Complex Number Calculator
Perform arithmetic on complex numbers and see results in both rectangular and polar form. For example, (3 + 4i) x (1 - 2i) = 11 - 2i, which has magnitude 11.18 and angle -10.3 degrees.
Complex Number Arithmetic
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i = sqrt(-1).
Addition and Subtraction
Just combine the real and imaginary parts separately:
- (3 + 4i) + (1 - 2i) = 4 + 2i
- (3 + 4i) - (1 - 2i) = 2 + 6i
Multiplication
Use the distributive property (FOIL) and the fact that i² = -1:
- (3 + 4i)(1 - 2i) = 3 - 6i + 4i - 8i² = 3 - 2i + 8 = 11 - 2i
Division
Multiply numerator and denominator by the conjugate of the denominator:
- (3 + 4i) / (1 - 2i) = (3 + 4i)(1 + 2i) / ((1)² + (2)²) = (-5 + 10i) / 5 = -1 + 2i
Polar Form
Every complex number can also be written as r(cos theta + i sin theta), where:
- r = sqrt(a² + b²) is the magnitude (distance from origin)
- theta = atan2(b, a) is the angle from the positive real axis
Polar form is especially useful for multiplication and division, since you just multiply/divide magnitudes and add/subtract angles.