1. 10.4 Products & Quotients of Complex Numbers in Polar Form

2. 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:

3. 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

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?
2.73V
real part

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

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

9. Homework
# Pg #1-51 odd