Formula and Method for Multiplying Complex Numbers
A complex number z = a + bi has a real part a and an imaginary part b, where i is the imaginary unit defined by i² = −1. To multiply two complex numbers z₁ = a + bi and z₂ = c + di, treat them like binomials and expand with FOIL (First, Outer, Inner, Last), then simplify using i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. This calculator also reports the modulus of the product, |z₁ × z₂| = |z₁| × |z₂| = √(a²+b²) × √(c²+d²).
How the calculation works
Expanding (a + bi)(c + di) term by term gives ac + adi + bci + bdi². The last term bdi² equals −bd because i² = −1, so it becomes a real number that combines with ac. The middle two terms, adi and bci, both carry a single factor of i, so they combine into (ad + bc)i. Collecting the real terms and the imaginary terms separately gives the final result: real part = ac − bd, imaginary part = ad + bc. For example, (3 + 2i)(1 + 4i) = (3×1 − 2×4) + (3×4 + 2×1)i = (3 − 8) + (12 + 2)i = −5 + 14i.
Common mistakes
- Forgetting i² = −1: the most common error is leaving the bdi² term as +bd instead of flipping its sign to −bd.
- Mixing up real and imaginary parts: keep a, c as real parts and b, d as imaginary-part coefficients; swapping them changes the answer.
- Sign errors with negative parts: when b or d is negative, carry the negative sign through the FOIL expansion carefully — do not drop it.
Real-world applications
- Electrical engineering multiplies complex impedances and phasors to analyze AC circuits, where the imaginary part represents reactance.
- Signal processing uses complex multiplication in the Fourier transform and filter design to combine magnitude and phase.
- Computer graphics and robotics use complex multiplication to represent 2D rotation and scaling in a single operation.
- Control systems and quantum mechanics use complex multiplication to combine transfer functions and probability amplitudes.
Frequently Asked Questions
What is the formula for multiplying two complex numbers?
To multiply z₁ = a + bi and z₂ = c + di, use the FOIL expansion together with i² = −1: z₁ × z₂ = (ac − bd) + (ad + bc)i. For example, (3 + 2i)(1 + 4i) = (3×1 − 2×4) + (3×4 + 2×1)i = −5 + 14i.
Why does i² = −1 matter when multiplying complex numbers?
Expanding (a+bi)(c+di) with FOIL gives ac + adi + bci + bdi². Since i² = −1, the bdi² term becomes −bd, a real number, so it combines with ac to form the real part of the product, while adi and bci combine into the imaginary part.
How do you multiply complex numbers in polar form?
In polar form z = r(cos θ + i sin θ), multiplying two complex numbers multiplies their moduli and adds their arguments: |z₁ × z₂| = |z₁| × |z₂| and arg(z₁ × z₂) = arg(z₁) + arg(z₂). This calculator reports the modulus of the product directly.
What does multiplying complex numbers represent geometrically?
Multiplying by a complex number z scales a point in the plane by |z| and rotates it by z's argument (its angle from the positive real axis). This is why complex multiplication is used to model rotation and scaling in engineering, signal processing, and computer graphics.
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
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| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
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