Complex number to polar form

The complex number to polar form converter takes a complex number given by its real part and imaginary part and returns its polar form: the magnitude, which is the distance from the origin in the complex plane, and the angle (or argument) measured counterclockwise from the positive real axis, reported in both radians and degrees. The angle is computed with the two-argument arctangent, so it always lands in the correct quadrant, and a negative imaginary part yields a negative angle. Both parts must be finite numbers; anything else is rejected with a clear validation error.

From a + bi to magnitude and angle

A complex number written as a + bi locates a point in the complex plane: the real part a measures how far along the horizontal axis it sits, and the imaginary part b how far up or down. Polar form describes the very same number by a distance and a direction instead. The distance is the magnitude, often written r or |z|, computed as the hypotenuse of the right triangle formed by the two parts, so 3 + 4i has magnitude exactly 5.

The direction is the angle, often written theta or arg(z), measured counterclockwise from the positive real axis, so a purely imaginary number like 2i sits at pi/2 radians, which is 90 degrees.

Sending the two parts to this complex number to polar form converter returns the magnitude and the angle in a single call, and once you have them you can write the number as r(cos theta + i sin theta) or compactly as r times e to the i theta, which is the form that makes multiplication, division, powers, and roots of complex numbers simple.

Because the magnitude comes from a hypotenuse calculation rather than a naive square root of summed squares, the result avoids overflow even when the parts are very large finite numbers.

Why the angle uses atan2 and comes in two units

A plain arctangent of the imaginary part divided by the real part cannot distinguish a number in the first quadrant from one in the third, because the ratio is identical, and it breaks entirely when the real part is zero. The two-argument arctangent, atan2, inspects the signs of both parts, so the angle always points into the correct quadrant and is defined for every input except the exact zero.

The result lies in the interval from negative pi to positive pi radians: positive angles mean the number sits above the real axis, negative angles mean below, exactly pi means the negative real axis, and zero means the positive real axis. This converter reports the angle twice, in radians and in degrees, because different consumers want different units: radians feed directly into the trigonometric and exponential functions of every programming language and into Euler's formula, while degrees are what most people read on homework, in textbooks, and in engineering specifications.

Computing both in one call saves you a conversion step and a rounding ambiguity, and both values are rounded to twelve decimal places so results compare byte-identically across machines and JavaScript engines.

Validation, determinism, and where it fits

Each part must be a finite number. The zero complex number is a deliberate special case: its magnitude is zero and its angle is mathematically undefined, so the converter returns an angle of zero in both units by convention rather than failing, which keeps pipelines total and predictable while documenting the choice. Typical callers are electrical engineering tools converting impedance between rectangular and phasor form, signal processing code moving between Cartesian and magnitude-phase representations, control systems doing frequency response analysis, and students checking complex arithmetic homework.