) ( 2 + 3 i ) . ( 3 − 4 i ) ( 2 + 3 i ) .

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 + b i a + b i is a − b i . a − b i .

Note that complex conjugates have a reciprocal relationship: The complex conjugate of a + b i a + b i is a − b i , a − b i , and the complex conjugate of a − b i a − b i is a + b i . a + b i . 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 + d i c + d i by a + b i , a + b i , 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.

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

Apply the distributive property.

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

The Complex Conjugate

The complex conjugate of a complex number a + b i a + b i is a − b i . a − b i . 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.

Finding Complex Conjugates

Find the complex conjugate of each number.

  • ⓐ 2 + i 5 2 + i 5
  • ⓑ − 1 2 i − 1 2 i
  • ⓐ The number is already in the form a + b i . a + b i . The complex conjugate is a − b i , a − b i , or 2 − i 5 . 2 − i 5 .
  • ⓑ We can rewrite this number in the form a + b i a + b i as 0 − 1 2 i . 0 − 1 2 i . The complex conjugate is a − b i , a − b i , or 0 + 1 2 i . 0 + 1 2 i . This can be written simply as 1 2 i . 1 2 i .

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 .

Given two complex numbers, divide one by the other.

  • Write the division problem as a fraction.
  • Determine the complex conjugate of the denominator.
  • Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.

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

We begin by writing the problem as a fraction.

Then we multiply the numerator and denominator by the complex conjugate of the denominator.

To multiply two complex numbers, we expand the product as we would with polynomials (the process commonly called FOIL).

Note that this expresses the quotient in standard form.

Substituting a Complex Number into a Polynomial Function

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

Substitute x = 3 + i x = 3 + i into the function f ( x ) = x 2 − 5 x + 2 f ( x ) = x 2 − 5 x + 2 and simplify.

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 .

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

Substituting an Imaginary Number in a Rational Function

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

Substitute x = 10 i x = 10 i and simplify.

Let f ( x ) = x + 1 x − 4 . f ( x ) = x + 1 x − 4 . 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.

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 .

Simplifying Powers of i i

Evaluate i 35 . i 35 .

Since i 4 = 1 , i 4 = 1 , we can simplify the problem by factoring out as many factors of i 4 i 4 as possible. To do so, first determine how many times 4 goes into 35: 35 = 4 ⋅ 8 + 3. 35 = 4 ⋅ 8 + 3.

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.

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

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

  • Multiply Complex Numbers
  • Multiplying Complex Conjugates
  • Raising i to Powers

Explain how to add complex numbers.

What is the basic principle in multiplication of complex numbers?

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

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

For the following exercises, evaluate the algebraic expressions.

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

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

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

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

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

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

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

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

1 − 2 i 1 − 2 i

− 2 + 3 i − 2 + 3 i

− 3 − 4 i − 3 − 4 i

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

( 3 + 2 i ) + ( 5 − 3 i ) ( 3 + 2 i ) + ( 5 − 3 i )

( − 2 − 4 i ) + ( 1 + 6 i ) ( − 2 − 4 i ) + ( 1 + 6 i )

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

( 2 − 3 i ) − ( 3 + 2 i ) ( 2 − 3 i ) − ( 3 + 2 i )

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

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

( 5 − 2 i ) ( 3 i ) ( 5 − 2 i ) ( 3 i )

( 6 − 2 i ) ( 5 ) ( 6 − 2 i ) ( 5 )

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

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

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

( 4 − 2 i ) ( 4 + 2 i ) ( 4 − 2 i ) ( 4 + 2 i )

( 3 + 4 i ) ( 3 − 4 i ) ( 3 + 4 i ) ( 3 − 4 i )

3 + 4 i 2 3 + 4 i 2

6 − 2 i 3 6 − 2 i 3

− 5 + 3 i 2 i − 5 + 3 i 2 i

6 + 4 i i 6 + 4 i i

2 − 3 i 4 + 3 i 2 − 3 i 4 + 3 i

3 + 4 i 2 − i 3 + 4 i 2 − i

2 + 3 i 2 − 3 i 2 + 3 i 2 − 3 i

− 9 + 3 − 16 − 9 + 3 − 16

− − 4 − 4 − 25 − − 4 − 4 − 25

2 + − 12 2 2 + − 12 2

4 + − 20 2 4 + − 20 2

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

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.

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

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

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

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

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

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

1 i 11 − 1 i 21 1 i 11 − 1 i 21

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

i −3 + 5 i 7 i −3 + 5 i 7

( 2 + i ) ( 4 − 2 i ) ( 1 + i ) ( 2 + i ) ( 4 − 2 i ) ( 1 + i )

( 1 + 3 i ) ( 2 − 4 i ) ( 1 + 2 i ) ( 1 + 3 i ) ( 2 − 4 i ) ( 1 + 2 i )

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

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

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

3 + 2 i 1 + 2 i − 2 − 3 i 3 + i 3 + 2 i 1 + 2 i − 2 − 3 i 3 + i

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Access for free at https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
  • Authors: Jay Abramson
  • Publisher/website: OpenStax
  • Book title: Precalculus 2e
  • Publication date: Dec 21, 2021
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
  • Section URL: https://openstax.org/books/precalculus-2e/pages/3-1-complex-numbers

© Jan 9, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 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.

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