Complex Number Calculator
Perform arithmetic on complex numbers with magnitude and argument.
All calculations happen locally in your browser — nothing is uploaded, stored or tracked.
Enter two complex numbers
Real and imaginary parts for z₁ and z₂.
Pick the operation
Add, subtract, multiply or divide.
Read the result
Including magnitude, argument and conjugate.
What Is a Complex Number Calculator?
A complex number combines a real part and an imaginary part: z = a + bi, where i is the square root of −1. Complex numbers appear throughout engineering, physics and signal processing — anywhere alternating current, waves or rotation are modelled.
The calculator performs addition, subtraction, multiplication and division of two complex numbers, and reports the magnitude (distance from the origin), the argument (angle in degrees) and the conjugate of the first number.
The Rules
Division multiplies by the conjugate of the denominator to make the denominator real. Results are displayed in standard a + bi form with up to 6 decimal places.
Your data never leaves your device.
Every calculation on this site runs in your browser using vanilla JavaScript. Nothing is uploaded, stored or tracked.
- 100% local, client-side calculations
- No data stored on any server
- No tracking, no analytics, no ads scripts
- Works offline once loaded
Frequently Asked Questions
What is i in complex numbers?
i is the imaginary unit, defined as the square root of −1, so i² = −1.
How do you multiply complex numbers?
Expand like binomials and replace i² with −1: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the denominator. This makes the denominator a real number.
What is the magnitude of a complex number?
Its distance from the origin in the complex plane: √(a² + b²).
What is the argument of a complex number?
The angle the number makes with the positive real axis, measured in degrees by this calculator.