How are continued fractions useful? On Wolfram Alpha, I see continued fractions being listed in the results. Although I understand continued fractions, and how they can be used for approximations, what is a better approximation than a decimal representation of the number?
For example,

31.999999999664

tells me so much more than

So, I do not get why continued fractions are important. Are they supposed to tell me something which I am missing all these days?
 A: How are continued fractions cool? Let me count the ways:


*

*If you know how to get the partial terms of the continued fraction expansion of an irrational number, you essentially have a way to find approximate values of the irrational number. This gives you one possible way to get a floating-point estimate for $\sqrt{2}$, $\pi$, etc.

*It is known that a number is irrational if and only if it has an infinite continued fraction expansion. This makes it a sort of irrationality test.

*Continued fraction expansions of irrational numbers exhibit sometimes surprising regularity. $\pi$ and $e$ have continued fraction representations that are simple, which is strange seeing as they are transcendental numbers. This can sometimes tell you things about these numbers, and other times the existence of such a regular expansion is just plain cool.

*As mentioned in the comments, continued fractions can be used to solve certain equations like the Pell equation.
And others. Continued fractions solve some other problems - a famous story relates that Mahalanobis and Ramanujan shared a room, and one day Mahalanobis posed Ramanujan a problem which he instantly solved for all cases via continued fractions.
A: I once encountered a situation where I had a floating-point number, probably 10 or 12 decimal places long, but I suspected that it was actually a rounded-off version of some rational number with not-too-big denominator (less than 1000).  How can I find (or at least guess) the rational number?  What I did was to start computing (in Excel, which tells you something about my programming skills) the continued fraction of my floating-point number.  For a while, the integers I got were reasonably small ones, but then I got one that was way bigger than the previous ones.  So I said: That must be the rounded-off version of $\infty$.  In other words, I replaced that big integer with $\infty$, which amounts to cutting off the continued fraction at that step.  The resulting finitely long continued fraction is, of course, rational, and it gave the answer I wanted.  (Later, I checked with a number theorist, and he confirmed that this is how one should attack such questions.)
A: Some commonly-encountered irrational numbers have "nice" representations as continued fractions:


*

*$e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, ...]$

*$\sqrt{2} = [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...]$

*$\phi = [1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...]$


By continuing the pattern to the desired level of precision, you can obtain an accurate rational approximation of the number.
A: Continued fractions are useful for converting decimals into fractions, or reducing fractions to their common divisor.  Often finding fractions between X and Y, like 8/11 for the logrithms of the chords of the heptagon, means that (in this case), you only need to consider 11 cases.
A: In your example, the huge number $2997044470$ tells you that you will make very little error stopping before it.  At the third step you know you are very close to $32$.
A: repeated fractions for prime square roots
You can give the exact value of the root of a non prime number. It is algorithmic.        √17 = √16+1 = 4+1/8 with the denominator 8 having infinite fractions added to it. Squaring this number equals exactly 17. The square root of 18 is 4+2/8 with the 2/8 being added to every 8 in the denominator.
We took √18 to equal √16+2 because 16 is the nearest prime. The +2 becomes 2 over 8 repeated to infinity. We have to add 2/8 because we need a number bigger than 4 because 18 is bigger than the perfect square we used.
It works the other way too. The √3 is ≈ 1.73205. We can't use a perfect square less than 3 because non exist, they are all prime numbers. So we: √3 = √4-1 = 2-1/4 with the 1/4 subtracted to the denominator to infinity. We subtracted this time instead of adding because  we took a perfect square of 4 that is larger than our targeted power of 3. 2-1/4...= √3 ≈ 1.73205.
Most people use calculators to find non prime numbers. People who MAKE calculators or software need to know number theory to program the logic. This isn't commonly taught because it is used for encryption of government and financial data. It is related to the study of large prime numbers and finding their roots and using it for encryption, programming logic for calculating software and so on. To make software for numerical analysis you need to understand what logic to program it with. This is a cornerstone of that logic.
The fraction is plus or minus the number of units away from the target prime over twice the root of the perfect square you used. 8 was double 4, and 1/x, and 2/x for 17 and 18 respectfully. Had to add the fractions to make a number slightly larger than 4^2. 4 was double 2 and 1/x. Had to minus the fraction since we had to make a number slightly less than 2^2.
There are many applications but mostly related to sensitive work.
