The Universe of Discourse

Approximations and the big hammer
In today's article about rational approximations to √3, I said that "basic algebra tells us that √(1-ε) ≈ 1 - ε/2 when ε 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 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 √(1-ε) is simple. First-year algebra tells us that (1 - ε/2)² = 1 - ε + ε²/4. If ε is small, then ε²/4 is really small, so we won't lose much accuracy by disregarding it.

This gives us (1 - ε/2)² ≈ 1 - ε, or, equivalently, 1 - ε/2 ≈ √(1 - ε). Wasn't that simple?

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