Complex Number Calculator
Add, multiply, divide and raise complex numbers, with polar form.
About this tool
Type two complex numbers, such as 3+4i, -2i, 5 or polar 5∠53.13, and choose an operation. The calculator shows the result and its modulus, argument in radians and degrees, conjugate, polar form, exponential form and square roots. Open the details to see the same properties for each input. Engineers can type j instead of i.
How complex number arithmetic works
A complex number a + bi has a real part a and an imaginary part b, where i is defined by i² = −1. To add or subtract, combine the real parts and the imaginary parts separately. To multiply, expand the brackets and replace i² with −1: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i. To divide, multiply the top and bottom by the conjugate of the divisor, which makes the denominator real: (3 + 4i) ÷ (1 − 2i) = −1 + 2i. On the complex plane, a + bi is a point. Its distance from the origin is the modulus, √(a² + b²), and the angle from the positive real axis is the argument. For 3 + 4i, the modulus is 5 and the argument is 53.13°, so its polar form is 5∠53.13° and its exponential form is 5e^(0.927i). Multiplying in polar form multiplies the moduli and adds the angles, which is why powers and roots are easiest there. Every non-zero number has two square roots; −4 has ±2i. Honest limits: non-integer powers use the principal value, so they return one of several possible answers, and results are rounded to ten decimal places. The quadratic solver produces complex roots, the matrix calculator handles linear algebra and the scientific calculator covers everything else.
Frequently asked questions
How do you multiply complex numbers?
Expand the brackets and use i² = −1. (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
What is the modulus of a complex number?
Its distance from zero on the complex plane: |a + bi| = √(a² + b²). |3 + 4i| = 5.
What is the square root of −1?
i, and also −i. The calculator shows both square roots of any number.
Is my data saved?
No. The calculation runs in your browser and nothing is stored or sent.