Multiplication Formulas for Complex Numbers
Complex number multiplication differs depending on the form you're working with. The rectangular approach uses the distributive property combined with the fact that i² = −1, while the polar method exploits the relationship between magnitude and phase.
Rectangular form (a + bi) × (c + di):
= ac + adi + bci + bdi²
= ac + adi + bci − bd
= (ac − bd) + i(ad + bc)
Polar form r₁∠φ₁ × r₂∠φ₂:
= (r₁ × r₂)∠(φ₁ + φ₂)
a, c— Real parts of the first and second complex numbersb, d— Imaginary coefficients of the first and second complex numbersr₁, r₂— Magnitudes (absolute values) of the two numbersφ₁, φ₂— Phase angles (arguments) measured in degrees or radians
Working with Rectangular Form
When both numbers are expressed as a + bi, apply the distributive property carefully. Expand all four terms from (a + bi)(c + di), then consolidate using i² = −1.
- Real part: Multiply real-to-real and imaginary-to-imaginary: ac − bd
- Imaginary part: Multiply real-to-imaginary and imaginary-to-real: ad + bc
For example, (3 + 2i)(1 − 4i) yields:
- Real: 3(1) − 2(−4) = 3 + 8 = 11
- Imaginary: 3(−4) + 2(1) = −12 + 2 = −10
- Product: 11 − 10i
Why Polar Form Simplifies Multiplication
Converting to polar form reveals elegant behavior: magnitudes multiply, and phases add. This property emerges from Euler's formula and makes certain calculations—especially cascaded operations—far simpler than rectangular arithmetic.
If z₁ = 5∠30° and z₂ = 2∠45°, then z₁ × z₂ = 10∠75°. No expansion, no combining like terms. This advantage becomes pronounced in electrical engineering when tracking impedance chains or in signal processing with Fourier transforms.
Conversion between forms:
r = √(a² + b²)andφ = arctan(b/a)(rectangular to polar)a = r cos(φ)andb = r sin(φ)(polar to rectangular)
Common Pitfalls and Practical Advice
Avoid these frequent mistakes when multiplying complex numbers.
- Forgetting the i² = −1 rule — When expanding (a + bi)(c + di), the term bdi² becomes −bd, not +bd. Double-check that you've applied this sign flip; it's the source of most rectangular-form errors.
- Mixing units in polar form — If one angle is in degrees and the other in radians, convert to a single unit before adding. A 30° phase plus a 0.785 rad phase requires consistency, or your result will be nonsensical.
- Magnitude and phase ambiguity — A complex number has infinitely many polar representations due to periodicity: 5∠30° = 5∠390° = 5∠(−330°). Use the principal argument [0°, 360°) or [−180°, 180°] to avoid confusion.
- Precision loss in successive conversions — Converting rectangular → polar → rectangular introduces rounding errors, especially with many operations. Stick to one form for a calculation sequence, then convert only the final result.
Practical Applications
Complex multiplication appears everywhere in technical fields:
- AC circuit analysis: Impedances are multiplied when combining series and parallel networks; voltage and current phasors multiply in power calculations.
- Signal processing: Filter design and convolution rely on multiplication in the frequency domain (via FFT).
- Quantum mechanics: Probability amplitudes are complex numbers whose products determine interference patterns.
- Control systems: Transfer function cascade multiplication uses complex pole-zero analysis.
In each domain, rapid, accurate computation prevents design errors and accelerates iteration.