Complex Number Calculator

How it works

A complex number a + bi has a real part a and an imaginary part b, where i is the square root of −1. Adding and subtracting work part by part, exactly like adding vectors: (a + bi) + (c + di) = (a + c) + (b + d)i.

Multiplying expands the brackets and uses i² = −1: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac − bd) + (ad + bc)i. Dividing uses the conjugate trick: multiplying the top and bottom of (a + bi) ÷ (c + di) by c − di turns the bottom into the real number c² + d², and the result is ((ac + bd) + (bc − ad)i) ÷ (c² + d²).

Every result is also described geometrically. Its modulus |z| = √(a² + b²) is its distance from the origin of the complex plane and its argument is the angle from the positive real axis, measured counter-clockwise, so z = r(cos θ + i sin θ), often abbreviated r cis θ. Multiplying complex numbers multiplies their moduli and adds their arguments, which is why polar form is the natural way to handle powers and roots.

The conjugate of a + bi is a − bi, the reflection across the real axis. A number times its conjugate is its modulus squared, a real number, which is what makes division possible.

Formula

(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) − (c + di) = (a − c) + (b − d)i
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
(a + bi) ÷ (c + di) = ((ac + bd) + (bc − ad)i) ÷ (c² + d²)
|z| = √(a² + b²)      arg z = atan2(b, a)      z̄ = a − bi
z = r(cos θ + i sin θ) = r cis θ

Example

(3 + 4i)(1 − 2i) = (3×1 − 4×(−2)) + (3×(−2) + 4×1)i = 11 − 2i. Its modulus is √(11² + 2²) = 11.18034 and its argument is atan2(−2, 11) = −10.3048°, so in polar form it is 11.18034(cos −10.3048° + i sin −10.3048°). The conjugate is 11 + 2i.

(3 + 4i) ÷ (1 − 2i): multiply top and bottom by the conjugate 1 + 2i to get ((3×1 + 4×(−2)) + (4×1 − 3×(−2))i) ÷ (1² + 2²) = (−5 + 10i) ÷ 5 = −1 + 2i. Check: (−1 + 2i)(1 − 2i) = 3 + 4i.

3 + 4i on its own has modulus √(9 + 16) = 5 and argument atan2(4, 3) = 53.1301° (0.927295 rad), the 3-4-5 triangle.

Assumptions and limitations

• The argument is reported in the range (−180°, 180°], the principal value that atan2 gives; add 360° if you need a positive angle. The argument of 0 is reported as 0.
• Results are cleaned to 15 significant digits to remove floating-point noise and shown to six decimal places. Inputs are ordinary decimals; fractions such as 3/17 appear as 0.176471.
• Dividing by 0 + 0i is undefined and reported as an error.
• This is arithmetic on two numbers. Powers, roots and solving polynomial equations are not covered.

Frequently asked questions