Complex number calculator
Add, subtract, multiply and divide complex numbers (a + bi), with modulus, argument and polar form.
How the calculation works
A complex number is z = a + bi, with i² = −1. Addition and subtraction work part by part. Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. Division: multiply top and bottom by the divisor's conjugate. The modulus is |z| = √(a²+b²) and the argument is the angle in the complex plane.
Examples
- (2 + 3i) + (1 − i) = 3 + 2i.
- |3 + 4i| = √(9 + 16) = 5.
Frequently asked questions
What are complex numbers used for?
Solving equations with no real solution (like x² = −1) and modeling electronics (impedance), waves and signal processing.
What is the imaginary unit i?
Defined by i² = −1, i.e. i = √(−1). It allows roots of negative numbers, impossible in the reals.
What is the modulus of a complex number?
The distance from (a, b) to the origin in the complex plane: |z| = √(a² + b²) — its "size".
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Updated on July 4, 2026 · by Rafael Rossi · Methodology & sources