Complex Number Calculator
Add, subtract, multiply and divide complex numbers exactly, raise them to powers by De Moivre's theorem, find nth roots on an Argand diagram, and convert to polar form.
About the Complex Number Calculator
Enter two complex numbers as a + bi and choose an operation (add, subtract, multiply, divide, power and more). The calculator gives the exact result with its modulus, argument and conjugate.
A complex number a + bi has a real part a and an imaginary part b, where i² = −1. Complex numbers describe AC circuits, signal processing and rotations. In polar form, r∠θ, multiplying multiplies the lengths and adds the angles, which is why powers are easy in polar form (De Moivre's theorem).
The formula
(a + bi)(c + di) = (ac − bd) + (ad + bc)i; |z| = √(a² + b²); arg z = atan2(b, a)
Worked example
(3 + 4i) × (1 − 2i)
- 3 − 6i + 4i − 8i²
- i² = −1, so 3 + 8 − 2i
- = 11 − 2i
Questions people ask
What is (3 + 4i)(1 − 2i)?
11 − 2i.
How do I divide complex numbers?
Multiply top and bottom by the conjugate of the bottom.