Complex Number Multiplication Calculator

Work out complex number multiplication instantly with clear inputs, formula shown and shareable results.

About Complex Number Multiplication Calculator

Multiplying uses i² = −1, giving (ac − bd) + (ad + bc)i. In polar terms the moduli multiply and the arguments add, so multiplication is a rotation combined with a scaling.

Formulas

Complex multiplication

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

Polar form

|z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂

Worked example

With the default values below, Complex Number Multiplication Calculator returns Product of -5 + 14i. Change any input above and every figure here is recalculated in your browser.

Inputs used

First real part: 3
First imaginary part: 2
Second real part: 1
Second imaginary part: 4

Results

Product: -5 + 14i
Modulus of the product: 14.86606875
Argument of the product: 109.653824 °
|z₁| × |z₂| (must match): 14.86606875

Step by step

1Values used
2Complex multiplication
3Polar form
4Product= -5 + 14i
5Modulus of the product= 14.86606875
6Argument of the product= 109.653824 °
7|z₁| × |z₂| (must match)= 14.86606875

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Multiply 3 + 2i by 1 + 4i.

(3 − 8) + (12 + 2)i = −5 + 14i, of modulus √221 ≈ 14.8660687.

Why do the arguments add?

Because eⁱᵃ · eⁱᵇ = eⁱ⁽ᵃ⁺ᵇ⁾, which is Euler's formula turning multiplication into rotation.