Evelyn Lamb of the University of Utah has a nice piece on continued fractions in her SciAm math blog.

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Let’s look at how it works for pi. John Heidemann at the Information Sciences Institute at USC has a list of all the best rational approximations (of the first kind) of pi with denominators up through about 50 million. The numbers 3/1, 13/4, 16/5, 19/6, and 22/7 are the first few fractions on this list. 13/4 is a little closer to pi than 3 is. It’s about 0.1084 away instead of 0.1416. But if we multiply the differences by the denominators, 13/4 doesn’t do so well. We get 0.1416×1=0.1416 and 0.1084×4=0.4336, so 13/4 loses pretty badly. The same is true for 16/5 and 19/6. Both of them are a little closer to pi, but they aren’t enough closer to make up for their larger denominators. Thus, 13/4, 16/5, and 19/6 are best approximations of the first kind but not of the second kind. But when we get to 22/7, things change. It’s quite close to pi. Its difference, 0.00126, is very small. If we multiply it by its denominator, we get 0.00126×7=0.00882. This beats 0.1416 pretty handily, so it’s a best approximation of the second kind. It’s also the next convergent in the continued fraction for pi.

My favorite convergent in the continued fraction for pi 355/113. It’s a really good approximation. It’s a continued fraction mic drop. The next best approximation of the first kind has the denominator 16,604. But it isn’t even that much better than 355/113. We don’t get another decimal digit of accuracy, but we have to increase our denominator by two decimal places. No thanks. We have to increase to the denominator 33,215 before we get a new approximation that’s worth bothering with.

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