How this calculator works
Complex arithmetic keeps real and imaginary parts separate using i^2 = -1. Division multiplies by a conjugate, while powers are evaluated as exp(w log z). The principal logarithm chooses an argument from -pi to pi. Magnitude is non-negative and phase uses the engine angle selector. Zero has no phase.
Formula
(a+bi)(c+di) = (ac-bd) + (ad+bc)i. log(z) = ln|z| + i arg(z). z^w = exp(w log(z)).
Worked example
(1 + 2i)(3 - 4i) = 3 - 4i + 6i - 8i^2 = 11 + 2i. For division, (3 + 4i)/(1 - 2i) = (-5 + 10i)/5 = -1 + 2i.
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Frequently asked questions
Why can powers have more than one value mathematically?
A complex logarithm can differ by multiples of 2pi i. This calculator consistently uses the principal branch.
Can I divide by zero?
No. Both parts of the divisor cannot be zero.
Are phase values degrees?
The default readout is radians. Select degrees or turns in the output unit control.