1. 10.3 Polar Form of Complex Numbers

2. We have explored complex numbers as solutions.
Now we connect to both the rectangular and polar planes.
Every complex number can be represented in the form
a + bi
real part corresponds to x-axis
imaginary part corresponds to y-axis
horizontal axis = real axis
vertical axis = imaginary axis P
a + bi
 (a, b)
 (r, θ)
argument θ r b
a = rcos θ
& b = rsin θ θ
a + bi
= rcos θ + irsin θ
= r(cos θ + isin θ) a
modulus
/ absolute value of a + bi

3. Ex 1) Graph each complex number and find the modulus.
A) i
B) –2i
(2, 3)
(0, –2)
modulus:
modulus: A B

4. The expression r(cos θ + isin θ) is often abbreviated r cis θ.
This is the polar form of the complex number.
(a + bi is the rectangular form)
We need to be able to convert between the forms.
Ex 2) Express the complex number in rectangular form. A) B)

5. Ex 3) Express each complex number in polar form. Use θ  [0, 2π)
A) z = 2 – 2i  x = y = –2
On your own
in QIV B)
in QII

6. Ex 3) Express each complex number in polar form. Use θ  [0, 2π)
C) z = –60 – 11i  x = –60 y = –11
in QIII D)
remember: cos (–π) = cos π
sin (–π) = –sin π
needs to be positive!

7. Ex 4) Show that the product of a complex number and its conjugate is
always a real number.
Let and its conjugate
Then,
which is a real number.

8. Homework
# Pg #1–27 odd (skip 7 & 11), 33, 46, 47, 48