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That cryptic constant is actually a composite of three bitfields, and twiddling it requires some understanding of what those fields are. It would be clearer, but a few more operations, to do that line as a pair of bitfield extract/inserts. But we're saving divides in the subsequent iterations, so the extra 1-cycle operations are a wash.
== Continued fraction expansion ==
This section is the longest in the article, and most laden with awkward symbolism. It might in large part, be the reason for the "Too Technical" maintenance tag. I do a lot of my work on a cell phone or tablet, and it's gobbledegook. I first question whether the section belongs in the presentation at all - any number, even irrational and transcendental ones (including the square root of any number), can be represented as a [[Continued fraction]], and there's a lead article on that topic, as well as [[Generalized continued fraction]] and [[Periodic continued fraction]]. There's nothing unique about the procedure with respect to the square root of a number. It's sometimes convenient, for mathematical foundations as well as estimating, to have a continued fraction representation of frequently used numbers like {{sqrt|2}}, {{sqrt|10}}, <math>e</math>, <math>\pi</math>, <math>\phi</math>. It's tedious to work out one of these; unless there's some motive besides obtaining a final numerical approximation, we do it by looking up the expansion in a table, and it's usually for some mathematics-related number like the above enumerated ones. The worked eample has a better place in one of the aforementioned articles. If we're going to give an example here, it's more useful to use a number like {{sqrt|2}}. I'd like to see the section merged into one of the other articles, and replaced with a much shorter useful presentation, rather than a derivation-based one.[[User:Sbalfour|Sbalfour]] ([[User talk:Sbalfour|talk]]) 18:27, 30 November 2019 (UTC)
I'm moving all the text associated with the {{sqrt|114}} to [[Periodic continued fraction]], for a number of reasons:
1) it's an unwieldy large example; 2) it's not actually very useful, not as useful as say a similar example for {{sqrt|3}}; 3) it's more about how-to (i.e. how to expand the fraction and find the repetend) rather than using a fraction to compute a value; 4) such extreme precision as 7 denominators will almost never in practice be done; 5) all those radicals are intimidating to much of an amateur non-mathematical audience. I note that this example was once before deleted from the article, and for related reasons. Expanding a continued fraction is mostly the province of mathematicians; using one, i.e. to compute a value, is actually rather straight forward. But neither reducing a continued fraction to a rational fraction, nor computing an actual value from it is covered in that text. Finally, by my so doing, we're not actually losing any information from the encyclopedia - it's a matter of locating the example in the most generally applicable article. [[User:Sbalfour|Sbalfour]] ([[User talk:Sbalfour|talk]]) 18:03, 13 December 2019 (UTC)
There are four different types of notation in this section:
:<math> \sqrt{S} \Rightarrow 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + ...}}} \equiv 1 + \frac{1 |}{| 2} +\frac{1 |}{| 2} + \frac{1 |}{| 2} + \cdots \equiv [1;2,2,2...] \equiv [1;\overline{2}]</math>
But then, we make the incredible leap <math>[1;2,2,2] = 17/12</math>. I gave the problem to a neighbor's son who's a math whiz in high school, and he was unable to demonstrate to me how to compute 17/12 from the given expression. He didn't recognize any of the 4 expressions for continued fractions, and was unable to correctly compute anything. The mathematical brevity here is inscrutable. I didn't get continued fractions until a second course in calculus, as a college math major. A non-math major will never see any of this notation. Only applied mathematicians use that kind of notation (and they don't source wikipedia for it), and only for computing reference continued fractions. The rest of us look up the fraction, and use it to compute with. And how do you do that? Hmmmm.... [[User:Sbalfour|Sbalfour]] ([[User talk:Sbalfour|talk]]) 19:08, 13 December 2019 (UTC)
I'm going to move all the dense mathematical formalisms into a mathematical sidebar or footnote; that'll shrink the section substantially. Then, add at least one sequence showing the progressive reduction of the continued fraction to a rational (numerical) fraction, and finally, computation of the value of the root from it. That should leave the section accessible at the high school level. [[User:Sbalfour|Sbalfour]] ([[User talk:Sbalfour|talk]]) 19:32, 13 December 2019 (UTC)
== Pell's equation? ==
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