Complex Numbers Calculator

Arithmetic, Modulus, Argument, Conjugate, Polar Form & De Moivre's Theorem

Quick Answer – What are Complex Numbers?

A complex number is of the form z = a + ib, where i = √(−1). The modulus |z| = √(a² + b²) gives the distance from the origin, and the argument arg(z) is the angle with the positive real axis. Polar form z = r(cosθ + i sinθ) enables easy multiplication and powers via De Moivre's theorem.

Key Formulas

|z| = √(a² + b²),   θ = tan−1(b/a) [adjusted for quadrant]
Conjugate: z̅ = a − ib
z1z2 = (a1a2 − b1b2) + i(a1b2 + a2b1)
Polar: z = r eiθ = r(cosθ + i sinθ)
De Moivre: (r eiθ)n = rn (cos nθ + i sin nθ)

Solved Examples

Example 1: (3 + 2i) + (4 − i) = 7 + i

Example 2: Modulus and argument of 3 + 4i

|z| = √(9+16)=5, arg(z) = tan−1(4/3) ≈ 53.13°

Example 3: (2 ei30°)3 = 8 ei90° = 8i

Frequently Asked Questions

Q1. How do I enter negative or decimal numbers?
Simply type with a minus sign or decimal point. Commas are converted to dots.
Q2. What is the range of argument output?
The principal argument is given in (−180°, 180°] (or (−π, π] radians). The calculator displays degrees by default.
Q3. Can I use fractions as exponents?
De Moivre works best for integer exponents. For fractional exponents (roots), the calculator gives one principal value; all n roots can be found by adding multiples of 360°/n.
Q4. How precise are the results?
Results are shown to 4 decimal places. Internal calculations use double precision.
Q5. Does the polar form support radians?
The argument input is in degrees. To use radians, convert manually or use the properties tab to get radians (by multiplying degree output by π/180).