11.2 – Geometric Representation of Complex Numbers

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Presentation transcript:

11.2 – Geometric Representation of Complex Numbers Essential Question: How do you multiply and divide complex numbers?

(Complex Plane) To represent complex numbers we use the complex plane. However, this graph has an imaginary axis and real axis to be able to graph numbers like 5-2i. Argand Diagram: (Complex Plane) real axis imaginary axis 2-4i -4-0i 4+3i 0+4i

z = a + bi with coordinates, (a , b) Polar Form: Rectangular Form: z = a + bi with coordinates, (a , b) Polar Form: z = r cos  +(r sin )i = r cis  Ex. 2 cis 30º = 2(cos 30º + isin 30º) The length of the arrow representing z is

Examples: a. Express 6 cis 30 in rectangular form. b. Express 1 + i in polar form. c. Express - i in polar form.

Product of 2 Complex Numbers in Polar Form: Division: 1. Divide their absolute values. 2. Subtract the angle measures. Product of 2 Complex Numbers in Polar Form: To multiply two complex numbers in polar form: 1) Multiply their absolute values. 2) Add their polar angles.  (r cis α )(s cis β) = rs cis(α + β)

Example: Express (3 cis 165)(4 cis 45) in polar and rectangular form.

Example: Express (8 cis 300)  (2 cis 30) in rectangular form.

Example: Let z1 = 2 – 2i and z2 = 3i. a. Find z1z2 in rectangular form.

b. Find z1 , z2 , and z1 z2 in polar form. c. Convert z1 z2 to rectangular form.