Convert the rectangular form of the complex number 2−2i into polar form. Show all work and label the modulus and argument.

Answer 1

To convert the complex number 2−2i from rectangular form to polar form, we first need to find the modulus and the argument.

Given complex number: z = 2−2i

Modulus (r):
r = √(2^2 + (-2)^2)
r = √(4 + 4)
r = √8
r = 2√2

Argument (θ):
θ = arctan(-2/2)
θ = arctan(-1)
θ = -π/4

Now, we can express the complex number in polar form:

z = 2√2*(cos(-π/4) + i*sin(-π/4))

Therefore, the polar form of the complex number 2−2i is:
z = 2√2*(cos(-π/4) + i*sin(-π/4))