Complex Number Multiplication Calculator
Multiplying two complex numbers (a + bi)(c + di) follows the ordinary distributive rule together with the defining identity i squared = -1. The result has real part ac - bd and imaginary part ad + bc. Enter the real and imaginary parts of both numbers and this calculator returns the product in rectangular form, along with the modulus and argument of the result, since multiplication scales by the moduli and adds the arguments. Every output is computed by the deterministic formula shown below.
Complex multiplication formula
(a + b*i)(c + d*i) = (ac - bd) + (ad + bc) * i
Real = ac - bd
Imaginary = ad + bc
Modulus = sqrt(Real^2 + Imaginary^2)
Argument = atan2(Imaginary, Real)
The product expands by distribution. The term bd carries a factor of i squared = -1, which is why it is subtracted in the real part.
About complex multiplication
- Multiplying multiplies the moduli and adds the arguments of the two numbers.
- Multiplying by i rotates a complex number by 90 degrees counterclockwise.
- Complex multiplication is commutative, associative and distributive.
- The product of a number and its conjugate is the real squared modulus.
- This operation underlies phasor arithmetic in alternating-current circuit analysis.
Complex multiplication: frequently asked questions
How do you multiply two complex numbers?
Use the FOIL rule and i squared = -1. For (a + bi)(c + di) the real part is ac - bd and the imaginary part is ad + bc. So the product is (ac - bd) + (ad + bc)i.
Why does i squared equal minus one?
The imaginary unit i is defined so that i squared equals -1. This single rule is what makes the cross terms in the product combine the way they do and gives complex arithmetic its structure.
What happens to modulus and argument when multiplying?
Multiplying complex numbers multiplies their moduli and adds their arguments. So |z1 z2| = |z1| times |z2|, and arg(z1 z2) = arg(z1) + arg(z2). This calculator reports both for the product.
Is complex multiplication commutative?
Yes. The order does not matter: (a + bi)(c + di) gives the same product as (c + di)(a + bi). Complex multiplication is commutative, associative and distributive over addition.
What is the geometric meaning?
Multiplying by a complex number both scales by its modulus and rotates by its argument. This is why complex numbers are a natural tool for two-dimensional rotations and signal phase shifts.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Variables.
- NIST Digital Library of Mathematical Functions: DLMF home.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.