What is a complex number?
A complex number has a real part and an imaginary part. It is commonly written as a + bi, where a and b are real numbers and i = √−1.
How to use the complex number calculator
Enter the first and second numbers, choose an operation, and select Calculate. The calculator returns the result in standard a + bi form and also shows its real part, imaginary part, modulus, and conjugate.
Complex number formulas
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Division
(a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
Example
(3 + 4i) + (1 − 2i)
= (3 + 1) + (4 − 2)i
= 4 + 2i
Frequently asked questions
What is a complex number?
A complex number combines a real part and an imaginary part and is written as a + bi.
What does i mean?
The symbol i is the imaginary unit and satisfies i² = −1.
Can I calculate complex number division?
Yes. The calculator divides two complex numbers and rationalizes the denominator internally.
What is the modulus of a complex number?
For a + bi, the modulus is √(a² + b²), which represents its distance from the origin on the complex plane.
Is this a full complex number solver?
It covers the operations most people need day to day, including addition, subtraction, multiplication, division, modulus, and conjugate. More advanced operations such as powers and roots of complex numbers are not included yet.