Complex Numbers 3
With a Little Imagination:
Understanding the Complex Numbers
Name
MAT 390
Complex Numbers
Date
Complex numbers are frequently occurring solutions for quadratic equations and the idea of imaginary numbers often poses challenges to students of mathematics. As a result, this lesson is designed be taught in a 10th grade classroom in order to take away the mystery from the concept of the complex number. Topics to be covered during this lesson include the history of the complex number system, operations with i, and graphing on the complex plane.
Student Name / Level: Beginner-IntermediateClass: 10th grade Algebra / Date
Topic
Complex Numbers
Lesson Essential Questions (LEQ’s)
What is a complex number?
How do we operate with complex numbers?
How can they be represented?
Materials to be UsedComputer with projector
Powerpoint and Powerpoint file
Whiteboard/Smartboard with markers or stylus
Wipe-off Graph board or Cartesian plane feature on Smartboard
Homework Worksheets for each student
Math Journals
One blindfold for in class activity
Behavioral Objectives
After participating in the lesson and completing both class work and homework, students will be able to:
1. Define the term “complex number” in their own words as well as provide a definition in mathematical form
2. Perform mathematical operations with complex numbers including addition, subtraction, multiplication, division with 80% accuracy on today’s class work and homework
3. Apply knowledge of the complex plane to graph complex numbers following the instructions on the homework sheet.
Assessments to be Used
Formative:
· In class questioning (1,2,3)
· Class work (2,3)
Summative:
· Homework: Graphing Complex Numbers worksheet (2,3)
Note: This lesson may need to be covered in two class periods to cover all the material.
Preview:
This lesson is to take place after a lengthy discussion on the solving of quadratic equations. Students will have been evaluating the discriminate of quadratic equations to determine the number of real solutions for x. They will already be familiar with the concept of i from earlier classes. This lesson is meant to extend on the concept of complex numbers and enhance student knowledge to boost their comfort level with the subject.
Motivation:
The motivation for this lesson is presented on the “So Far…” slide (slide number 3) after the Warm-up. It has students remember what has been discussed and consider why negative discriminates don’t make sense. It then piques their interest by asking them how we can make sense of imaginary roots. Another motivation for students is presented on Slide 4 where uses for i are discussed (i.e. Electromagnetism). This is done to motivate students that may be heading into vocational programs or engineering.
Slide 1
This is only the cover slide for the lesson’s presentation. Before getting into the material, be sure to take role and answer any homework questions that the students have from the previous lesson’s homework.
Warm-up: Slide 2
The warm-up for this lesson requires students to recall information from the last unit on evaluating the discriminant of quadratic equations to determine the number of roots or x-intercepts for a given function. The questions are:
◦ x2 -2x +1=0
◦ x2 – 4x -3 =0
◦ -2x2 + 3x +1= 13
The answers for the warm up are:
◦ One real rational solution, x = -1
◦ Two real rational solutions, x = -3 and x = -1
◦ Two imaginary solutions, x = ¾ ± (√6)i
Notice that the last question requires manipulation before the discriminant can be evaluated. Give students about three minutes to complete the warm-up and then go over the answers with them for another two minutes.
So Far…:Slide 3
This slide acts as a recap of the warm-up and of previous material. It asks questions instead of lecturing about material to assess student progress. It asks students to explain what a negative discriminant implies and to explain why this happens. The last bullet point “How can we make sense of this?” acts as a transition into the lecture material by piquing student interest.
“What is “i”?: Slide 4
This slide may be a review for some students, but it necessary to reinforce this material as it will be useful further into the lesson. Here √(-1) is set equal to “i”. It is explained that it is easier to think of √(-1) as a letter to avoid some of the confusion. It explains that we consider solutions involving imaginary numbers because it has applications in other fields such as electromagnetism, engineering, and applied mathematics.
What is a Complex Number?: Slide 5
Here the definition of the term “complex number” is defined as any number that takes the form of a ± bi, where “a” and “b” take on real number values. This slide defines “a” to be the “real part” of the number and “b” to be the “imaginary part”. Here, ask students to consider why “b” would be the imaginary part of the complex number. They should be able to respond that it is multiplied by i, which is imaginary. This acts as a check for understanding before moving into the historical part of the lesson.
History: Slide 6
(From Sources 1 and 5)
This slide starts out in ancient history with the Ancient Greek historian Heron of Alexandria. It discusses that he did mathematics and often ran into negative square roots. Instead of considering the possibilities of imaginary numbers, the custom was to disregard the answer, say there were no solutions, or to replace the number by its additive inverse. Explain that this continued for centuries because mathematicians refused to accept the existence of imaginary numbers. Even in Heron's day, irrational numbers went against the mathematical community’s conceptions about the world. Ask students why replacing the negative number with its positive opposite would be a bad practice.
History: Slide 7
(From Sources 1, 3, and 6)
Here the first mathematician to really consider the possibility of the imaginary numbers is discussed. Cardano in 1545 published the Ars Magna. In his book, he presented information that had been passed down to him through a series of events by the mathematicians del Ferro and Tartaglia and he cited his sources. Stress to the students that even in math, plagiarism is a major issue. Continue the discussion with Cardano’s uncertainty about the topic and ask students to consider why he may have felt this way. Students should respond that the mathematical community was still ruled by many of the prior Greek conceptions of the natural world and going against these norms was often considered heresy. In addition, he was unsure of the usefulness of the new numbers.
History: Slide 8
(From Sources 3, 6, 8, 9, 10, and 11)
On this slide, the contributions of Rene Descartes and Wallis are considered. Rene Descartes, inventor of the Cartesian plane, devised the notation for complex numbers (a ± bi). In addition, his plane would later be modified to allow for the visual depiction of complex numbers. Wallis was the first to attempt interpreting imaginary numbers. He approached the subject using the conventional methods of his day, conic sections and proportionality.
History: Slide 9
(From Sources 1, 2, 3, 4, 7, 8, 9, and 11)
Slide 9 presents some of the early significant contributors to the topic, Wessel and Argand. Wessel was the first to give a geometric representation of the complex numbers. This was done in a surveying report he authored and the information was later published in 1799. This publication did not gain much attention and was lost until it was found again my Jean-Robert Argand.
Argand plotted complex numbers on the Cartesian plane using the x-axis for the real numbers and the y-axis as the imaginary. This way all complex numbers could be represented visually. His contribution was also not widely used and was lost again until Carl Gauss used it in 1831.
History: Slide 10
(From Sources 3, 4, 6, 7, and 8)
This is the last slide of history presented in the lesson. Carl Gauss’ contributions are discussed here. Gauss was the first to use the term “complex number” and used the new numbers to revise the Fundamental Theorem of Algebra. Previously, students have been working with the version of the Fundamental Theorem of Algebra that says an nth degree polynomial has at most n many solutions. Gauss’ doctorial dissertation included the complex solutions and he proved that an nth degree polynomial has exactly n many solutions. Ask students to explain what the difference between the two versions is and why it changed to “exactly”. They should respond that the new complex solutions are now included in the list of zeroes for the equation.
Working with Complex Numbers: Slide 11
This slide transitions from the history portion of the lesson into the mathematical section. It tells students that complex numbers can be treated as any other number and that operations with them are indeed possible. This lesson will include the addition, subtraction, and multiplication of complex numbers.
Adding Complex Numbers: Slides 12 and 14
Slide 12
Here, students are asked to consider the addition of two complex numbers, 7 + 3i and 2 + 5i. It first asks students to evaluate what the process of addition might be. Then, the process of treating the real and imaginary parts separately, performing the operations, and then combining the answers is discussed.
Slide 13
The math of the previous problem is performed on this slide. Ask students to list the real parts of the number along with the imaginary parts. The parts of the complex numbers are then considered separately and are added together yielding 9 for the real part and 8i for the imaginary part.
Slide 14
The answers from the previous slide are brought to the top of this slide and then the two pieces are combined to give the answer 9 + 8i. After this, show students on the board how to add the numbers together by using the standard algorithm of stacking the numbers and adding vertically. Make sure to stress that students can perform the operation either way that is comfortable for them.
Subtracting Complex Numbers: Slide 15
This slide considers the subtraction of the same two complex numbers from the previous example and follows the same process of breaking up into real and imaginary parts, operating and then combining. Again, show the standard algorithm of subtraction on the board and stress the acceptance of either method. Either way, the answer will be 5-2i.
Your Turn: Slide 16
Here, students are given some guided practice in adding and subtracting complex numbers. The process is written on the slide for students to follow along with in case they get stuck. The problems are:
◦ 5 + 9i + 6 + 7i
◦ -6 + 4i + 3 – i
◦ (11 – 8i ) – (8 – 2i )
◦ (5 + 9i ) - (-6 + 7i )
The answers to the problems are 11 + 16i, -3 + 3i, 3- 6i, and 11 + 2i. At the end of the slide, it asks students to consider what this process resembles. They should respond that it acts similar to combining like terms in the addition or subtraction of polynomials.
Multiplying Complex Numbers: Slides 17-19
Slide 17
Here, the material transitions form addition and subtraction to multiplication. Students are asked to consider the problem: (5 + i) (3 + 2i ). The question of how to perform this operation is then posed to the students. Some of the students may consider the separate multiplication of the real and imaginary parts. However, the true process is by “foil-ing” like in the multiplication of binomials. Be sure to address the need for the application of distributive property.
Slide 18
The process of “foil-ing” is considered on this slide. The multiplication is completed in pieces which are then combined to give the answer, 2i 2 + 13i + 15. However, this is not the true or complete answer. The real answer is given on the next slide.
Slide 19
The answer from the last slide is posted at the top of this slide. The first question posed to students is what is i2? The process of squaring i is then presented and the answer is -1. This is then substituted into the previous answer of 2i 2 + 13i + 15 yielding the complete answer of 13 + 13i.
Your Turn 2: Slide 20
Guided practice is applied in this slide. Students are told to FOIL the following complex numbers:
◦ (5 + 9i )(6 + 7i )
◦ (-6 + 4i )(3 – i )
◦ (11 – 8i )(8 – 2i )
◦ (5 + 9i )(-6 + 7i )
The students should come up with -33 + 89i, -14 + 18i, 72 - 86i, and -93 -19i. This may take some time for students, so, be patient.
Graphing Complex Numbers: Slides 21-22
Slide 21
Here, the method for graphing a complex number on the complex plane is presented. It begins with the recall of how graphing is performed on the Cartesian plane with x and y co-ordinates. Thanks are given to Argand and Gauss for producing the complex plane where the horizontal axis is the real number line and the vertical axis is the imaginary number line.
Slide 22
The plotting of the following complex numbers is required:
◦ 5 – 2i
◦ -3 + 4i
◦ -8 – 9i
◦ 6 + 7i
There is no instructional information on this slide as this will be done on the wipe off graph or the Cartesian plane on the Smartboard. Show students that the real part of the number becomes the x co-ordinate and the imaginary part becomes the y co-ordinate. Be sure to write the point as an ordered pair to encourage reinforcement of the comparison between making graphs on the Cartesian and complex planes. Show the process for the first two complex numbers and allow students to direct the plotting of the next two numbers, scaffolding off of what has already been done on the board. Ask students to consider where the real number 6 would be plotted on the complex plane. Students should reply that is will be plotted 6 units to the right of the origin on the x-axis. Ask them why it would be placed there. They should respond that 6 has no imaginary part and does not need to be raised off of the x-axis. After the graphing is completed, move into the In Class Activity.