Complex Number Calculator - Add, Multiply, Divide, Powers, Roots & Polar Form
Complex Number Calculator
Arithmetic, powers and roots in rectangular and polar form
Every Basic Operation
Add, subtract, multiply and divide, raise to any real power, or list all n roots of a number.
Polar and Exponential
Each result is shown as a + bi, as r∠θ and as r·e^(iθ), with the argument in degrees and radians.
Flexible Input
Type 3+4i, -2i, 4i-1, engineering-style 3-4j, or polar values like 5∠53.13.
Private
All arithmetic runs in your browser.
Rectangular and Polar Form
A complex number a + bi has a real part a and an imaginary part b, where i2 = −1. Plotted on a plane, it sits at the point (a, b); its distance from the origin is the modulus |z| = √(a2 + b2) and its angle from the positive real axis is the argument θ. Adding and subtracting are easiest in rectangular form, while multiplying and dividing are easiest in polar form: the moduli multiply and the angles add.
That is De Moivre's theorem: (r∠θ)n = rn∠nθ. It also explains why a number has exactly n different n-th roots, spaced evenly around a circle 360°/n apart. For example, the three cube roots of 8 are 2, −1 + 1.732i and −1 − 1.732i. The argument shown here is the principal value, between −180° and 180°.
Key Takeaways
- Exact integer powers: Whole-number powers use repeated multiplication, so (1+i)^2 is exactly 2i.
- All the roots: Root mode lists every n-th root, not just the principal one.
- Clean output: Floating-point noise like 1e-16i is rounded away.