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Sun, 26 Mar 2006
Approximations and the big hammer
A lot of people I know would be tempted to invoke calculus for this, or might even think that calculus was required. They see the phrase "when ε is small" or that the statement is one about limits, and that immediately says calculus. Calculus is a powerful tool for producing all sorts of results like that one, but for that one in particular, it is a much bigger, heavier hammer than one needs. I think it's important to remember how much can be accomplished with more elementary methods. The thing about &radic(1-&epsilon) is simple. First-year algebra tells us that (1 - &epsilon/2)^{2} = 1 - ε + ε^{2}/4. If ε is small, then &epsilon^{2}/4 is really small, so we won't lose much accuracy by disregarding it. This gives us (1 - &epsilon/2)^{2} ≈ 1 - ε, or, equivalently, 1 - &epsilon/2 ≈ √(1 - ε). Wasn't that simple?
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