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> Is there’s something more to that? Yes: this is about building the quotient field (field of fractions) [1] for some integral domain, or more generally, build

• wakawaka28 16d ago
  It's mighty pretentious to say that one needs all that theory to simply answer the question lol. For many questions, only the most rudimentary theory is plenty to get an answer, that is exactly the same answer as a more elaborate theory would yield.
  • aleph_minus_one 16d ago
    If you just want to do some stupid computations: sure. But this is not what mathematics is centrally about. The central point is the kind of thinking about the respective topics (and understanding it) which these more abstract definitions encode. Understanding the topic just enough to do some elementary computations does not give you the kind of thinking that is often near a transcendental experience. Just to give one example: the reason why the localization of a commutative ring (a generalization of the field of fractions) is introduced is that many properties of ring hold if and only if they hold for all of its local rings; see for example [1]. This means to understand some property of a commutative ring R, we "just" have to understand its (simpler) local rings. This is an example why one wants to study such ideas; on the other hand, I can imagine sooo many more exciting things to do with my life than dividing numbers by each others to form fractions. :-) [1] https://en.wikipedia.org/w/index.php?title=Localization_(commutative_algebra)&oldid=1371034380#Localization_at_primes https://en.wikipedia.org/w/index.php?title=Localization_(com...
    • wakawaka28 16d ago
      Those "stupid computations" comprise the bulk of useful work in the world. If one learns enough to do that, there may be no reason to go further. You're proving my point about the pretentiousness of insisting on the theory when one doesn't need it. Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician. Imagine arguing that the only way to understand or appreciate basic set logic is to know all about infinite sets and ZF axioms... Most people, even mathematicians, will not understand all of that and have only heard about it in the most basics if at all. A similar phenomenon happens with philosophy. Imagine arguing that simple logic is "stupid" and that one can only reason well if they have a total understanding of epistemology. I happen to think epistemology matters, and that people can benefit from at least being aware of it, but it is really a separate topic from actual mechanical logic and argumentation.
      • jgerrish 15d ago
        > Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician. What a coincidence this came up today. I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me. For most people, this might be nonsense. But it doesnt have to be. I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me. To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right. First, just learning more theory shows me we can learn and grow in our old age. This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more. And the parent's comment about quotient fields and rings is directly related to a question I asked myself about the
      • jgerrish 15d ago
        > Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician. What a coincidence this came up today. I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me. For most people, this might be nonsense. But it doesnt have to be. I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me. To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right. First, just learning more theory shows me we can learn and grow in our old age. This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more. And the parent's comment about quotient rings is directly related to an active learning question I asked myself about whether the set with only the zero element is a linear space. I don't think it is a field if 0 is the multiplicative identity, 0 can't be 1, so it can't be a linear space, right? But it works. I guess the set of the field for the scalar in a linear space is always assumed to contain more elements. It's a different set than the linear space. It seems like, duh, of course it is. But you don't see it until you work through it. And I'm guessing my experience can inform teaching others. It's confusing to me, maybe because the notation is sparse in explicitly defining the set of the linear space and the set of the associated linear space. But this helps me truly understand linear codes down the line, and Linear Feedback Shift Registers and FFTs. It's not just theory to me, I can now understand what my peers are saying and contribute my own thoughts.
• ndriscoll 16d ago
  Don't we still teach kids that e.g. 3/4=6/8, that they need to make a common denominator to add, and that they should cross multiply to check equality? I suppose we don't teach zero divisors, but otherwise, jargon aside, I'd be hard pressed to explain how we don't teach kids that fractions are members of ZxZ* mod (ad-bc). Lies to children are like... time-reversal symmetry.
  • charlieyu1 16d ago
    I don’t think it is lying, it’s just simplified so they could learn one skill at a time.
• bluecheese452 16d ago
  I thought we were adding 2 fractions? This seems completely unnecessary. It is like explaining how to kick a ball and you busting out string theory.