How to convert into polar form?

Question

z = 1 - i

w = 1 - √3i

Answer 1

The polar form expresses a complex number as:

z = |z| exp(i theta)

theta is the angle with the positive real axis.

For z = 1 - i, we have:

|z| = sqrt(2)

Then it's easy to see that you can take theta = -pi/4:

exp(-i pi/4) = cos(pi/4) - i sin(pi/4) =
1/sqrt(2) (1-i).

Then in case of

w = 1 - √3i

you have:

|w| = sqrt(4) = 2

And you easly see that theta = -pi/3

You can compute theta directly by taking arctan of imaginary part divided by the real part, but you may then need to add pi to this. If you multiply z by -1, theta changes by plus or minus pi, while the ratio stays the same.

Answer 2

In polar form, the first is (1/sqrt(2),-pi/4)

The second is (2,-pi/3)