Complex Number Calculator
Add, subtract, multiply, divide, and more – with instant results!
First Complex Number
Second Complex Number
Complex number calculator
Add, subtract, multiply, divide, and find the modulus, argument, conjugate, and polar form of complex numbers — instantly, with step-by-step solutions.
What is a complex number?
A complex number is a number of the form a + bi, where a is the real part and b is the imaginary part. The symbol i represents the imaginary unit, defined by the property i² = −1. Complex numbers extend the real number line into a two-dimensional plane, enabling solutions to equations that have no real roots — such as x² + 1 = 0.
All operations this calculator performs
Visualizing complex numbers on the Argand diagram
The Argand diagram (complex plane) plots the real part on the horizontal axis and the imaginary part on the vertical axis. Every complex number corresponds to a unique point — or equivalently a vector from the origin.
The modulus is the length of the vector (r), and the argument is the angle θ measured counterclockwise from the positive real axis. These two values define the polar form of any complex number.
Common operations — worked examples
Multiplying two complex numbers
Dividing complex numbers
Finding modulus and argument of 3 + 4i
Polar form vs rectangular form — when to use which
| Task | Rectangular a+bi | Polar r(cosθ+i sinθ) |
|---|---|---|
| Addition / subtraction | ✓ Easy — add components | ✗ Requires conversion first |
| Multiplication | ✓ Possible via FOIL | ✓ Faster — multiply r, add θ |
| Division | ✓ Possible via conjugate | ✓ Faster — divide r, subtract θ |
| Powers / roots (De Moivre) | ✗ Very tedious | ✓ Trivial with De Moivre |
| Graphical interpretation | ✓ Direct plot on x-y plane | ✓ Direct magnitude and direction |
| AC circuits (impedance) | ✓ Adding impedances in series | ✓ Phase angle representation |
Operation difficulty chart
Where complex numbers are used — real-world applications
How to use this calculator — step-by-step guide
- Choose your input format: rectangular (a + bi), polar (r, θ), or Euler form.
- Enter the real part (a) and imaginary part (b) for the first complex number.
- If performing a two-operand operation (add, multiply, etc.), enter the second complex number.
- Select the operation you want to perform from the dropdown or operation buttons.
- Click Calculate — the result is shown instantly in all three forms.
- Expand the Step-by-step section to see the full working for your problem.
- The Argand diagram updates to plot both input numbers and the result.
Pro tip: When computing powers or roots, switch to polar form — De Moivre’s theorem makes the calculation dramatically simpler, and the calculator handles the conversion automatically.
De Moivre’s theorem — powers and roots
De Moivre’s theorem is one of the most powerful tools in complex number arithmetic. It states that for any complex number in polar form and any integer n:
Finding the nth roots of a complex number z = r·e^(iθ) gives exactly n distinct roots, evenly spaced around a circle of radius r^(1/n) in the complex plane:
Euler’s identity — widely considered the most beautiful equation in mathematics — is a direct consequence of complex exponential form:
e^(iπ) + 1 = 0
This connects five fundamental constants: e, i, π, 1, and 0.
Frequently asked questions
Why use an online complex number calculator?
Manual complex arithmetic is error-prone, especially for multiplication with four terms or division requiring conjugates. This calculator eliminates arithmetic mistakes, shows complete working so you can learn from each step, handles angle quadrant issues automatically in polar conversion, and lets you focus on understanding rather than mechanics. Students, engineers, and researchers all benefit from a reliable, always-available tool that handles the full range of complex operations.
Whether you’re solving AC circuit problems, working through a complex analysis textbook, or just need to quickly find the argument of a complex number — this calculator is designed to be the only tool you need. Bookmark it and come back whenever complex arithmetic slows you down.