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 has a real part and an imaginary part, written a + bi where i is the square root of -1. Complex numbers extend the real line into a 2D plane and let you solve equations like x^2 = -1.
How do you multiply complex numbers?
Use (a + bi)(c + di) = (ac - bd) + (ad + bc)i. You can also multiply the magnitudes and add the arguments (the polar form approach), which is cleaner for repeated multiplication or for taking powers.
How does complex division work?
Multiply numerator and denominator by the conjugate of the denominator. (a + bi) / (c + di) becomes ((ac + bd) + (bc - ad)i) divided by (c^2 + d^2). Division is undefined when both c and d are zero.
What is the argument of a complex number?
The argument is the angle the vector (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. arg(0 + 0i) is undefined.
When should I use polar form instead of rectangular?
Polar form (r times (cos theta + i sin theta)) makes multiplication, division, and powers easier. Rectangular form (a + bi) is easier for addition and subtraction. The calculator shows both so you can choose.