Math
Complex Number Calculator
Add, subtract, multiply, or divide complex numbers and get rectangular and polar form, magnitude, argument, and conjugate.
Complex number operation
Rectangular form
4 + 6i
Polar: 7.211103 ∠ 56.3099°
Frequently Asked Questions about the Complex Number Calculator
What is a complex number?
A complex number combines a real part and an imaginary part in the form a + bi, where i is defined as the square root of -1. Because no real number squares to -1, complex numbers extend arithmetic into a 2D plane, letting you work with equations like x^2 = -1 that have no real solutions.
How do you multiply complex numbers?
Apply the FOIL rule and substitute i^2 = -1: (a + bi)(c + di) = (ac - bd) + (ad + bc)i. For example, (2 + 3i)(1 - i) gives (2 - (-3)) + (-2 + 3)i = 5 + i. The calculator uses this formula directly and rounds results to 6 decimal places.
How does complex division work?
Multiply both numerator and denominator by the conjugate of the denominator. (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2). Division is undefined when c and d are both zero, so the calculator returns no result in that case.
What is the argument of a complex number?
The argument is the angle the point (a, b) makes with the positive real axis, measured counterclockwise. The calculator returns it in degrees in the range (-180, 180] using the atan2 function. The argument of 0 + 0i is undefined, so the calculator returns no result for that input.
When should I use polar form instead of rectangular?
Use polar form (magnitude r and angle theta) when multiplying, dividing, or raising a complex number to a power, since you multiply magnitudes and add angles instead of expanding brackets. Stick to rectangular form (a + bi) for addition and subtraction. The calculator shows both forms for every arithmetic result so you can switch between them freely.