Complex Number Calculator and Polar Converter

What is a complex number?

A complex number is written z = a + bi. Here a is the real part, b is the imaginary part and i is the imaginary unit whose square is −1. You can picture a complex number as a point in a plane: the horizontal axis shows the real part and the vertical axis the imaginary part.

Rectangular and polar form

The same point can be given by its distance from the origin and the angle it makes with the horizontal axis. The distance is the modulus (r) and the angle is the argument (θ).

• r = √(a² + b²) and θ = atan2(b, a)
• a = r · cos θ and b = r · sin θ
• Exponential form: z = r · e^(iθ)

Example: for z = 3 + 4i, r = √(9 + 16) = 5 and θ = arctan(4/3) = 53.13°. The same conversion turns a point (x, y) in the plane into polar coordinates.

The four operations

Addition and subtraction work on the real and imaginary parts separately. To multiply, expand the brackets and replace i² with −1: (−1 + 3i)(2 − 5i) = 13 + 11i. To divide, multiply the numerator and the denominator by the conjugate of the denominator: (−1 + 3i) / (2 + 5i) = (13 + 11i) / 29.

Powers and roots

By De Moivre's formula zⁿ has modulus rⁿ and angle nθ. For (1 + i)⁸ take r = √2 and θ = 45°: (√2)⁸ = 16 and 8 · 45° = 360°, so the result is 16. Every number other than zero has n nth roots, spaced evenly around a circle. The cube roots of 8 are 2, −1 + √3 · i and −1 − √3 · i.

How to type a number

You can type expressions such as 3 + 4i, −2i or (1 + i)^8. Values such as √3/2 and pi/3 are understood as well. In polar input you enter the modulus and the angle separately and choose degrees or radians.

Limits

The argument is given as the principal value between −180° and 180°. Results are shown as decimals and exact surd forms such as √3/2 are not produced. Complex values of exponential, logarithmic and trigonometric functions are outside the scope of this tool.