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 absolute value of a complex number b a+bi 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 . Then express it in rectangular form.

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