9-6: The Complex Plane and Polar
Form of Complex Numbers
Objectives
•Graph complex numbers in the complex plane.
•Convert complex numbers from rectangular
to polar form and vice versa.
Complex Numbers
Rectangular form of complex numbers: a + bi
Sometimes written as an ordered pair (a,b).
Example
Solve the equation
3x + 2y – 7i = 12 + xi – 3yi
for x and y where x and y
are real numbers.
Complex Plane
Complex plane (or Argand plane)
i
R
real axis
imaginary axis
Distance
Distance from origin: z = a+bi
z  a b
2
z 
2
2
a b
2
2
absolute value
of a complex
number
b
a
Example
Graph each number in the complex plane and find
its absolute value.
1. z = 4+3i 2. z = 2.5i
Polar Coordinates
a+bi can be written as rectangular coordinates (a,b). It can
also be converted to polar coordinates (r,θ).
r: absolute value or modulus of the complex number
θ: amplitude or argument of the complex number (θ is not
unique)
Polar Coordinates
So a=rcosθ and b=rsinθ.
z=a+bi
=rcosθ +(rsin θ)i
=r(cosθ +i sinθ)
=rcisθ
Polar form (or
trigonometric form)
Example
Express the complex number 1 – 4i in polar form.
Example
Express the complex number -3 – 2i in polar form.
Example
Graph
form.
 5 
2 cis 

 6 
. Then express it in rectangular
Homework
9-6 p. 590
#15-42 multiples of 3
#48