1. 9-6: The Complex Plane and Polar Form of Complex Numbers

2. Objectives Graph complex numbers in the complex plane.
Convert complex numbers from rectangular to polar form and vice versa.

3. Complex Numbers Rectangular form of complex numbers: a + bi
Sometimes written as an ordered pair (a,b).

4. Example Solve the equation 3x + 2y – 7i = 12 + xi – 3yi
for x and y where x and y
are real numbers.

5. Complex Plane Complex plane (or Argand plane) i R real axis
imaginary axis

6. Distance Distance from origin: z = a+bi
absolute value of a complex number b
a+bi a

7. Example
Graph each number in the complex plane and find its absolute value.
1. z = 4+3i 2. z = 2.5i

8. 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)

9. 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)

10. Example
Express the complex number 1 – 4i in polar form.

11. Example
Express the complex number -3 – 2i in polar form.

12. Example
Graph Then express it in rectangular form.

13. Homework
9-6 p. 590
#15-42 multiples of 3
#48