Roots of Complex Numbers
Lesson 9 of 11
Objective: SWBAT find the nth roots of complex numbers.
Since we have discussed how to find a power it seems logical to also learn how to find the root of a complex number.
This is a little more difficult for students to understand.
The bell work today begins the lesson by asking students to find the roots for x^4-1 . I explain that each of the answers is a 4th root of the expression.
As student work the above problem I discuss the meaning of root. This term is sometimes not used much in other math courses. I also discuss the Fundamental Theorem of Algebra which states that a polynomial equation of degree n has n solutions in the complex number system. If a polynomial has n solutions than it must have the same number of roots. Students have been told that roots, solutions and x intercepts are the same just used in different situations so reminding students of the connection helps students connect prior knowledge.
Once we have gone over the bell work give students the definition of nth root of a complex number. We compare this to the nth root of an equation to see if they are similar.
As we move on I begin pondering to the class: "z=a+bi is a complex number it can be written as z=r(cos(theta) i sin(theta)). So if I have z^n how can I find the nth roots?" I think aloud to help students with their reasoning. Most students will begin by saying we need to take the nth root of the number.
Once the students have considered how we might find the nth root, I prove nth root of a complex number formula with the class this takes several minutes as we discuss how the angle can be a coterminal angle of theta which can be confusing to some students.
Now that students have developed the formula for the nth root I ask students to work together to find the fifth roots of 1. Before the students begin working I ask "How many roots will you have? Is 1 written as a complex number in trigonometric form?"
As students work I move around and answer questions. Some struggles will be finding the trigonometric form of 1 and what numbers should be used as k. As students work I many need to stop the group for a time and ask students to share what they have found so far. I will do this once most students have found the trigonometric form of 1.
Once students have the solution I ask the students how we might verify that the solutions are correct. Many students do not realize that 2 is the cube root of 8 because 2*2*2=8. I remind students that should be able to multiply the roots and get the original number.
Another problem I give students to complete is to find all solutions for x^4+16i=0. This is a little different since the students will first solve for x^4 then find the 4 roots of -16i.
Once most students have completed the problems, students share their process for finding the answers.
As class ends I give student a problem to extend some of their reasoning. This problem connects the ideas we have worked on throughout the year. This problem allows students to see the visual representation of roots of complex numbers. By doing this problem I am able to assess which students are able to extend their reasoning and interpret mathematics in written form.