How this calculator works
Polar inputs first become real and imaginary components. Multiplying numerator and denominator by the divisor conjugate removes the imaginary denominator. The step table uses a scaled divisor to reduce overflow when its components are large. A zero numerator is valid, but has no phase. The phase readout uses the principal angle from -π to π.
Formula
(a + bi)/(c + di) = [(ac + bd) + (bc - ad)i]/(c² + d²). In polar form, divide magnitudes and subtract phases. The table sets s = max(|c|,|d|), C = c/s, D = d/s before applying the same identity.
Worked example
(3 + 4i)/(1 - 2i) has numerator -5 + 10i after conjugate multiplication and real denominator 5. The quotient is -1 + 2i, with magnitude √5.
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Frequently asked questions
Can the two inputs use different forms?
Yes. Either the numerator or divisor can be polar while the other is rectangular.
Why does the conjugate change the sign?
The conjugate of c + di is c - di. Their product is c² + d² because i² = -1.
Can a polar magnitude be negative?
Use a non-negative magnitude and express direction with the phase. The divisor magnitude must be strictly positive.