Complex Number Calculator

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.

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.

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.

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 Argument and Polar modes return no result for zero. Other valid operations can still produce 0 + 0i.

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.