I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:
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$S^n$ is never contractible, but $S^{\infty}$ is.
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The vanishing viscosity method of PDE's.
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Higher-dimensional topology as opposed to low-dimensional topology (in some specific cases)
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Singular homology as opposed to simplicial homology
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Cube complexes as opposed to 3-manifolds
etc.
What other examples are there where a more complex object is simpler to analyze than a 'simpler' object? I realize that you could say that if it is easier to analyze, then it is less complex, so let me restate it this way:
What examples are there where one object seems much more complicated than another, but in fact has a simpler structure?
I've been thinking about things like the Ising model for magnetic phase changes and also about Navier-Stokes; perhaps the simplifications used to derive them make them harder to analyze in the end.
Best Answer
The complex object $\mathbb C$ is in some respects easier to understand that the apparently simpler object $\mathbb R$. (E.g., if we consider zeroes of polynomials, or the behavior of power series.)