Talk:Dyadic rational/GA1

GA Review
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Reviewer: Bilorv (talk · contribs) 18:09, 17 August 2021 (UTC)

No issues with sourcing, comprehensiveness or the majority of the material, so it's just nitpicks below really. It's a very well-written article and I learned a fair few things from it. It's a little bit rare for me to venture into maths articles at the GA-and-above level so apologies for any conventions that I'm unfamiliar with. — Bilorv ( talk ) 18:09, 17 August 2021 (UTC)
 * Is it usual for the body to not repeat the definition of "dyadic rational" from the lead (perhaps in more formal terms if the lead is "one level down")?
 * I added a more detailed set of definitions to the "arithmetic" section. But I didn't want to push the definitions into a separate earlier section for reasons of technicality: the "applications" section is the least technical, and I think the most informative to non-mathematically-inclined readers, so I wanted to put it first before getting into even the easier mathematical parts. —David Eppstein (talk) 19:22, 17 August 2021 (UTC)
 * The lead image looks to be just the dyadic rationals in [0, 1] with denominator at most 64 (in simplest terms). Obviously we can't demonstrate the whole set, but as the 64th lines are still quite tall on my screen (not a huge one) is it possible to add a few more layers (unlabelled, like the 32nd and 64th lines)? Or if not, maybe a tweak to the caption for accuracy.
 * Added 1/128th lines. There's a little moiré visible on my screen with that many subdivisions but maybe that's acceptable. More lines than that (in the same style as the current image) would require making the other lines even taller. —David Eppstein (talk) 19:22, 17 August 2021 (UTC)
 * Not sure how it looks on your screen but on mine it looks great, thanks for the change. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * In "Arithmetic", why is the subtraction equation by cases when the addition equation is with max/min? Additionally, "difference" might be a confusing term as it sometimes refers to the absolute value of subtraction ("the difference between 2 and 3, or 3 and 2, is 1"). Then, and this could just be me, I think it's a little bit odd to have addition and subtraction presented as completely independent when the simplest underlying properties (to me) are closure under addition and closure under taking the negative of a number (so that subtraction, $$\frac{a}{2^b}-\frac{c}{2^d} = \frac{a}{2^b}+\frac{-c}{2^d}$$, is just a combination of the two).
 * Made the subtraction equation more parallel, and used "addition, subtraction, and multiplication" instead of "sum, product, and difference". Although subtraction is really just addition+negation, I think it is less technical to present it as its own operation rather than describing it as produced by a closure of other operations. —David Eppstein (talk) 19:22, 17 August 2021 (UTC)
 * Alright, that's fair. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * Giving an example of non-closure under division (simplest is just 1 and 3) might be illustrative.
 * Ok, added. —David Eppstein (talk) 19:22, 17 August 2021 (UTC)
 * Can you spell out to the reader exactly how every integer and half-integer are dyadic (cf Rational number)?
 * Added, but without a source. I think it would be extremely difficult to find a source spelling out this basic explanation. —David Eppstein (talk) 19:48, 17 August 2021 (UTC)
 * Yep, more than happy that this is fine under Scientific citation guidelines. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * Around $$\textstyle \frac{\left\lfloor 2^i x \right\rfloor}{2^i}$$, is it worth defining the floor function symbol and giving a hyperlink? I think a reader (maybe a high schooler) could be following along up to this point but not know the notation.
 * Grr, the vertical fractions were added by a persistent IP editor with an obsession about avoiding horizontal fractions. Horizontal works much better for inline formulas. Anyway, explanatino added. —David Eppstein (talk) 19:48, 17 August 2021 (UTC)
 * "For a fractal subset of the real numbers, this error bound is optimal ...": it took me a lot of re-reading and thinking to parse this sentence. Am I right in saying that the point is that $$x-\textstyle \frac{\left\lfloor 2^i x \right\rfloor}{2^i}\not\in o\left(\textstyle \frac{1}{2^i}\right)$$ for some fractal subset $$x\in\mathbb{R}$$? If so, what exactly is the definition of "fractal" here and is the error term non-optimal everywhere outside of the fractal?
 * I don't think the use of o-notation exactly matches the sense here. It's more like $$\liminf_{i\to\infty} \frac{x-\left\lfloor 2^i x \right\rfloor / 2^i}{1/2^i}>0.$$ The difference is that your formulation with o-notation expresses the idea that it's not possible for the whole sequence of approximations to converge more quickly, but the actual idea is a stronger one, that even if you pick out a subsequence of unexpectedly good approximations from the whole sequence, you still won't get better by more than a constant. And "fractal" is used here in its standard technical sense: having fractional Hausdorff dimension (I think more precisely, having a dimension that depends on how far away from zero you want the lim inf to be). Given that this is still prior to the "advanced mathematics" section, I'm hesitant to spell all of this out in more technical detail in the article, though. —David Eppstein (talk) 19:57, 17 August 2021 (UTC)
 * Okay, I think it now makes sense to me. The circles/Cantor set image helps with this. I'll just suggest as a possibility shunting the technicalities (not necessarily the fractal definition) to a footnote, but I won't require that for the review. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * (Not required for GA.) Not sure how great it'd look in practice but I've got an image idea for the dyadic rational approximation paragraph: you could plot on a numberline with 0 at one end, a value $$x$$ at the other and then show points (with annotations of the values for the first few) of the approximations for each $$i$$. For instance, if you picked the non-dyadic rational $$\frac{1}{3}=0.010101..._2$$ then the series of approximations (occurring twice each) are $$0,\frac{1}{4},\frac{1}{4}+\frac{1}{16},\frac{1}{4}+\frac{1}{16}+\frac{1}{64},...$$ (so it looks like a geometric series).
 * I'll have to think about this one, rather than just answering quickly. An irrational number like pi might make for a better visualization. It might also be possible to come up with an image showing a fractal set of badly-approximable numbers; again, I'll have to think about it. —David Eppstein (talk) 20:10, 17 August 2021 (UTC)
 * Ok, two new images added. —David Eppstein (talk) 22:42, 17 August 2021 (UTC)
 * Both looking great—the $$\sqrt{2}$$ image is particularly neat. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * Linking infinite series in the equation is a bit of an Easter egg link, and I'd be surprised if a reader followed the whole paragraph up to what an infinite series is. If you were willing to switch the first example to $$1_2=0.111..._2$$ then you could link 1/2 + 1/4 + 1/8 + 1/16 + ⋯, or maybe you could say in parentheses "(a geometric series"). If it's not original research, maybe drawing out the connection between this binary case of terminating expansions and the decimal case (with a link to 0.999...) might help.
 * I took out the link, as part of some edits addressing earlier issues. I don't know that the expansion into geometric series is really necessary here; people who already understand geometric series will already understand its implicit presence in the binary representation, and people who don't will just be confused. —David Eppstein (talk) 20:02, 17 August 2021 (UTC)
 * "by evaluating polynomials with integer coefficients, at the argument $1⁄2$" – I think the comma is not desirable here.
 * Really? I thought it was important to avoid ambiguity. The point is that "at the argument" modifies "evaluating", but without the comma it looks like it could modify "integer coefficients". —David Eppstein (talk) 20:02, 17 August 2021 (UTC)
 * On second thoughts, I think you're right about this, no change needed. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)
 * "the dyadic rationals form a dense subset of the 2-adic numbers, the set of 2-adic numbers with finite binary expansions" – Could the comma be changed to a colon? Took me a minute to parse.
 * Instead, I added "and are" at the start of the second clause, since I think the two ideas in the two clauses are more or less independent rather than one being a consequence of the other. —David Eppstein (talk) 20:04, 17 August 2021 (UTC)
 * I think I've addressed all your comments — time to take another pass? —David Eppstein (talk) 22:42, 17 August 2021 (UTC)
 * Yep, I'm now ready to pass for GA. Thanks for the quick and thorough responses. A few comments above, mostly positive, but with one suggestion of an optional change that could happen outside of the GA process. — Bilorv ( talk ) 12:24, 18 August 2021 (UTC)