The Unit Circle & Complex Numbers
DATE: 1/28 CLASS PERIOD: PreCalc UNIT: U.C.-1 .
LESSON OBJECTIVES:
In this extension lesson, students will extend their knowledge of the unit circle to include complex numbers. Students will explore Pythagorean Triples and the n complex roots of xn-1=0. This lesson corresponds with Common Core State Standards-The Complex Number System N-CN 5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
Schedule / ActivitiesWarm-up (10’) / As they enter the room, hand students a warm-up worksheet.
Activity (20’) / See attached outline.
Worktime (15’) / Hand out WS 1-5.
I. Review – Go over warm up.
A. Absolute Value. Review definition but share that Absolute Value can be thought of as “distance from the origin.”
B. Equation of the Unit Circle is x2+y2=1.
C. Graphing Complex Numbers in the plane as points or vectors (Argand Diagram).
II. Lesson
A. Absolute Value. How do we find the absolute value of 1+2i? The absolute value of a complex number z=a+bi is |z|=a2+b2. So |1+2i| is 12+22=5. This is literally the distance from the origin.
B. Can you think of another equation for the unit circle? What about |z|=1? Here are some example points on the unit circle: ±1, ±i. See how they correspond to
(1,0), (-1, 0), (0, 1), & (0, -1)?
C. What other points do you think would be there? What other points have
a2+b2=1? Do any of the points on the unit circle fit that equation? Remember from before i = 22+22i ? Isn’t that point on the unit circle? So i is 22+22i. Note a2+b2=1. Similarly, cos2+sin2=1 for the same point (π4). In fact, all four ±22±22i are on the unit circle.
SKIP THIS NEXT SECTION IF TIME IS SHORT. SAVE FOR TOMORROW.
D. Pythagorean Triples. A “Pythagorean Triple” is a set of three integers that make up a right triangle. The most famous is 3:4:5. 6:8:10 doesn’t count. Another one is 5:12:13. There are many more – 15:8:17, 7:24:25, 10:24:26, 21:20:29, 9:40:41, 35:12:27, and 11:60:61. In fact, there are infinitely many of them.
So we have a2+b2=c2. If we divide by c2, we have ac2+bc2=1. So, 35+45i, 513+1213i, and 4041+941i are all on the unit circle. (Verify.)
E. Here’s another application of complex numbers on the unit circle. Let
z=12+32i. Plot this and other powers of z as shown in this table. (see text, p. 408)
Power of z / Coordinates / qz / 12+32i / π3
z2 / -12+32i / 2π3
z3 / -1 / p
z4 / -12-32i / 4π3
z5 / 12-32i / 5π3
z6 / 1 / 2π
Show how this table shows the six complex roots of the equation x6-1=0.
III. Homework. Worksheet 1-5 attached.
Unit Circle Worksheet 1-5 (Complex Numbers and the Unit Circle)
Name ______
1. Are the following complex numbers on the unit circle (|z|=1)? Answer Y or N.
a. 32+12i b. 1+i
c. 2+2i d. 35+45i
e. 2425+725i f. 2735+1235i
2. Let z=i. Show z, z2, z3 and z4 on the unit circle below. What do z5 and z11 equal?
3. Let z=32+12i. Calculate z2 and z3. Show z, z2, and z3 on the unit circle below.
Unit Circle 1-5 Warm Up Name ______
1. Remember absolute value? Solve the following.
a. |5| b. |-p|
c. -|-7| d. (|-1|)(x)
2. Plot 1+2i on the coordinates below. 3. What is the equation of the unit circle?
Unit Circle 1-5 Warm Up Name ______
1. Remember absolute value? Solve the following.
a. |5| b. |-p|
c. -|-7| d. (|-1|)(x)
2. Plot 1+2i on the coordinates below. 3. What is the equation of the unit circle?