We could go on for a long time, mentioning a multitude of small effects that might be important, depending on how accurate an answer we needed. Indeed, the real art of being a good physicist is "finding the physics" -- deciding what is important to pay attention to and what it is OK to ignore. The classic "physicist joke" relies on this insight.
When a physical factor is important or dominant, we try to treat it as correctly as we can; exactly, if possible. The less important effects, we either ignore or we use approximation method in mathematics to treat "as carefully as is appropriate for what we want to know." I believe that Einstein had something like this in mind when he said: Physics should be made as simple as possible, but not simpler.
These "approximate" answers can, in fact, be as good as the "analytic" answers we are used to generating. For example, we may be perfectly happy to write our analytic answers in terms of sines or cosines, but these are transcendental functions. That is, they cannot be exactly generated for any arbitrary value (only for some special cases). We are presently accustomed to pushing a button and having a calculator or computer spit out 8 or 10 decimal values for a sine or cosine (as a previous generation was accustomed to looking them up in a paper table). But generating those decimal values (or creating the previous generation's numerical tables) requires both an approximation process and a way of controlling our errors.
This plays quite well with how physics looks at the world. In physics (or in science in general), we never make exact statements, but we make approximate statements that have controlled errors. Thus, if we say, "The world is a sphere," that statement is wrong. But if we say, "There is a point inside the earth such that if you measured the distances to the surface in all different directions, all of your results would agree to better than 1%," we have a result that can be seen to be correct and that we can expect to remain correct for millions of years (barring some unforseen catastrophe).
To get some idea of how we bound results in approximations, read the chapter on series and convergence in Boas (chapter 1).
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Last revision 24. September, 2005.