Complex Number to Polar Form Calculator

A complex number in standard rectangular form is written as z = a + bi, where 'a' is the real component and 'b' is the imaginary component. To convert this into polar form (r θ), we compute two distinct parameters: the magnitude (r) and the phase angle (θ). The magnitude represents the absolute distance from the origin to the point on the complex plane, calculated using the Pythagorean theorem: r = √(a² + b²). The phase angle is the counterclockwise angle measured from the positive real axis, determined using the four-quadrant inverse tangent function: θ = atan2(b, a). Together, they express the complex number as r(cos θ + i sin θ) or simply in exponential shorthand as r e^(iθ).

Let us convert the rectangular complex number 3 + 4i into its polar form. Here, our real part is a = 3 and our imaginary part is b = 4. First, we find the magnitude by squaring both components, adding them together, and taking the square root: r = √(3² + 4²) = √(9 + 16) = √(25) = 5. Next, we determine the phase angle using the two-argument arctangent: θ = atan2(4, 3) ≈ 0.9273 radians (or about 53.13 degrees). Therefore, the polar form of 3 + 4i is 5 with a phase angle of 53.13°, representing a vector of length 5 pointing at a 53.13-degree angle.

When performing manual checks or setting up problems, always pay close attention to the quadrant in which your complex number lies. Using a standard arctangent function (atan(b/a)) can yield incorrect angles if 'a' is negative, which is why the two-argument arctangent (atan2) is essential for correct quadrant placement. Additionally, ensure your calculator is set to your preferred output unit—radians or degrees—before interpreting the phase angle.