10.4 Products & Quotients of Complex Numbers in Polar Form

z1z2 = r1r2[cos(θ1 + θ2) + isin(θ1 + θ2)] Exploration! We know how to multiply complex numbers such as… (2 – 7i)(10 + 5i) = 20 + 10i – 70i – 35i2 = 20 + 10i – 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:

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

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.

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

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

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

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

Homework #1005 Pg 513 #1-51 odd