Complex number calculator

Expand the brackets and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 4i)(1 − 2i) that is (3 + 8) + (−6 + 4)i = 11 − 2i. In polar form the rule is shorter: multiply the moduli and add the angles, so 5∠53.13° × 2.236∠−63.43° = 11.18∠−10.30°.

Multiply the top and bottom by the conjugate of the divisor, which makes the denominator a real number. (3 + 4i)/(1 − 2i) = (3 + 4i)(1 + 2i)/(1² + 2²) = (3 + 6i + 4i + 8i²)/5 = (−5 + 10i)/5 = −1 + 2i. Division by 0 + 0i is undefined.

The modulus is r = √(a² + b²) and the argument is θ = atan2(b, a), which picks the correct quadrant. For 3 + 4i, r = √(9 + 16) = 5 and θ ≈ 53.130°, so 3 + 4i = 5∠53.130°. Plain arctan(b/a) gives the wrong angle when the real part is negative: −3 − 4i has an argument of about −126.870°, not 53.130°.

For z = r(cos θ + i sin θ) and a whole number n, zⁿ = rⁿ(cos nθ + i sin nθ). So (1 + i)⁸, with r = √2 and θ = 45°, equals (√2)⁸ = 16 at an angle of 8 × 45° = 360°, which is the real number 16. The same idea gives n evenly spaced nth roots: the cube roots of 8 are 2 and −1 ± 1.732051i, 120° apart.

The imaginary unit i, defined by i² = −1; −i is the other square root. Every negative number has two imaginary square roots, so √−4 = ±2i, and the principal root, the one with argument 90°, is 2i. Powers of i repeat every four steps: i, −1, −i, 1, then i again.

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “(3 + 4i)(1 − 2i)” is checked against 3 − 6i + 4i − 8i² = 11 − 2i (hand calculation; Python (3+4j)*(1-2j)).

Wolfram MathWorld — Complex Number; Wolfram MathWorld — de Moivre's Identity.