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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 z x y i be a complex number a. What is z ? 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. 2 cos 45 i sin 45 a. 4 cos120 b. 5. B. i sin120 Convert the polar coordinates above into rectangular form. Euler’s Formula 1. Fill in the following chart: x e ix - rectangular form e ix - polar form 0 2 3 2 2 6 4 2. What conclusions can you make from the above analysis? 3. Convert the following into polar form i i 3 5e 3 8e 4 a. b. 4. Write the following in exponential form 2 cos 45 i sin 45 a. b. Math 120 Notes 11.3 4 cos120 i sin120 Page 1 of 2 III. Operations on complex numbers A. Addition / subtraction 1. Rectangular form a. 2 3i 5 8i b. 2. B. 7 6i 3 4i Multiplication 1. Rectangular form: 2. Polar form: 2 cos 45 i sin 45 5 cos 30 i sin 30 C. D. [HUH?] 3 3i 4 8i Polar form 2 cos 45 i sin 45 5 cos 30 i sin30 a. b. So in general, r1 cos 1 i sin 1 r2 cos 2 i sin 2 = ? Use the formula to find 2 2 cos 35 i sin35 10 cos 70 i sin70 Try multiplying 3 3i 4 8i by converting to polar form first. c. 3. Division 1. Rectangular form: 2. Polar form 4 7i 3 2i a. What do you think is the formula for c. Use the formula to find 8 cos 20 r1 cos 1 i sin 1 =? r2 cos 2 i sin 2 i sin 20 12 cos 75 i sin 75 Power – De Moivre’s Theorem 1. Using the multiplication rule, if z r cos i sin , what is z 2 , z 3 , and z n ? 2. 3. Prove your result for z n using exponential notation. Use De Moivre’s Theorem to write the following in standard form. 2 cos 20 i sin20 b. i sin 4 cos 10 10 Regular Coordinates Complex #s Vectors Math 120 Notes 11.3 a. 3 5 Rectangular Polar r , Exponential - x yi r cos i sin re i r - x , y x, y xiy j Page 2 of 2