Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Complex Multiplication Calculator.
Read the complete guideFOIL Method
(a+bi)(c+di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i. In polar form: z₁·z₂ = r₁r₂ · cis(θ₁ + θ₂), which shows that moduli multiply and arguments add.
Examples
Multiply (3+4i)(1−2i)
Compute the product of 3+4i and 1−2i.
FOIL: (3)(1) + (3)(−2i) + (4i)(1) + (4i)(−2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i.
Key takeaway: The key step is replacing i² with −1, which turns the −8i² term into +8.
Efficient Complex Multiplication
Choose the best method for the situation:
- Use FOIL for small numbers where the algebra is straightforward
- Use polar form when you need the modulus and argument of the product
- Multiplying by i rotates 90° counterclockwise — a useful mental shortcut
Frequently Asked Questions about Complex Multiplication Calculator
Why does multiplication rotate complex numbers?
When you multiply z₁ = r₁∠θ₁ by z₂ = r₂∠θ₂, the result is r₁r₂∠(θ₁+θ₂). The argument (angle) of the product is the sum of the arguments — this is a rotation by θ₂.
What happens when you multiply a complex number by its conjugate?
z · z̅ = |z|², which is always a non-negative real number. This is the basis for rationalizing complex denominators.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Operation
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