I took a course in college called 'Numerical Analysis'. I don't even know if it is taught any more. NA was about devising algebraic structures that simulated the techniques of real analysis. It was used to mimic some of the tools available to the mathematician that could not be programmed, such as limits, differentials, integrals, and so on. For example, if you could not solve a definite integral, you divided up the area under the curve into little rectangles and added them up. How many terms should a series solution contain to approach a formal solution. How did you determine that? We even studied Newton's "fluxions", one of his approximations to the derivative. Part of the content of the course was studying the behavior of these numerical approximations as they approached a "real" solution. Did the curves converge? Were they asymptotic? Did the approximation jump around the true value, did it oscillate about it, did it suddenly diverge? In the interest of computational expense, how close did your approximation and the true function have to approach in order to make it worthwhile?

We used computers to investigate some of these relationships, but it was already becoming clear that the use of machines was rapidly making NA techniques obsolete, it was often easier to using computing power "brute force" rather than devise an algorithm which would give a precision that was "good enough". Even so, the limited precision of some computers, plus space and time limitations, often caused machine derived values to diverge extravagantly from reality. Part of the rationale behind the course was to devise approximations that tended to approach the correct values, and how to identify situations where they might be expected to diverge.

Before computers were readily available, complex calculations on vast masses of data were carried out by groups of individuals using tables, slide rules and ten-key adding machines (usually undergrad students). Each worked on parts of the calculation, and their results were combined into more complex formulas using paper forms. These folks were called "computers". In order to make their calculations cheaper, and less prone to error, the equations they were solving were often replaced with "approximations" that were simpler to compute and which achieved sufficient precision. Numerical analysis put all this on a somewhat more formal basis.

That's all computing machines do. They are fast and accurate. They are useful, even invaluable. But they are not magic. They don't tell us anything we don't already know.