Posted by Lynn on Thursday, October 6, 2011 at 7:47pm.
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.
In polar form, the first is (1/sqrt(2),-pi/4)
The second is (2,-pi/3)
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