TITLE: What are the applications of continued fractions?
QUESTION [38 upvotes]: What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?

REPLY [6 votes]: Application #1: The first proof that $e$ is irrational, due to Euler, was based on showing its continued fraction expansion is infinite: $e = [2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,...]$ with the obvious repeating pattern of even numbers separated by two $1$'s.
Application #2: A year consists of $365.24219$ days. If we want to create a calendar with a block of $Y$ years containing $L$ leap years, then $Y$ years will contain $365Y+L$ days. The average number of days in a year of each block is $(365Y+L)/Y = 365 + L/Y$, so we want $L/Y$ to be a good rational approximation to $.24219$, whose continued fraction expansion is $[0,4,7,1,3,24,6,2,2]$. The first six continued fraction convergents are $0$, $1/4$, $7/29$, $8/33$, $31/128$, and $752/3105$. The convergent $1/4$ leads to the Julian calendar having $1$ leap year every $4$ years. The choice $L/Y = 8/33$ is used in Iran: $8$ leap years in a block of $33$ years is part of the Jalil calendar proposed by Omar Khayyam in the year $1079$. If we look at intermediate convergents between $31/128$ and $752/3105$, which by definition are fractions $(31a + 8)/(128a+33)$ for $a = 1,2,\dots,23$, then at $a = 6$ we get $194/801$, and $801$ is nearly $800$, a nice round $8$ centuries. So let's modify this a little to consider $194/800 = 97/400$: this is the Gregorian calendar, which has $97$ leap years every $400$ years, that is a leap year every $4$th year except for years divisible by $100$ but not by $400$ (a total of $100 - 3 = 97$ leap years).
Application #3: As spin wrote in a comment, being able to make a good guess for the value of a rational number when you know only some approximation to it (even if the approximation is itself rational!) is a key part of Shor's algorithm for factoring integers on a quantum computer. Guessing rational numbers from numerical approximations is also important in doing computations related to elliptic curves, as William Stein explained to me: if an elliptic curve $E$ is defined over $\mathbf Q$ and you use complex analysis to compute a coefficient of a Weierstrass equation for $E$, or to compute $L(E,1)/\Omega$, all you get is a decimal estimate for a number that is supposed to be rational and you want to decide what that rational number ought to be (which might then be rigorously proved by other methods). Typically the decimal approximation of the rational number may have far fewer digits than the full period length, so you can't guess the fraction as simply as you can if someone asks you what fraction $.333333333$ probably is (namely $1/3$).
Here is an example of how continued fractions let you guess rational numbers from decimal expansions, which I did as a trick at the end of a talk I gave today on continued fractions. I asked a student in the audience to compute a fraction as a decimal on a calculator and tell me what digits showed up on the calculator, but make sure there was no obviously periodic part shown by the calculator and not to tell me the original fraction and make sure to write down the original fraction for comparison with my guess later. The student gave me $.1391200951$. Using a computer algebra package, I converted $.1391200951$ into a continued fraction: it is $[0,7,5,3,6,1,56891,1,103,2]$. The $56891$ is so huge compared to what comes before that my educated guess for the student's original fraction is $[0,7,5,3,6,1]$, which is $117/841$. That indeed was the number the student  wrote down earlier. The audience was impressed. The actual decimal period of $117/841$ is $812$, so I was able to figure out a fraction having a repeating decimal period length over $800$ knowing only the first $10$ digits of its decimal expansion.