How this calculator works
Convert polar factors to rectangular components before expanding. The real terms combine as ac-bd because the last term contains i². The other two terms combine as (ad+bc)i. In polar language the magnitudes multiply and the phases add, followed by wrapping to the principal interval from -π to π. A zero product has no defined phase. If an intermediate term exceeds the numeric range the tool returns an error.
Formula
(a+bi)(c+di)=(ac-bd)+(ad+bc)i. r₁e^(iφ₁) × r₂e^(iφ₂)=r₁r₂e^[i(φ₁+φ₂)].
Worked example
(1+2i)(3-4i)=3-4i+6i-8i²=11+2i. Its magnitude is √125. Multiplying i by i gives -1, a pure real result.
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Frequently asked questions
Can one factor be polar and the other rectangular?
Yes. Each factor has its own form selector and units for its phase.
Why is bd subtracted?
The product bi×di is bd i², and i² equals -1.
Why does the phase disappear at zero?
Zero has no direction from the complex origin. Every phase with magnitude zero represents the same number.