What are the applications of continued fractions?

One very nice elementary application is Gosper's batting average problem: if a baseball player's (3-digit rounded) batting average is $.334$, what's the smallest number of at-bats that player could have? (Batting average is computed as (number of hits)/(at-bats).)

The solution proceeds by noting that a rounded average of $.334$ corresponds to an actual number in the range $[.3335, .3345)$; finding the continued fractions for these values yields $.3335 = 667/2000 = [0; 2, 1, 666]$ and $.3345 = 669/2000 = [0; 2, 1, 94, 1, 1, 3]$. This implies that the 'simplest' number within the range is $[0; 2, 1, 95] = \frac{96}{287}\approx 0.334495$.

Since no one mention it, I think it's really pleasant knowing that every rational tangle correspond to some continued fraction(Conway showed that).

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For further information see:

• An Enumeration of Knots and Links, and Some of Their Algebraic Properties by Conway

• Another proof On the classification of rational tangles

Here is good expository article:

• The Power of Mathematics by Conway

• Conway’s Rational Tangles, a good educational article

Note:Image source: http://rationaltangle.wordpress.com/what-are-tanglesrational-tangles/

There is this interesting article about the application of continued fractions in Phyllotaxis, the research on leaves, which I found pretty interesting. Maybe you can find more over google? ${}{}{}{}{}{}$