At first glance, continued fractions look more like a curiosity than like useful mathematics. And yet they come up surprisingly often in applications.
For an irrational number x, the numbers you get by truncating the infinite continued fraction for x are the optimal rational approximations to x given the size of their denominators. For example, since
π = 3.141592…
you could obviously approximate π by 31/10, but 22/7 is more accurate and has a smaller denominator. Similarly, if you wanted to approximate π using a fraction with a four-digit denominator, you could use 31415/10000 = 6283/2000, but 355/113 is much more accurate and uses a smaller denominator. Continued fractions are how you find optimal approximations like 22/7 and 355/113. More examples here.
Optimal rational approximations have widespread uses. They explain, for example, why some complicated calendar systems are the way they are. The ratios of astronomical periods, such as that of the earth’s orbit around the sun to that of the earth’s notation, are not rational, and calendar systems amount to constructing rational approximations. More on that here.
Aside from constructing rational approximations, continued fractions are often used to efficiently evaluate mathematical functions. For example, I’ve written about how continued fractions are used in computing hazard functions and entropy.