Are Cartesian coordinates unable to explore multi-valued functions beyond the principle value?

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By expressing the domain of a function in polar coordinates it is possible to take the function beyond it’s principle value (ie beyond 2$\pi$) to reveal the full beauty & complexity of it’s Riemann surface. Am I right in saying that Cartesian co-ordinates are inherently limited in this regard in a similar way that real numbers hide the greater truths revealed by using complex numbers?