It's not immediately clear how to begin with this puzzle, since the numbers involved are so seemingly arbitrary and varied that there's no obvious way to extract a simple message. This is also partly because the numbers have multiple components (numerator, denominator, surd) which don't clearly correlate with each other in anyway. So perhaps the best we can do for now is simply take the bluntest approach and calculate each number as a decimal.
Doing so actually proves promising, as the integer part of each number turns out to always be a nice number between 1 and 26. If we go ahead and convert these integer parts to letters (bolded in the table below), we get the message CONT FRACTION TO DECIMAL. If not certain what the CONT can stand for here, the title helps - we're talking about continued fractions. So we want to convert each number to a continued fraction, and then, somehow, to a decimal.
Converting to continued fractions can be done fairly easily, either by hand (with calculator in tow), or as it turns out, using various online solvers. There is a convenient way to record continued fractions that avoids drawing all the fraction parts, namely writing them as the list of integer parts in square brackets, with a semicolon separating the original number's integer part. For example, the first term (21 - √6)/5 = [3;1,2,2,4]. The message told us to convert this to a decimal, and obviously simply calculating the decimal form of this will have made the continued fraction step unnecessary. So the most intuitive conversion we can then do is to replace the semicolon in this continued fraction representation with a decimal point, and concatenate everything. For example, our term [3;1,2,2,4] can be come 3.1224.
Now that we've converted our original numbers into actually new ones, the question remains how to extract a final answer from these. Checking integer parts does not help us this time, but one thing we could try is undoing the original step, converting the new number values into continued fractions. For example, 3.1224 as a continued fraction is [3;8,5,1,16]. This string of numbers is actually exciting, because they are always finite, and converting them to letters in the usual way, we get the word CHEAP. (The only place this doesn't work entirely by the book is for the second entry OCTE, whose original decimal's repeating string contains a 0 and therefore needs to consider it attached to the 2 before it, interpreted as a 20).
In other cases we don't get words though, like OCTE and TABL. What is clear though is that all these strings want to have one letter added at the end to make them a new word (they want "To Be Continued"), and most of the time this added letter is unique when considering only common words. (Adding S to make a plural should be considered uninteresting and therefore not the sought-after letter, leaving only ICE_ as ambiguously D or R). This is all summarised below:
So adding the final letters seems to give us a new message, namely OTHER TERM FOR THREE DOTS. The most obvious answer to this description is an ellipsis, which is also what you'd expect to see after the title "To Be Continued".
|The answer is: ellipsis|