The Riemann surface of the complex numbers

The Riemann surface is a way mathematicians visualize complex numbers and their functions, especially when those functions are multi-valued, like the square root or logarithm. Imagine extending the complex plane into multiple layers or sheets connected along branch cuts, so each point has a well-defined value on one of these sheets. This structure unwraps the complexities of multi-valued functions, making them single-valued and easier to analyze. Essentially, the Riemann surface provides a mathematical "map" that captures all possible values of complex functions, offering a clearer understanding of their behavior across different regions.