Analog Angle Article

The art of approximation

A large part of science and engineering work requires equations which model and analyze various situations. In many cases, these equations soon become complex and hard to resolve or solve analytically. While we can almost always number-crunch and solve a given equation numerically, that approach does not provide the insight than an equation-based, analytical solution offers.

That's were approximations help. The most basic and useful one is that we can replace sin x with x itself for small angles (x, of course, is in radians). Using this approximation, many complex equations can be simplified and solved analytically. (There are many other commonly used approximations, of course, but “sin x = x” is among the most common ones.)

Of course, the challenge in using any approximation is to understand its limits. One person's “small angle” is not so small in another application. The sin x = x approximation has about 1% error at about π /16 or 0.2 radians (about 13°), and 1% may be good enough for a valid solution in some situations, and terrible in others. You also have to watch out for cumulative effects of approximations, and cases where the analysis is very sensitive to errors at certain points, with sensitivities that the approximation may aggravate. If you use multiple approximations, that's a potential problem as well, as the errors can cascade and cause a far greater final error result. On the other hand, if you looking for the nearest standard 5% resistor value, your results may be just fine even when using a few modest approximations and simplifications.

Understanding what the numerical analysis and any approximations you employ actually imply is what good engineering is all about. It's all about careful use and understanding of the tools you have at hand when doing analysis, measurement, and drawing conclusions.

Not all simplifications are for equations alone, and by this I don't mean the helpful Cliff's Notes versions of Shakespeare's works. For example, you really come to appreciate the accurate but greatly simplified discussions of Einstein's various theories that appear in any good physics textbook after you try to read his original papers (in English, of course).

For example, the book Einstein's Miraculous Year by John Stachel has faithful translations of Einstein's five papers of 1905 (including relativity and Brownian motion), and they are still fairly hard to follow; the E = mc2 message is almost lost in the thicket. Even with the excellent and lengthy explanations of each paper by the Stachel, it can be hard to follow Einstein's analysis. But a good physics book can summarize and approximate his papers clearly and accurately, and open the door to understanding.

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