About this tool
Do arithmetic with two complex numbers a+bi and c+di. Choose add, subtract, multiply, divide or power, and the calculator shows the result in rectangular form a+bi, its polar form r∠θ, and the modulus and argument. Everything is computed in your browser.
Addition and subtraction work component by component. Multiplication is (ac - bd) + (ad + bc)i. Division multiplies by the conjugate: ((ac + bd) + (bc - ad)i) / (c² + d²). Powers use De Moivre's theorem: with modulus r = √(a²+b²) and argument θ = atan2(b, a), the result is rⁿ(cos nθ + i sin nθ). The modulus of any result is √(re² + im²) and its argument is atan2(im, re), which is also the θ shown in the polar form.
For the power operation the exponent is taken from the real part of the second number, so set c to the power you want (for example c = 2 to square a+bi). Division by zero (0 + 0i) is rejected rather than returning infinity.