Notes 7
Math 120 Notes 11.3 –The Complex Plane
I. The complex plane
A. Define: real axis, imaginary axis
1. Plot the following complex numbers
a. 3 + 3i b. -2 – i
c. -3 d. -2i
2. Let [pic] be a complex number
a. What is [pic]?
b. Find the magnitude for the complex numbers above.
3. Find a formula to convert z from rectangular to polar form, then covert the above points into polar form (degrees).
4. Plot the following polar coordinates.
a. [pic]
b. [pic]
5. Convert the polar coordinates above into rectangular form.
B. Euler’s Formula
1. Fill in the following chart:
|x |[pic] - rectangular form |[pic] - polar form |
|0 | | |
|[pic] | | |
|[pic] | | |
|[pic] | | |
|[pic] | | |
|[pic] | | |
|[pic] | | |
2. What conclusions can you make from the above analysis?
3. Convert the following into polar form
a. [pic] b. [pic]
4. Write the following in exponential form
a. [pic]
b. [pic]
III. Operations on complex numbers
A. Addition / subtraction
1. Rectangular form
a. [pic]
b. [pic]
2. Polar form: [pic] [HUH?]
B. Multiplication
1. Rectangular form: [pic]
2. Polar form
a. [pic]
b. So in general, [pic] = ?
c. Use the formula to find [pic]
3. Try multiplying [pic] by converting to polar form first.
C. Division
1. Rectangular form: [pic]
2. Polar form
a. What do you think is the formula for [pic] = ?
c. Use the formula to find [pic]
D. Power – De Moivre’s Theorem
1. Using the multiplication rule, if [pic], what is [pic], [pic], and [pic]?
2. Prove your result for [pic] using exponential notation.
3. Use De Moivre’s Theorem to write the following in standard form.
a. [pic]
b. [pic]
| |Rectangular |Polar |Exponential |
|Regular |[pic] |[pic] |- |
|Coordinates | | | |
|Complex #s |[pic] |[pic] |[pic] |
|Vectors |[pic] |[pic] |- |
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