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comment Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?
@AlJebr, certainly "algebra" is used in studying algebraic numbers, but not only! Also complex analysis, Fourier analysis, and many other things.
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awarded  complex-analysis
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comment is $dx$ greater than $\frac{dx}{2}$?
@Bye_World, well, ok, if you insist. I myself find that viewpoint too negative. Mileages vary, and all that. I've been amused over the years by a few "colleagues"'s complaints that I'd made things "understandable by weaklings". I don't take the hard-line Spartan viewpoint, insofar as I think that reduction of difficulty probably benefits all, "not only the weaklings". :)
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comment is $dx$ greater than $\frac{dx}{2}$?
Dear Mohamed, I think the point is two-fold: your curiosity is completely reasonable, but/and the ultimate issues are subtle. That is, we can drive a car without understanding the thermodynamics of an internal combustion engine... we can use the internet without knowing how it works, or even how a transistor works. Yes, it is honorable to be curious about those things... but it would be foolish to refuse to use the internet until one knew how transistors, ... worked. The point is that the question is substantial, fully "weighted"... and in fact cannot be trivially answered. :)
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answered is $dx$ greater than $\frac{dx}{2}$?
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Ennar, a more important watershed, to my mind, is whether what we write is a narrative of some trans-semantic events, or whether the written account is the thing. The latter is brittle and fragile, the former elusive but adaptive. Thus, a flawed account of events/phenomena that have their own reality is different from the failure of prose to conform to rule for_prose_itself. Information is always incomplete, context is always a little ambiguous, etc. Why draw a thing to look like a square if it did not look like a square "in life"? :)
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Ennar, the average human has quite wonderful spatial visualization, avoiding walking in to walls, chewing gum and walking at the same time, hitting balls with bats, all kinds of stuff. Semantic things, especially purely semantic, are not so good. Humans do not use language with any facility. So I argue against divorcing mathematics from physical reality and making it purely semantical. That kind of thing. I do tell all my students to trust their physical intuition (note, I don't say "semantic intuition").
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Ennar, I absolutely do not see any disconnect between "mathematical thinking" and "natural thinking".
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awarded  Nice Answer
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comment Casino turns 50% of your losses into “free play”, are odds in your favor?
This is just a thinly-veiled version of "gambler's ruin". :)
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Rob, by the idealized standard, there are no square walls, etc. I'm not sure that airtight reasoning is better taught as adherence to rules, as opposed to exercise of judgement about what is important. If all details are equally important, then, in effect, none is, and all that. It is very tricky to draw a useful line between pedantry and carefulness, etc. For my own kids, and undergrads, and grad students, I emphasize "sanity checks", and I tend to declare a long, line-by-line computation or proof "unpersuasive" if there's no over-arching cause delineated. Tastes vary.
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comment Integers divisible by 4 but not by 3 and 16
How can an integer be divisible by 16 but not by 4?
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Rob, there is a large range of opinion about whether mathematics should be written as though it had no context (that is, denied any notion of "common sense"). In practice, it mostly does admit a huge "overhead" of prior experience and resulting sensibilities, contrary to what is often portrayed or claimed in school and in textbooks. Other people have the opposite opinion, etc.
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comment Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?
@AlonsodelArte, I do think that "algebraic number theory" and "analytic number theory" are not optimally distinguished by use of "algebra" versus use of "analysis", despite the words. The supposed "algebra/analysis" schism does not reflect practice, although it is echoed in standard curricula, and, therefore, in textbooks written to fit into that curriculum, etc.
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Kevin, yes, also it is a bad question, but/and to analyze it as though there were underlying truth is a waste of energy.
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
@Strants, oh, yes, indeed, sometimes things misrepresent themselves, and it's good to be alert to that. For that matter, sometimes one deceives oneself via inadvertent semantic boo-boos. But, unlike a contrived testing situation, in those cases we behave as though there were a genuine underlying truth, rather than a truth accessible only to a capricious authority.
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answered Identification of a quadrilateral as a trapezoid, rectangle, or square
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
"To fool the unwary"? Genuine mathematics is not created by pranksters to try to fool each other. Genuine problems are often difficult enough without nincompoops creating fake issues. This is why I am ever more disaffected with "school mathematics", despite having considerable enduring affection for real mathematics. I am not at all surprised that many kids grow up with a bad feeling about math, since it is too-often used as a way to prove to kids that the caprices of authority ... win every time... and that there's no sense to it, and that they're trying to prank you. Ugly.
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answered Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?
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comment Identification of a quadrilateral as a trapezoid, rectangle, or square
Exactly! Such questions are perverse. We should not teach students to be paranoid about "gotchas", which are not at all the primary issues and ideas in mathematics. Many pranks and traps are not even of much secondary interest, except to illustrate that the test maker(s) do not know any genuine mathematical questions to ask. Tsk! :)