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**10.4 Products & Quotients of Complex Numbers in Polar Form**

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**z1z2 = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)]**

Exploration! We know how to multiply complex numbers such as… (2 – 7i)(10 + 5i) = i – 70i – 35i2 = i – 70i + 35 = 55 – 60i So we can also multiply complex numbers in their polar forms. Let’s try sum identities! *this pattern will happen every time, so we can generalize it: If z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2) are complex numbers in polar form, then their product is z1z2 = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)] product:

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**Express in rectangular form. **

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

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**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.

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**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? 2.73V real part

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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

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**Ex 6) Find the multiplicative inverse of 5 – 14i**

quotient: If z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2) are complex numbers in polar form, then the quotient

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

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Homework # Pg #1-51 odd

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Slide 7.5 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Slide 7.5 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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