10.3 Polar Form of Complex Numbers

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

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

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)

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

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!

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.

Homework #1004 Pg 506 #1–27 odd (skip 7 & 11), 33, 46, 47, 48