Complex Number Calculator
Enter two complex numbers and choose an operation.
This result is an estimate only. Double-check important figures before relying on them.
How it is calculated
- Add: (a + c) + (b + d)i
- Subtract: (a − c) + (b − d)i
- Multiply: (ac − bd) + (ad + bc)i
- Divide: multiply top and bottom by the conjugate of the divisor, (c − di)
- Modulus = √(re² + im²), argument = atan2(im, re)
For the square root of a negative number, start from the root calculator.
Worked example
(3 + 4i) × (1 − 2i):
- Real part = 3 × 1 − 4 × (−2) = 11
- Imaginary part = 3 × (−2) + 4 × 1 = −2
- Result = 11 − 2i
Why your number may differ
Results are rounded to 6 decimal places in the text, and the angle is given between −180° and 180°.
Frequently asked questions
What is a complex number?
A number of the form a + bi, where a is the real part, b the imaginary part and i is the square root of −1.
How do I multiply complex numbers?
(a + bi)(c + di) = (ac − bd) + (ad + bc)i, because i² = −1.
What are the modulus and argument?
The modulus is the distance from the origin, √(a² + b²). The argument is the angle with the real axis, in degrees.