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10.4 Products & Quotients of Complex Numbers in Polar Form.

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Presentation on theme: "10.4 Products & Quotients of Complex Numbers in Polar Form."— Presentation transcript:

1 10.4 Products & Quotients of Complex Numbers in Polar Form

2 Exploration! We know how to multiply complex numbers such as… (2 – 7i)(10 + 5i) = i – 70i – 35i 2 = i – 70i + 35 = 55 – 60i So we can also multiply complex numbers in their polar forms. Lets try sum identities! *this pattern will happen every time, so we can generalize it: If z 1 = r 1 (cosθ 1 + isinθ 1 ) and z 2 = r 2 (cosθ 2 + isinθ 2 ) are complex numbers in polar form, then their product is z 1 z 2 = r 1 r 2 [cos(θ 1 + θ 2 ) + isin(θ 1 + θ 2 )] product:

3 Ex 1) Find product of Express in rectangular form. Draw a diagram that shows the two numbers & their product. w z zw

4 Ex 2) Interpret geometrically the multiplication of a complex number by i general polar: r (the modulus) is the same but angle is rotated counterclockwise Ex 3) Use your calculator to find the product.

5 Electricity Application V = IZ voltage = current impedance magnitude of voltage is real part of V Ex 4) In an alternating current circuit, the current at a given time is represented by amps and impedance is represented by Z = 1 – i ohms. What is voltage across this circuit? real part 2.73V

6 To understand the rule for dividing polar numbers, first we practice dividing rectangular complex numbers. Ex 5) Find the quotient of Remember the multiplicative inverse of 2 is ½ since 2 ½ = 1. The multiplicative inverse of a + bi is. By getting rid of i in denominator, we would get In general, the multiplicative inverse of z is

7 Ex 6) Find the multiplicative inverse of 5 – 14i If z 1 = r 1 (cosθ 1 + isinθ 1 ) and z 2 = r 2 (cosθ 2 + isinθ 2 ) are complex numbers in polar form, then the quotient quotient:

8 Ex 7) Find the quotient Convert back to rectangular. polar

9 Homework #1005 Pg 513 #1-51 odd


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