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Complex form

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.5*sqrt(2) + .5*sqrt(2)*i      or     -.5*sqrt(2) - .5*sqrt(2)*i

 

An interesting thing I was shown was that, if you look at the complex plane, multiplying two complex numbers results in adding the angles and multiplying the magnitudes of the vectors representing the complex numbers being multiplied.

This makes finding an nth root of complex numbers fairly simple.  Simply take the nth root of the magnitude, and divide the angle by n.  Then you just calculate the real and imaginary parts from the resulting vector using sine and cosine.  Finding all the other values for the nth roots is done by simply adding 2pi/n repeatedly to the angle of any found nth roots.

 

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