Complex numbers extend the real numbers with i, where i² = −1. They are essential in electrical engineering (impedance), signal processing (Fourier transforms), quantum mechanics and anywhere rotations and oscillations appear.
How to use the Complex Number Calculator
- z₁ real part (a)
- z₁ imaginary part (b)
- z₂ real part (c)
- z₂ imaginary part (d)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Addition and subtraction work component by component. Multiplication uses i² = −1, and division multiplies by the conjugate of the denominator. Powers and logarithms use polar form, z = |z|e^{iφ}, with the principal argument between −180° and 180°.
Formula
Sum = (a + c) + (b + d)i
Product = (ac − bd) + (ad + bc)i
Quotient = [(ac + bd) + (bc − ad)i] / (c² + d²)
ln z = ln|z| + i·arg z
Worked example
Inputs — z₁ real part (a): 3; z₁ imaginary part (b): 4; z₂ real part (c): 1; z₂ imaginary part (d): -2.
Results — z₁ + z₂ = 4 + 2i; z₁ − z₂ = 2 + 6i; z₁ × z₂ = 11 − 2i; z₁ ÷ z₂ = −1 + 2i; |z₁| = 5; Argument φ₁ = 53.13010235° (0.92729522 rad).
Key terms
- Imaginary unit i
- A number with i² = −1.
- Argument
- The angle of a complex number in the complex plane.
Frequently asked questions
How do I multiply complex numbers?
Expand like binomials and use i² = −1: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i.
How do I divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom so the denominator becomes real.
What is the argument?
The angle φ of the point (a, b) measured from the positive real axis.
Is z₁^z₂ unique?
No – complex powers have many values; the calculator gives the principal value.