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Why can the argument of a complex number have more than one value?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

The argument of a complex number can have more than one value because it is defined by the angle of rotation from the positive real axis to the vector representing the number in the Argand diagram.1

Because trigonometric functions such as sine and cosine are periodic, adding any integer multiple of $2 π$ radians (or 360 degrees) to an angle results in the same point on the unit circle. Consequently, if $θ$ is an argument of a complex number, then $θ + 2 k π$ for any integer $k$ is also an argument. This multi-valued nature is a fundamental property of the complex logarithm and related functions in complex analysis. To address this, mathematicians often define a "principal value" of the argument, typically restricted to the interval ((-\pi, \pi]], to ensure a unique representation.1