Multiply Complex Numbers Calculator
Enter two complex numbers z₁ = a + bi and z₂ = c + di and instantly get their product (ac − bd) + (ad + bc)i, magnitudes, and argument.
- 1
First: a × c
3 × 1 = 3 - 2
Last: b × d (subtracted — i² = −1)
2 × 4 = 8Because i² = −1 this term is subtracted from the real part. - 3
Real part: ac − bd
3 − 8 = -5 - 4
Outer: a × d
3 × 4 = 12 - 5
Inner: b × c
2 × 1 = 2 - 6
Imaginary part: ad + bc
12 + 2 = 14
How does this calculator work?
Multiply (a + bi)(c + di) = (ac − bd) + (ad + bc)i using FOIL. The real part subtracts the imaginary cross products (since i² = −1); the imaginary part sums the other two cross terms. Geometrically: magnitude multiplies (|z₁| · |z₂|) and argument adds (arg(z₁) + arg(z₂)).
Formula
How this is calculated
Two complex numbers z₁ = a + bi and z₂ = c + di are multiplied by applying the distributive law (FOIL). The real part of the product is ac − bd — the "real × real" and "imag × imag" terms, with the sign flipped on the second because i² = −1. The imaginary part is ad + bc, from the two cross terms. The full product is (ac − bd) + (ad + bc)i.
Geometrically, multiplying complex numbers in the Argand plane combines scaling and rotation: the magnitude of the product equals |z₁| · |z₂| (verified by |z₁ · z₂|² = (a² + b²)(c² + d²) = (ac−bd)² + (ad+bc)²), and its argument equals arg(z₁) + arg(z₂). The vector diagram draws both inputs and the resultant from the origin — a large imaginary component tilts the result away from the real axis; a larger magnitude stretches it outward.
Complex multiplication is both commutative (z₁ · z₂ = z₂ · z₁) and associative. Results are exact for finite floating-point inputs. The argument is reported in degrees in the range (−180°, 180°].
Frequently asked questions
By definition, the imaginary unit i satisfies i² = −1. Expanding (a + bi)(c + di) using FOIL gives ac + adi + bci + bdi². The last term bdi² = −bd, which flips the sign and becomes part of the real component, giving real part ac − bd.
In the complex (Argand) plane, multiplying z₁ by z₂ scales z₁ by |z₂| and rotates it counter-clockwise by arg(z₂). For example, multiplying any complex number by i (magnitude 1, argument 90°) rotates it 90° counter-clockwise without changing its length.
Yes — the modulus is multiplicative: |z₁ · z₂| = |z₁| · |z₂|. This follows algebraically from (ac−bd)² + (ad+bc)² = (a²+b²)(c²+d²), which equals |z₁|² · |z₂|².
Also known as
TG we-Calculate Editorial Team. (2026). Multiply Complex Numbers Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/multiply-complex-numbers-calculator
TG we-Calculate Editorial Team. "Multiply Complex Numbers Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/multiply-complex-numbers-calculator.
TG we-Calculate Editorial Team, "Multiply Complex Numbers Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/multiply-complex-numbers-calculator
@misc{wecalculate_multiply_complex_numbers_calculator, title = {Multiply Complex Numbers Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/multiply-complex-numbers-calculator}}, year = {2026}, note = {TG we-Calculate} }
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