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

1. Modulus (r):
r = √(a^2 + b^2)
where a = 2 (real part) and b = -2 (imaginary part)

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

2. Argument (θ):
θ = arctan(b/a)
where a = 2 and b = -2

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

Now we can express the complex number 2 - 2i in polar form as:
2 - 2i = 2√2 * cis(-π/4)

Therefore, in polar form, the complex number 2 - 2i is:
2√2 cis(-π/4)