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Sun, 26 Mar 2006

Approximations and the big hammer
In today's article about rational approximations to √3, I said that "basic algebra tells us that &radic(1-&epsilon) &asymp 1 - &epsilon/2 when &epsilon is small".

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 &epsilon 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 + ε2/4. If &epsilon is small, then &epsilon2/4 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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