How this calculator works
The real component is the horizontal projection of a vector of length r, and the imaginary component is its vertical projection. Multiplying a complex number by i turns that vector anticlockwise by 90 degrees, sending (a,b) to (-b,a). The table shows four positions without changing magnitude. Exact multiples of a quarter turn receive exact zero axis components; other angles retain full numerical precision.
Formula
a=r cos(θ); b=r sin(θ); z=a+bi. iz=-b+ai; i²z=-a-bi; i³z=b-ai.
Worked example
Magnitude 2 and phase 60° give a=1 and b=√3, so z=1+√3i. A quarter turn produces -√3+i. Magnitude 5 and phase -90° instead give -5i.
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Frequently asked questions
Can the magnitude be negative?
This tool uses the standard non-negative magnitude convention. To represent the opposite direction, add 180 degrees to the angle.
Can I enter degrees or radians?
Yes. Select the phase unit beside the field. Calculations use radians internally.
Does a phase matter when magnitude is zero?
Every phase then produces real and imaginary parts equal to zero. No unique phase can be recovered from that result.