Complex Numbers Teacher Resources

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Perform Arithmetic Operations with Complex Numbers: High School Math Common Core

For Teachers 9th - 12th Standards
Imagine a world where all the numbers live together harmoniously. Here you have it! Check out these resources that introduce the concept of an imaginary number, go through the operations, and model how to utilize the conjugate. 
35 Items in Collection
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Algebra: High School Common Core Math

For Teachers 9th - 12th Standards
All Algebra, all the time! Here is a compiled list of resources that address all of the standards regarding Algebra Common Core for high school. Each resource has the main standard it addresses under the notes section of the item. 
31 Items in Collection
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Number and Quantity: High School Common Core Math

For Teachers 9th - 12th Standards
This collection has a resource for each standard in the Number and Quantity selection of the CCSS. While several of the resources cover multiple standards, the focus standard is listed in the notes section of each resource for easy viewing.
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Complex Numbers

For Teachers 10th - 12th Standards
Learners are introduced to the concept of imaginary unit and complex numbers. They are taught how to add and subtract complex numbers. Young scholars define a complex number. They comprehend at least two applications of complex numbers....
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Classifying Complex Numbers

For Teachers 10th - 12th Standards
Imaginary numbers are a real thing. Scholars learn about complex numbers, real numbers, and imaginary numbers. They classify given numbers as strictly complex, strictly real, or strictly imaginary in an individual or group activity.
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Complex Numbers: Plotting and Polar Form

For Teachers 9th - 12th Standards
Explore the concept of, and use the Ti-Nspire to, convert complex numbers into polar form. Then practice graphing complex numbers in the polar coordinate plane.
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Powers and Roots of Complex Numbers

For Teachers 9th - 12th Standards
Explore a few powers of complex numbers numerically and decide if there is a pattern and use parametric equations to represent a complex number graphically. Use De Moivre's theorem to represent the roots of the complex number...
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Complex Numbers

For Teachers 9th - 12th Standards
The class practices, on paper and/or on a TI graphing calculator the concepts of how to add, multiply, divide and subtract complex numbers using the correct property.
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Complex Numbers in Polar Form; DeMoivre’s Theorem

For Teachers 10th - 12th Standards
The iconic links between trig functions and complex numbers are introduced in a lesson on the basic properties of complex numbers in polar form. Quickly progressing through the polarization of complex coordinates, a bulk of the worksheet...
EngageNY

An Appearance of Complex Numbers 2

For Teachers 11th - 12th Standards
Help the class visualize operations with complex numbers with a lesson that formally introduces complex numbers and reviews the visualization of complex numbers on the complex plane. The fifth installment of a 32-part series reviews the...
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Complex Numbers

For Students 9th - 12th Standards
Are complex numbers and binomials similar? This stack of slides provides an introduction to complex numbers and shows how to operate with them. The worked examples show a connection between operating with binomials and operating with...
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Trigonometric Form of Complex Numbers

For Teachers 11th - 12th Standards
This lesson is written from the perspective of a student teacher. A step by step lesson in which the class works on the trigonometric form of complex numbers. They convert between trigonometric and complex forms. They will also practice...
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Trigonometric Form of Complex Numbers

For Students 11th - Higher Ed Standards
Represent complex numbers in two ways. Pupils use the interactive to convert polar and rectangular representations of complex numbers. The learners drag an overlay back and forth over the coordinate plane to reveal the polar coordinates...
8:58
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Complex Numbers (Part 2)

For Students 9th - 11th
This video continues looking at dividing complex numbers by looking at the conjugate of a complex number. The instructor then uses the conjugate to rationalize the denominator of a rational expression with a complex number in the...
7:21
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Complex Numbers (Part 1)

For Students 9th - 11th
In this video, a complex number is defined and graphed on the complex plane. Sal also shows how to add, subtract, and multiply two complex numbers. He starts showing how to divide two complex numbers, but runs out of time and continues...
10:10
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Complex Numbers (Part 1)

For Students 9th - 11th
In this video, a complex number is defined and graphed on the complex plane. Sal also shows how to add, subtract, and multiply two complex numbers. He starts showing how to divide two complex numbers, but runs out of time and continues...
8:58
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Complex Numbers (Part 2)

For Students 9th - 11th
This video continues looking at dividing complex numbers by looking at the conjugate of a complex number. The instructor then uses the conjugate to rationalize the denominator of a rational expression with a complex number in the...
5:05
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How Do You Graph Complex Numbers?

For Teachers 10th - 11th
Complex numbers have two parts and can be graphed on a complex plane. After an explanation of the complex plane, this video shows three examples of graphing complex numbers.
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Complex Numbers Addition Worksheet

For Students 10th - 12th Standards
In this complex numbers activity, learners find the sum of complex numbers. This four-page activity contains approximately 20 multi-step problems.
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Complex Numbers

For Teachers 9th - 12th Standards
Students examine complex numbers. For this Algebra II lesson, students investigate two programs that involve complex numbers: the MANDELER Program and SYNDIV program.
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EngageNY

Complex Numbers and Transformations

For Teachers 9th - 12th Standards
Your learners combine their knowledge of real and imaginary numbers and matrices in an activity containing thirty lessons, two assessments (mid-module and end module), and their corresponding rubrics. Centered on complex numbers and...
EngageNY

Distance and Complex Numbers 1

For Teachers 11th - 12th Standards
To work through the complexity of coordinate geometry pupils make the connection between the coordinate plane and the complex plane as they plot complex numbers in the 11th part of a series of 32. Making the connection between the two...
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Can You Hear Me Now? Using Cell Phone Signals to Divide Complex Numbers

For Students 9th - 12th Standards
Without complex numbers, cell phones could not function. Scholars learn to divide complex numbers by applying them to the wavelengths of cell phone signals. They work to develop a method of division incorporating the conjugate. Then,...
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Distance and Complex Numbers 2

For Teachers 11th - 12th Standards
Classmates apply midpoint concepts by leapfrogging around the complex plane. The 12th lesson in a 32 segment unit, asks pupils to apply distances and midpoints in relationship to two complex numbers. The class develops a formula to find...
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Trigonometry and Complex Numbers

For Teachers 11th - 12th Standards
Complex numbers were first represented on the complex plane, now they are being represented using sine and cosine. Introduce the class to the polar form of a complex number with the 13th part of a 32-part series that defines the argument...
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Complex Numbers as Vectors

For Teachers 11th - 12th Standards
Show your math class how to use vectors in adding complex numbers. Vectors represent complex numbers as opposed to points in the coordinate plane. The class uses the geometric representation to add and subtract complex numbers and...
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An Appearance of Complex Numbers 1

For Teachers 11th - 12th Standards
Complex solutions are not always simple to find. In the fourth lesson of the unit, the class extends their understanding of complex numbers in order to solve and check the solutions to a rational equation presented in the first lesson....
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Computations with Complex Numbers

For Teachers 9th - 12th Standards
This quick set of problems provides a brief refresher on the arithmetic of complex numbers. Learners need to multiply, add and subtract, and remember features of i when raised to a power. Included solutions are clear enough that learners...
EngageNY

Complex Number Division 1

For Teachers 11th - 12th Standards
Conjugating in the math classroom — and we're not talking verbs! The seventh lesson in a series of 32 introduces the class to the building blocks of complex number division. During the instruction, the class learns to find the...
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Complex Number Properties

For Teachers 10th - 12th Standards
High schoolers experiment with complex numbers to see which properties of real numbers apply to the system of complex numbers and practice computations. The study of these properties is first explored by looking at rotational...
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Plotting, Simplifying, and Describing Complex Numbers

For Students 10th - 12th
In this complex numbers instructional activity, students simplify, plot, iterate, and state bounded or bounded for complex numbers in nine problems.  The solutions are provided.
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Computation with Real and Complex Numbers

For Students 9th - 12th Standards
In this computation with real and complex numbers worksheet, students use addition, subtraction, multiplication and division to solve 26 problems with complex numbers to win a bingo game.
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Properties of Complex Numbers

For Students 11th - 12th Standards
For this worksheet, students evaluate complex (signed and imaginary) numbers.  In addition, they are asked to graph complex numbers as well as identify a complex number on a presented graph.
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Complex Numbers

For Students 9th
In this complex numbers learning exercise, 9th graders solve 10 different problems that include determining the absolute value of each equation. First, they convert the number to the a+bi form. Then, students select the coefficients for...
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Multiplying and Dividing Complex Numbers

For Teachers 8th - 11th Standards
Students solve quadratic using multiplication and division. In this algebra instructional activity, students perform operation with complex numbers. They factor quadratics using the quadratic formula.
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Complex Numbers

For Students 9th
In this complex numbers worksheet, 9th graders solve 10 different problems that include addition and subtraction of these numbers. First, they add or subtract the coefficients of similar terms algebraically. Then, students remove the...
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Representation of Complex Numbers

For Teachers 9th - 12th
Learners write complex numbers in standard form. They add, subtract, multiply and divide complex numbers correctly.
3:38
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Imaginary Numbers Are Real (Part 6: The Complex Plane)

For Students 9th - 12th Standards
How do addition and subtraction work on the complex plane? A short video presentation provides a clue on how to add complex numbers geometrically. The video ends with four problems to determine the rules for multiplication on the complex...
2:57
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Imaginary Numbers Are Real (Part 4: Bombelli's Solution)

For Students 9th - 12th Standards
Is the square root of negative one crucial to the process of finding other solutions? Using the properties of the newly discovered square root of negative one, historical mathematician Bobelli is able to solve Cardan's problem. His...
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Complex Numbers

For Teachers 10th - 12th
Build on your class' understanding of real numbers as they begin working with complex numbers. Pupils begin with an exploration of i and the patterns in the powers of i. After developing a definition for i, they simplify complex number...
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Complex Numbers

For Students 9th
In this complex numbers learning exercise, 9th graders solve 10 different problems related to various complex numbers. First, they multiply and divide various types of complex numbers. Then, students determine the multiplicative inverse...
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Complex Numbers

For Students 9th
In this complex numbers instructional activity, 9th graders solve 10 different problems that include either adding or subtracting complex numbers. First, they determine the sum of the real components. Then, students determine the sum of...
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5-9 Skills Practice: Complex Numbers

For Students 10th - 12th
In this complex numbers activity, students simplify given radicals and complex numbers. This one-page activity contains 26 problems.
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The Solution is Complex

For Teachers 9th - 12th Standards
Consider complex numbers, roots, and quadratic equations. Use the discriminate as a way to determine the nature of a quadratic's roots. Then discuss the similarities and differences between quadratics with two, one, or no real roots....
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Rational Expressions and Complex Numbers

For Students 11th
In this rational expressions and complex numbers worksheet, 11th graders solve and complete 24 various types of problems that include determining logarithms and polynomials. First, they determine the remaining roots of a polynomial given...
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Complex Numbers and Operations

For Teachers 8th - 10th
In this algebra worksheet, students add, subtract and multiply using complex numbers. They apply the correct property of i as they solve. There are ten questions with an answer key.
7:22
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Complex Numbers - Multiplying and Dividing

For Teachers 8th - 12th
Algebra In action! Watch the step-by-step process needed to multiply and divide complex numbers. The FOIL method is used to solve two different practice problems involving multiplication and multiplying by the conjugation is used to...
2:02
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Complex Numbers - Algebra Help

For Teachers 7th - 10th
The foil method had been used for years as a way to remember how to simplify complex numbers. This process is demonstrated step-by-step, slowly so the learner will be able to follow along as the instructor solves the problem. Intended...
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Operations with Complex Numbers

For Teachers 9th - 11th Standards
This activity is a great introduction to the operations of complex numbers.  Combining their knowledge of algebra and the use of a calculator, your class can discover and apply the rules for adding, subtracting, and multiplying complex...
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As If Numbers Weren't Complex Enough Already!

For Teachers 9th - 12th Standards
The class explores the concept of complex numbers on a website to generate their own Mandelbrot sets. They will practice performing operations with complex numbers and then to get a visual understanding, graph the absolute value of a...
EngageNY

Complex Number Division 2

For Teachers 11th - 12th Standards
Individuals learn to divide and conquer complex numbers with a little help from moduli and conjugates. In the second activity on complex number division, the class takes a closer look at the numerator and denominator of the...
EngageNY

The Geometric Effect of Some Complex Arithmetic 1

For Teachers 11th - 12th Standards
Translating complex numbers is as simple as adding 1, 2, 3. In the ninth lesson plan in a 32-part series, the class takes a deeper look at the geometric effect of adding and subtracting complex numbers. The resource leads pupils into...
EngageNY

The Geometric Effect of Some Complex Arithmetic 2

For Teachers 11th - 12th Standards
The 10th lesson in a series of 32, continues with the geometry of arithmetic of complex numbers focusing on multiplication. Class members find the effects of multiplying a complex number by a real number, an imaginary number, and another...
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Linear Transformations Review

For Teachers 11th - 12th Standards
Time for matrices and complex numbers to come together. Individuals use matrices to add and multiply complex numbers by a scalar. The instructional activity makes a strong connection between the operations and graphical transformations.
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Two Variable Function Pump

For Students 9th - 12th Standards
Use a function to operate on two variables. Pupils look at operating with complex numbers as a function of two variables. The interactive squares the input and adds a constant to it. Learners visualize the resulting output and its...
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Complex Numbers

For Students 10th - 12th
A straightforward approach to teaching complex numbers, this lesson addresses the concepts of complex numbers, polar coordinates, Euler's formula, De moivres Theorem, and more. It includes a practice problems set with odd answers and a...
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Complex Numbers - Week 5

For Teachers 10th - 11th
In this complex numbers worksheet, students solve problems involving polynomials, real constants containing imaginary numbers, and find the complex number that satisfies a given equation. This two-page worksheet contains seven problems.
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Complex Numbers

For Students 10th - 12th
In this complex numbers activity, students write numbers in standard form.  They identify real, imaginary and complex numbers.  Students find the absolute value and conjugate of a number.  This two-page activity contains 45 problems.
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Radicals: Addition & Subtraction of Complex Numbers

For Teachers 10th - 12th Standards
In this radical worksheet, students simplify complex numbers.  They find the sum and difference of complex numbers.  This two-page worksheet contains 11 multiple-choice problems.  Answers are given on the last page.
4:25
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Imaginary Numbers Are Real (Part 7: Complex Multiplication)

For Students 9th - 12th Standards
Multiplying complex numbers geometrically is complicated. Find an easier way with a short video that explains the process of multiplying complex numbers on the complex plane. The connection with the process and the angles and moduli sets...
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Complex Numbers Solutions

For Students 9th - 12th Standards
Complex doesn't have to mean harder! Learners experiment with online software to determine the quadratic equations with complex solutions. They use the quadratic formula to solve equations with both real and complex solutions.
4:42
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Imaginary Numbers Are Real (Part 3: Cardan's Problem)

For Students 9th - 12th Standards
Were complex numbers discovered or invented? A video presentation makes the case for the discovery of square roots of negative one. In order for complex numbers to be real, then they must behave like other numbers, which they do in terms...
5:41
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Imaginary Numbers Are Real (Part 9: Closure)

For Students 9th - 12th Standards
The last video in a series about imaginary numbers demonstrates the need for complex numbers. Math pupils learn that they need to include the square root of negative one, the building block of the complex numbers, to have a set of...
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Imaginary Numbers? What Do You Mean Imaginary?

For Teachers 9th - 12th Standards
Don't worry, this resource actually exists. Scholars learn about imaginary numbers and work on problems simplifying square roots of negative numbers. As an extension, they research the history of imaginary numbers.
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Complex Numbers as Solutions to Equations

For Students 10th - 12th Standards
Quadratic solutions come in all shapes and sizes, so help your classes find the right one! Learners use the quadratic formula to find solutions for quadratic equations. Solutions vary from one, two, and complex. 
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DeMoivre's Theorem and nth Roots: The nth Roots of Unity

For Students 11th - Higher Ed
The nth roots of unity all have a magnitude of one. Scholars use the unit circle and DeMoivre's Theorem to find the complex roots of one and discover that the complex numbers all lie on the unit circle and are equally spaced around it...
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Application of Complex Numbers

For Teachers 9th - 12th
Students apply operations with complex numbers to electrical circuit problems, real-world situations, utilizing TI-83 Graphing Calculators. They perform basic operations of addition, subtraction, division and multiplication with complex...
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Complex Numbers

For Students 9th - 12th
In this algebra worksheet, learners simplify complex numbers ranging from performing basic operations to distribution with a few combine like terms. There are 44 questions.
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Introduction to Complex Numbers

For Students 10th - 12th Standards
In this complex number activity, learners explore the properties of imaginary numbers to find the square roots of negative numbers. Explanations and examples are given prior to the exercise. This two-page activity contains ten problems.
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Complex Equations

For Students 12th - Higher Ed
In this complex equations instructional activity, students solve and complete 40 various types of problems. First, they solve each of the equations over the set of complex numbers shown. Then, students solve each inequality and state...
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Complex Numbers

For Students 11th
For this algebra worksheet, 11th graders solve complex numbers using division. multiplication, addition and subtraction. There are 11 questions with an answer key.
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Complex Numbers

For Students 10th - 11th
In this algebra worksheet, students solve complex numbers using multiplication and division properties. There are 10 questions with an answer key.
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Complex Numbers

For Students 10th - 11th
In this algebra learning exercise, students solve complex numbers using multiplication and division properties. There are 10 questions with an answer key.
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Complex Numbers

For Students 11th
In this algebra activity, 11th graders solve complex numbers using division. multiplication, addition and subtraction. There are 11 questions with an answer key.
8:04
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Complex Conjugates

For Students 10th - 12th
Sal discusses the usefulness of the conjugate of the complex number is this video. First, he shows both graphically and algebraically adding a complex number and its conjugate. Then, he shows how to simplify a rational expression with...
4:07
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How Do You Add Complex Numbers by Graphing in the Complex Plane?

For Teachers 10th - 11th
Graphs usually add an extra dimension to ones understanding of mathematical concepts and graphing complex numbers is no exception. This video shows to graph complex numbers on the complex plane and what the results of adding complex...
2:45
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How Do You Multiply Complex Numbers Using FOIL?

For Teachers 10th - 11th
How do you multiply complex numbers? The instructor in this video uses the FOIL method of multiplying binomials to multiply two complex numbers. She finishes the problem, by using prior knowledge of the definition of the imaginary unit,...
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Worksheet 7.5 DeMoivre's Theorem and Complex Numbers

For Students 10th - 12th
In this DeMoivre's Theorem and complex numbers worksheet, students change complex numbers to real numbers. They determine square roots of complex numbers and represent them geometrically. This two-page worksheet contains eight problems.
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Vectors

For Teachers 10th - 12th Standards
Complex numbers bridge so many concepts as learners edge into higher-level math. Here, their polar and rectangular forms link through vector notation, and the arithmetic of vectors is born. These basic math skills translate into a...
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Complex Distance

For Teachers 10th - 12th Standards
Have learners graph complex numbers to gain a visual and mathematical understanding of the distance and the midpoint between two complex numbers. This lesson is short, but to the point, and addresses an important complex number concept....
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Steady-State Circuit Analysis & Filter Design

For Teachers 9th - Higher Ed Standards
Scholars identify how the TI-89 implements phasors to perform sinusoidal steady-state analysis. The focus is on how to enter and display complex numbers. They also show how a typical steady-state application how to use the Numeric Solver...
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Complex Analysis: Polar Form

For Students Higher Ed
For this polar form worksheet, students write complex numbers in polar form. They draw an Argand diagram containing vectors and draw sketches representing in a coordinate plane. This two-page worksheet contains thirteen multi-step...
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Solving Quadratic Equations with Complex Roots

For Students 8th - 9th
In this quadratic equations learning exercise, students solve 10 different equations that include complex roots. First, they solve for the indicated variable and express the roots of the equation in simplest form. Then, students...
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Quadratic Equations

For Teachers 10th - 12th Standards
Through a variety of physical and theoretical situations, learners are led through the development of some of the deepest concepts in high school mathematics. Complex numbers, the fundamental theorem of algebra and rational exponents...
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Application of Complex Numbers

For Teachers 9th - 12th
Students apply operations with complex numbers to electrical circuit problems. They use electrical current equations and graphing calculators to solve electrical engineering problems.
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Complex numbers in polar form

For Teachers 9th - 12th
Young scholars plot complex numbers in the polar plane. They convert complex numbers in polar form to rectangular form. Students plot coordinates in the polar plane. They plot complex numbers in the complex plane. Young scholars are...
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Complex numbers in polar form

For Teachers 9th - 12th
Students plot complex numbers in the polar plane. They are asked to list what they believe are the five most famous numbers in mathematics. Students are explained that generally accepted that the five most famous numbers in mathematics...
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Complex Numbers

For Students Higher Ed
In this complex numbers worksheet, students simplify 9 problems involving the addition, subtraction, multiplication, and division of complex numbers.
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Graphing and Absolute Value of Complex Numbers

For Students 10th - 12th
In this graphing and absolute value worksheet, students graph complex numbers and find the absolute value of complex numbers. Students complete 12 problems total.
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Sixty-Eight Solving Equations and Performing Operations With Complex Numbers

For Teachers 10th - 12th
For this complex numbers worksheet, students classify numbers, perform operations on complex numbers, and find complex solutions to equations, in sixty-eight problems.
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Complex Numbers

For Students 9th
In this complex numbers worksheet, 9th graders solve 10 different types of problems that include complex numbers in standard form. First, they state the values of the a and b components for each complex number when the number is in a...
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Operations with Complex Numbers

For Students 10th - 11th
In this complex numbers worksheet, students simplify thirty complex number expressions and answer two critical thinking questions.  The solutions are provided.
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Imaginary and complex Numbers

For Students 11th
In this Algebra II worksheet, 11th graders simplify expression involving imaginary and complex numbers and determine the quadratic equation that would have the given complex roots.  The on page worksheet contains thirty-four problems. ...
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Twenty Equations Involving Imaginary and Complex Numbers

For Students 10th - 12th
In this complex numbers worksheet, students solve twenty equations involving imaginary and complex numbers.  Four of these problems involve two variables.
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Math 321 Spring 98. Complex Numbers

For Students Higher Ed
In this complex numbers worksheet, students find the solutions to polynomials and write them in rectangular and exponential form. This two-page worksheet contains two problems.
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Complex Numbers

For Students 10th - 12th
In this complex numbers worksheet, students simplify the expressions of complex numbers and use the answers to answer a riddle. Students complete 15 problems.
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Algebra II Test covering Radicals, Complex Numbers, and Finding Roots of Quadratics

For Students 10th - 12th
In this algebra two worksheet, students solve twenty-five problems in test format.  The worksheet covers radicals, complex numbers, and finding roots of quadratics.
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Topic 5.9 and 5.10 - Complex Numbers

For Students 7th - 9th
In this complex numbers worksheet, students solve problems in which they either simplify or solve algebraic expressions. There are 15 questions on this worksheet.
EngageNY

A Surprising Boost from Geometry

For Students 10th - 12th Standards
Working with imaginary numbers — this is where it gets complex! After exploring the graph of complex numbers, learners simplify them using addition, subtraction, and multiplication. 
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Discovering the Geometric Effect of Complex Multiplication

For Teachers 11th - 12th Standards
Does complex number multiplication have the class spinning? Here's a resource that helps pupils explore and discover the geometric effect of multiplying complex numbers. In the 14th installment in the 32-part unit groups look at the unit...
EngageNY

Exploiting the Connection to Trigonometry 1

For Teachers 11th - 12th Standards
Class members use the powers of multiplication in the 19th installment of the 32-part unit has individuals to utilize what they know about the multiplication of complex numbers to calculate the integral powers of a complex number. Groups...
EngageNY

Mid-Module Assessment Task - Precalculus (module 1)

For Teachers 11th - 12th Standards
Individuals show what they know about the geometric representations of complex numbers and linearity. Seventeen questions challenge them to demonstrate their knowledge of moduli and operations with complex numbers. The assessment is the...
1 In 1 Collection
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Complex Number Operations

For Teachers 10th - 12th Standards
What do animated videos have to do with mathematics? Using operations of complex numbers and their representations on the complex plane, high schoolers observe how mathematics could be used to move animations. The lesson provides an...
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Name the Property

For Students 10th - 12th Standards
In this property activity, students examine given problems containing complex numbers.  They identify the property used in the problem.  This one-page activity contains 10 problems.
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Multiplying Complex Numbers

For Students 10th - 11th
In this complex number worksheet, students multiply binomials containing imaginary numbers. They simplify imaginary numbers to their lowest terms. This two-page worksheet contains ten problems.
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Worksheet 7.5 Trigonometric Form for Complex Numbers

For Students 10th - 12th
In this trigonometric form worksheet, learners represent complex numbers geometrically and express complex numbers in trigonometric form. Students solve equations containing complex numbers. This two-page worksheet contains 13 problems.
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Complex Numbers

For Students 9th
In this complex numbers worksheet, 9th graders solve and graph 10 different problems that include various complex numbers. First, they represent each of the problems shown as complex numbers graphically. Then, students graphically add...
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Math 312 Spring 2004: Complex Numbers

For Students 12th - Higher Ed
In this complex numbers worksheet, students complete proofs for complex numbers, sketch points which solve an equation, compute and sketch an Argand diagram. This two-page worksheet contains five multi-step problems.
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Math 312 Spring 95 - Complex Numbers

For Students Higher Ed
In this complex number learning exercise, students explore complex numbers and sketch a set of points which solve in the equation. This one-page learning exercise contains five problems.
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Complex Analysis: Properties of Complex Numbers

For Students Higher Ed
In this properties of complex numbers worksheet, students list and explain the properties and operations used to compute the value of expressions. This one-page worksheet contains six definition questions and three equations.
12:58
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Complex Determinant Example

For Students 10th - 12th
Sal solves a complex determinant problem from the 2010 IIT entrance test. Knowledge of determinants, the exponential form of complex numbers, trigonometric ratios, and algebraic thinking are all utilized here.
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Iterating the Function and Complex Numbers

For Teachers 7th - 10th
Students identify the end behavior of polynomial graphs. In this algebra lesson, students factor and solve quadratic and complex equations. They factor out negative roots and identify the real and imaginary parts of an equation.
4:38
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Imaginary Numbers Are Real (Part 5: Numbers are Two Dimensional)

For Students 9th - 12th Standards
If the square root of negative one exists, where is it on the number line? A math video presents a geometric representation of complex numbers and the complex plane, and then compares it to the geometric representation of real numbers —...
4:40
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Imaginary Numbers Are Real (Part 8: Math Wizardry)

For Students 9th - 12th Standards
How does the Fundamental Theorem of Algebra apply to equations with complex roots? Using recently learned concepts about the complex plane, learners determine all the complex roots of polynomials. They can find the same roots as...
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Imaginary Numbers

For Students 9th - 12th Standards
Here is a worksheet that takes all aspects of quadratic functions and incorporates imaginary numbers into the problems. The problems range from simplifying to graphing and solving by using a variety of methods with imaginary and complex...
EngageNY

End-of-Module Assessment Task — Precalculus (Module 1)

For Teachers 11th - 12th Standards
A transformational assessment determines how far pupils are advancing toward mastering complex and matrix standards. The assessment checks the learners' understanding of linear transformations, complex numbers and the complex plane,...
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Complexity

For Teachers 10th - 12th Standards
Students perform mathematical operations on complex numbers. After a lecture/demo, students utilize a worksheet imbedded in this plan to gain practice performing operations on complex numbers.
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Working With Radicals and Complex Numbers

For Teachers 9th - 11th
Learners solve problems with radicals and complex numbers. In this algebra lesson, students add, subtract, multiply and divide radicals and complex numbers using the conjugate.
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Preliminaries and a Review of Some Geometry and Complex Numbers

For Students 9th - 11th
In this review worksheet, students explore the rules of exponents and the Pythagorean Theorem.  They examine formulas for the area of polygons.  Students review the topic of complex numbers.  This five-page worksheet includes extensive...
EngageNY

Matrix Multiplication Is Not Commutative

For Teachers 11th - 12th Standards
Should matrices be allowed to commute when they are being multiplied? Learners analyze this question to determine if the commutative property applies to matrices. They connect their exploration to transformations, vectors, and complex...
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Complex Number Operations

For Teachers 9th - 10th
Students solve problems with complex numbers. In this algebra lesson, students use addition, subtraction, multiplication and division of complex numbers.
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The Complex Numbers

For Teachers 9th - 10th
In this algebra worksheet, students solve 16 problems involving imaginary numbers. In the first three, students find the square root of 3 negative numbers in terms of i. Nine problems involve addition, subtraction, multiplication and...
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Exploiting the Connection to Trigonometry 2

For Teachers 11th - 12th Standards
The class checks to see if the formula for finding powers of a complex number works to find the roots too.  Pupils review the previous day's work and graph on the polar grid. The discussion leads the class to think about how to reverse...
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Justifying the Geometric Effect of Complex Multiplication

For Teachers 11th - 12th Standards
The 14th instructional activity in the unit has the class prove the nine general cases of the geometric representation of complex number multiplication. Class members determine the modulus of the product and hypothesize the relationship...
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The Geometric Effect of Multiplying by a Reciprocal

For Teachers 11th - 12th Standards
Class members perform complex operations on a plane in the 17th segment in the 32-part series. Learners first verify that multiplication by the reciprocal does the same geometrically as it does algebraically. The class then circles back...
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Exploiting the Connection to Cartesian Coordinates

For Teachers 11th - 12th Standards
Multiplication in polar form is nice and neat—that is not the case for coordinate representation. Multiplication by a complex number results in a dilation and a rotation in the plane. The formulas to show the dilation and rotation are...
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Absolute Value

For Students 11th - 12th
In this absolute value worksheet, students solve 10 different equations that include solving absolute values. First, they determine whether their equation is expressed in the a+bi form. Then, students select the coefficients for a and b....
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Complex Numbers

For Teachers 10th - 12th
In this complex numbers worksheet, students write numbers as a multiple of i. They add and subtract imaginary numbers. Students write complex quotients in standard form. This three-page worksheet contains six problems.
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Complex Analysis: Graphical Representations

For Students Higher Ed
For this graphical representation worksheet, students draw an Argand diagram depicting complex numbers. They draw in vectors that represent complex numbers. This three-page worksheet contains approximately ten problems.
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Complex Analysis: Graphical Representations

For Students Higher Ed
For this graphical representation worksheet, students draw an Argand diagram depicting two complex numbers in a complex plane. They draw vectors to represent complex numbers. This two-page worksheet contains five equations to solve.
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Quaternions

For Students 11th - 12th
For this quaternions worksheet, students solve and complete 7 different problems. First, they read and identify quaternions as expressions in a given form of ordinary real numbers. Then, students add and subtract a few problems by doing...
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Operations of Complex Numbers and Intro to DeMoivre's Theorem

For Teachers 9th - 10th
Learners solve problems with complex numbers. In this algebra lesson, students factor complex numbers and simplify equations using DeMoivre's Theorem. They add, subtract, multiply and divide using negative roots.
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Complex Analysis: Complex Numbers

For Students Higher Ed
In this complex numbers worksheet, students compute the value of given expressions and write the answer in complex number form. This one-page worksheet contains three equations.
5:16
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Imaginary Numbers Are Real (Part 2: A Little History)

For Students 9th - 12th Standards
In some cases, a square root of a negative number must exist in order to determine the roots of a cubic equation. An educational presentation provides a specific example of a cubic with a known root to have an understanding of a square...
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Saxon Math: Algebra 2 (Section 7)

For Teachers 9th - 12th Standards
Section seven of twelve turns the mathematical eye toward higher-level concepts. From radians in the unit circle to the complex number system, logarithms to advanced factoring methods, this lesson—just past the halfway point in the...
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Real-World Applications to Imaginary and Complex Numbers

For Teachers All
Authenticate imaginary numbers through real-life applications in science, math, and literature.
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Complex Numbers

For Teachers 9th - 10th
Students define sine waves as it relates to digital signal. In this algebra instructional activity, students investigate the importance of complex numbers as it relates to math and observed the distortion caused by a digital signal,...
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Complex Number

For Students 8th - 10th
In this algebra instructional activity, students identify the vertex of a quadratic functions. They identify complex numbers. There are 6 questions.
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Complex Analysis: Polar & Exponential Forms

For Students Higher Ed
In this polar and exponential form worksheet, students use Euler's Formula to write complex numbers in exponential form. They write numbers in rectangular form. This two-page worksheet contains seven multi-step problems.
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Easy Worksheets: Number Sets

For Students 9th - 12th Standards
In this number sets worksheet, high schoolers solve 6 multiple choice problems. Students determine the number set that the given number belongs to such as whole, integers, rational, real, complex, etc.
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Easy Worksheet: Properties

For Students 9th - 12th Standards
In this properties worksheet, 10 short answer problems are solved. Learners name the property illustrated such as the inverse property or associative property of multiplication.
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Polynomials and Complex and Imaginary Numbers Review

For Students 9th - 12th
In this polynomials review worksheet, high schoolers answer 20 questions regarding polynomials, synthetic division, and complex and imaginary numbers. Students also complete a 10 question spiral review of parent graphs, systems,...
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Test Chapter 8 Algebra II

For Students 10th - 12th
In this radical equations and expressions test, students answer 4 multiple choice and 21 short answer questions.
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Order in the Classroom

For Teachers 1st
First graders define and identify integers, rational, irrational, real, and complex numbers. They place number cards in ascending and descending order.
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Complex Numbers

For Students 9th - 11th
In this complex number worksheet, students simplify given equations and solve for the unknown. Students add, multiply and divide algebraic expressions. This two-page worksheet contains 23 problems.
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Exploring Harmonic Conjugates

For Teachers 9th - 10th
High schoolers investigate harmonic conjugates using Cabri Software or the Ti Calculator to analyze harmonics. They explore different angles measurements as they move their circles around and solve problems.
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Wishful Thinking—Does Linearity Hold? (Part 1)

For Teachers 11th - 12th Standards
Not all linear functions are linear transformations — show your class the difference. The first lesson plan in a unit on linear transformations and complex numbers that spans 32 segments introduces the concept of linear transformations...
5:07
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A Brief History of Numerical Systems

For Students 8th - 12th Standards
As people and civilizations have become more complex, numbers have evolved into what they are today. Watch a video that walks viewers through the history of the development of symbols into numbers and positional notations.
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Tens and Ones with Unifix Cubes

For Students K - 2nd
Use Unifix Cubes to make math more tangible for your young pupils! A simple activity will get learners talking and collaborating...and all about MATH! In partners, learners pick a number card (these are provided, and can be simply cut...
3:55
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Calculating Molar Volume Using Experimental Data

For Students 9th - 12th Standards
Calculating molar volume from experimental data often means the additional challenge of complex numbers and slight errors in measurement. The 23rd video in a 30-part series offers two types of experimental data to work from and models...
5:20
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Introduction to I and Imaginary numbers

For Students 9th - 11th
Students are always fascinated by imaginary numbers. In this video, Sal defines the imaginary unit, i, and shows when raising i to an exponent it has a cyclic nature.
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Quiz 1: Types of Functions

For Students 9th - 12th Standards
Sometimes the best things are already done for you! Here is a six-problem quiz that has a variety of problems ranging from solving quadratic equations to interpreting a function. The piece-de-resistance is the worked out answer key in...
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Quiz 2: Types of Functions

For Students 9th - 12th Standards
Here is a resource that provides the structure of an assessment with the convenience of a full answer key. The focus is on rational, exponential, and logarithm functions with a few questions on solving polynomials. 
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Complex Number Multiplication

For Students 10th - 12th
In this multiplication worksheet, students multiply complex numbers.  This interactive one-page worksheet contains 10 problems.  Answers are provided.
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Arithmetic Complex Numbers

For Teachers 10th - 11th
Students convert quadratic functions from standard form to vertex form. In this algebra instructional activity, students solve polynomials using synthetic and long division. They derive and apply the remainder theorem and factor theorem.
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Complex Solutions?

For Students 9th - 12th Standards
Complex numbers don't need to be so ... complex. Young mathematicians determine conditions for when quadratic equations have complex solutions. Their work should rely on the quadratic formula.
2 In 2 Collections
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Polynomial, Rational, and Radical Relationships

For Students 9th - 12th Standards
This assessment pair goes way beyond simple graphing, factoring and solving polynomial equations, really forcing learners to investigate the math ideas behind the calculations. Short and to-the-point questions build on one another,...
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Factoring Extended to the Complex Realm

For Students 10th - 12th Standards
A solution will work one way or another: find solutions, or use solutions to find the function. Learners use polynomial identities to factor polynomials with complex solutions. They then use solutions and the Zero Product Property to...
1 In 1 Collection
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Unit 8 Review Packet: Polar Coordinates, Polar Graphs, and Vectors

For Teachers 10th - 12th Standards
Tying together an introductory unit on polar coordinates, graphs, and vectors is often a daunting task. So many nuanced math skills develop through these lessons that making sure an assessment hits them is difficult, to say the least....
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Obstacles Resolved—A Surprising Result

For Students 10th - 12th Standards
The greater the degree, the more solutions to find! Individuals find the real solutions from a graph and use the Fundamental Theorem of Algebra to find the remaining factors. 
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Imaginary Numbers

For Teachers 11th
For this Algebra II worksheet, 11th graders explain how every real number is also a complex number.  The one page worksheet contains one problem with solution. 
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Number Theory: Numbers and Geometry

For Students 9th - 12th
In this math worksheet, high schoolers explore properties of the number line, both standard and imaginary. This thorough worksheet has many problems for the students to solve, and includes an answer sheet.
7:16
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How Do You Find Higher Powers of i?

For Teachers 10th - 11th
What is �i raised to the 45th power? This video shows you how to break up the problem into a number of �i raised to 4th power, and calculate what power is left over to calculate.
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Complex Analysis: Rectangular Form

For Students Higher Ed
In this rectangular form instructional activity, students write numbers in rectangular form. They solve complex numbers, and find the value of a function. This two-page instructional activity contains approximately 12 multi-step problems.
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Equations and Inequalities

For Teachers Higher Ed
Students in a college algebra class use graphs to solve inequalities. Using the coordinate graphing system, they represent the relationship between x and y intercepts. They practice solving equations with complex numbers and locate...
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Complex Number Maze

For Students 9th - 12th Standards
This is a to-the-point complex number skills worksheet with a fun addition of completing a maze. Learners solve problems to determine if answers are imaginary or not, in order to work their way through the maze. An average level of...
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Complex Number Subtraction

For Students 10th - 12th
For this subtraction worksheet, students find the difference of complex numbers.  This one-page worksheet contains 10 problems.  Answers are provided.
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Complex Analysis: Polynomial Equations

For Students Higher Ed
In this polynomial equation activity, students explore real fractional powers of complex numbers. This three-page activity contains examples, explanations and approximately seven problems.
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Representing Reflections with Transformations

For Teachers 11th - 12th Standards
In the 16th lesson in the series of 32 the class uses the concept of complex multiplication to build a transformation in order to reflect across a given line in the complex plane. The lesson breaks the process of reflecting across a line...
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Find Imaginary Solutions: Imaginary Zeros

For Students 10th - 12th Standards
The resource is the real deal. Individuals investigate the imaginary zeros of f(x) = x^2 + 1. They accomplish this task by using an interactive that shows input values x = a + bi and output values x^2 + 1 on a complex plane.
13:04
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Basic Complex Analysis

For Students 10th - 12th
This video gives an excellent overview of the different representations of a complex number. First, Sal reviews a pictorial representation of a complex number as an Argand diagram, then shows how to find the modulus and the argument of a...
11:53
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Exponential Form to Find Complex Roots

For Students 10th - 12th
Sal solves the problem x^3 = 1 for all of the real and complex roots using the exponential form of the complex number. He shows how the roots found in exponential form are the same as those in a more traditional form of a complex number...
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Intermediate Algebra

For Students 9th - 12th Standards
Algebra concepts are all wrapped up in one nice bow. The resource combines all the concepts typically found in Algebra I and Algebra II courses in one eBook. The topics covered begin with solving linear equations and move to linear...
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College Algebra

For Students 9th - Higher Ed Standards
Is there more to college algebra than this? Even though the eBook is designed for a college algebra course, there are several topics that align to Algebra II or Pre-calculus courses. Topics begin with linear equation and inequalities,...
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Precalculus

For Students 11th - Higher Ed Standards
Take a step beyond Algebra 2. Learners use the eBook to learn concepts from the typical Precalculus course. Content starts off with a short review of functions in general and moves on to the basic functions, finishing up with more...
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Simplifying Radicals/Imaginary Numbers

For Students 9th - 12th Standards
Looking for straightforward practice on simplifying radicals and imaginary numbers? Here are 20 problems to practice these skills and an answer key for the odd-numbered questions. 
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Algebra and Trigonometry

For Students 10th - Higher Ed Standards
Move on into trigonometry. An informative eBook takes the content of a College Algebra course and adds more relating to trigonometry and trigonometric functions. The content organization allows pupils to build upon their learning by...
5:20
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Introduction to I and Imaginary numbers

For Students 9th - 11th
Students are always fascinated by imaginary numbers. In this video, Sal defines the imaginary unit, i, and shows when raising i to an exponent it has a cyclic nature.
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Complex Number Division

For Students 9th - 12th
In this online math activity, students practice complex number division using positive and negative integers. When complete, students submit their answers and get instant feedback regarding their accuracy.
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Share this page Facebook Page Complex number addition

For Students 9th - 12th
In this online math worksheet, students practice adding positive and negative integers. When complete, students submit their answers and get instant feedback regarding their accuracy.
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Absolute Values

For Teachers 8th - 12th
In this absolute value worksheet, students determine the absolute value of each complex number. Students complete 10 problems.
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Exponents - Week 1

For Teachers 10th - 12th
In this exponent worksheet, students add, subtract, multiply and divide exponents. They solve problems containing negative exponents, fractional exponents, and polynomials. This two-page worksheet contains eight problems.
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The Mandelbrot Set

For Teachers 10th - 12th
Students explore the concept of Mandelbrot sets and Julia sets. In this Mandelbrot and Julia set lesson plan, students use a function integrator applet to investigate two-variable function iterations. Students use Julia set and...
6:50
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Imaginary Numbers

For Students 9th - 12th Standards
Scholars learn to use their imagination in math class! The tutorial explains the origin of the number i and then simplifies expressions that include imaginary numbers. The examples include i raised to different powers, giving viewers...
7:04
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Rationalize an Imaginary Denominator with Conjugate Method

For Students 9th - 12th Standards
Explore a procedure for rationalizing a denominator that includes imaginary terms with a video that shows how to use a conjugate when working with fractions. The narrator explains the process from start to finish with each step shown in...
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Reptile and Amphibian Anatomy

For Teachers 4th - 12th
Learners identify and interpret the various differences between reptiles and amphibians. Then they draw an amphibian and reptile and correctly label their body parts. Students also invent new reptiles and amphibians, sketch how they...
13:04
Lesson Planet

Basic Complex Analysis

For Students 10th - 12th
This video gives an excellent overview of the different representations of a complex number. First, Sal reviews a pictorial representation of a complex number as an Argand diagram, then shows how to find the modulus and the argument of a...
11:53
Lesson Planet

Exponential Form to Find Complex Roots

For Students 10th - 12th
Sal solves the problem x^3 = 1 for all of the real and complex roots using the exponential form of the complex number. He shows how the roots found in exponential form are the same as those in a more traditional form of a complex number...
Lesson Planet

Irrational and Imaginary Root Theorems

For Students 9th - 11th
In this irrational and imaginary root theorem learning exercise, learners identify the number of complex zeros for a given function. They write a polynomial function of leas degree with integral coefficients. This four-page learning...
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Plots of Polar Coordinates: Submarine Sonar

For Students 11th - Higher Ed Standards
Find the location of enemy submarines with a sonar display that doubles as a polar coordinate system. Pupils find polar coordinates that would be the same location of plotted submarines. They then determine multiple polar coordinates...
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Roots, Multiplicities, and Factoring Polynomials

For Students 10th - 12th
In this polynomials worksheet, students find zeros, state multiplicities, and factor twenty-two polynomials.  Students then find a polynomials that satisfies the stated conditions in eighteen problems,  Finally, students factor...
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Radicals, Complex and Absolute Value Numbers

For Teachers 9th - 11th
Students solve problems with exponents, radicals, absolute value and complex numbers. In this algebra lesson, students use properties of exponential functions to solve equations and word problems.
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Absolutely Radical and Complex

For Teachers 9th - 11th
In this algebra lesson, students solve problems with radicals and complex numbers. In this algebra lesson, students relate exponential functions to the real world as they discuss growth and decay.
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Solving Quadratic Equations by Completing the Square

For Students 9th - 12th
In this solving quadratic equations activity, students complete the square to tell the possible answers for 18 equations. 4 answers involve the square root of -1 (i).
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Mathematics B: High School Exam

For Students 11th - 12th
In this mathematics instructional activity, learners solve problems about numerous high school topics including trigonometry, algebra, and pre-calculus. This instructional activity contains, but is not limited to, radicals, inverse...
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Chapter 10 - Objective 3.1 Quadratic Formula

For Students 11th - 12th
In this quadratic formula instructional activity, students solve quadratic equations. They use the discriminant to determine the solution or solutions to the problem. This one-page instructional activity contains
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Quadratic Formula with Imaginary Numbers

For Students 8th - 10th
In this quadratic equation learning exercise, students use the quadratic formula to solve 6 fill in the blank problems that involve imaginary numbers.
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Final Spring Semester Review

For Students 9th - 12th
In this advanced algebra semester review worksheet, students answer 63 questions spiraling a review of topics including combinations, systems of equations, absolute value, parent graphs, conics, complex numbers, and binomial theorem.
3:24
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Introduction to Polar Coordinates - Concept

For Students 9th - 12th Standards
Learning polar coordinates does not require visiting the North Pole, though explaining igloos is much easier using polar coordinates. The lesson begins with a video explaining how to find coordinates on a spherical plane and how to...
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Polynomial Divisions

For Teachers 8th - 10th
Young scholars factor polynomials and use long division t solve problems. In this algebra lesson, students find the zeros of polynomials by synthetic and long division. They perform operation using complex numbers.

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