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Lesson Plan Worksheet

Complex numbers in polar form

Introduction The complex number z = a + bi can be represented in rectangular coordinates, (a, b), or in polar coordinates, (r, θθ ). Let’s explore complex numbers in polar form. Below are several questions designed to get you thinking about the activity. Answer them on a separate sheet of paper.

The absolute value of complex numbers activity

Go to complex absolute value activity at ExploreMath.com.

Converting complex numbers from rectangular form to polar form

Plot the number z = 6 + 8i using the activity. This can be accomplished by typing a 6 next to the a slide bar, and typing a 8 next to the b slide bar.

Question 1a. What is needed in order to plot this point on the polar plane? Question 1b. How could the r-value be determined? Hint: Select the “show right triangle” box.

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Question 1c. What is the r for this point? Check your answer by selecting the “ruler mode” box and stretching the ruler along the hypotenuse of the triangle. Question 1d. How could θθ be determined? Question 1e. What is θθ for this point? Question 1f. What would z = 6 + 8i look like on the polar plane? Question 2a. The point z = -2 – 2i lies in what quadrant? Question 2b. What are the coordinates of z = -2 – 2i in polar form? Question 2c. What would be the polar coordinates of the point graphed below?

Converting complex numbers from polar form to rectangular form

Go to the Complex numbers in polar form activity at ExploreMath.com. Graph (4.24, 2250) using the activity.

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Question 3a. How could you convert polar coordinates to rectangular coordinates? Hint: Multiply both sides of Euler’s formula by r. Question 3b. What are the rectangular coordinates for (4.24, 2250)? Question 3c. What are the rectangular coordinates for the point graphed below in polar coordinates?

ExploreMath.com Lesson Plan >> Complex numbers in polar form (Worksheet)>> Page 4 of 4

Conclusion The complex number z = a + bi can be represented in rectangular coordinates (a, b) or in polar coordinates (r, θθ ). The following formulas can be used to convert rectangular coordinates to polar coordinates: r = √√ (a2 + b2) and θθ = tan-1(b/a) Complex numbers can also be represented using the equation z = reiθθ discovered by Leonhard Euler.