9 ms·

Algebra

• Taiwan_No_1 3y ago
  [dead]
• turtleyacht 3y ago
  From The Feynman Lectures on Physics: https://www.feynmanlectures.caltech.edu/index.html https://www.feynmanlectures.caltech.edu/index.html
• lambdasquirrel 3y ago
  I think it's worth revisiting elementary algebra, because the way most of us learned it in grade school doesn't really do it justice. With that said, we abandoned `ab = ba` because it's not useful for e.g. linear algebra. Elementary algebra is a very specific (but very useful) mathematical "DSL" over the reals. It's also not necessarily going to help you learn to reason about the kinds of abstractions you have whilst programming, per se, because we can't reverse `a . b` to `b . a` when we code either. Make no mistake, this is a knowledge for knowledge's sake endeavor. A liberal arts of the STEM fields, if you will.
  • yakubin 3y ago
    Commutativity is important for e.g. parallel programming. See e.g. <https://youtu.be/dPK6t7echuA https://youtu.be/dPK6t7echuA>. In the context of linear algebra, it would also be good for folks to know that matrix multiplication (/linear transformation composition) is not commutative.
    • MereInterest 3y ago
      Isn’t it more often associativity that is required for parallel programming? Being able to parallelize “a+b+c+d” only requires associativity to order the operations as “(a+b) + (c+d)”. Sure, there’s additional benefits for memory locality if you can rearrange the term to “(a+c) + (b+d)”, assuming that the four terms are stored in contiguous memory and each sum is computed with vector operations, but that’s not strictly required for parallelization.
      • phlakaton 3y ago
        In parallel programming, subcomputations may complete out of order. The ability to combine those subcomputations in any order can be pretty useful!
        • BeetleB 3y ago
          That, though, is associativity, not commutativity. In FP arithmetic, operations are commutative, not associative.
          • phlakaton 3y ago
            If you receive b before a and combine them as b*a, that's commutative. Both properties are useful in parallel programming, but you may only need one in a specific case, depending on your application.
      • GrumpySloth 3y ago
        Both are important. I replied to a comment about commutativity, so I focused on that.
  • sriram_malhar 3y ago
    Commutativity has been abandoned? What?! Nothing could be farther from the truth! The requirement for order independence is literally everywhere … where would we be without being able to add or do set unions or lattice joins without fear of getting the order wrong?! Imagine getting two different waveforms when adding one to the other, depending on the order!
    • lambdasquirrel 3y ago
      Order independence is not a property of general-purpose programming. As much as we'd love for that to go away, it is very much a reality of day-to-day programmers.
      • persnickety 3y ago
        There's plenty of commutative operations in everyday programming. a = f(c) b = g(d) can be rearranged if f and g take different input data, and calling independent functions is something seen literally everywhere in programming. Optimizing compilers often take advantage of that.
      • sriram_malhar 3y ago
        Not sure what you mean by it not being a property of general-purpose programming. Do you mean it as, "I don't have to account for it in my day-to-day work"? If so, I agree with you, but not quite in the way you mean (I think). We don't account for it just as fish don't have to account for water. The fact that you can swap two rows in an excel file and have the results be unchanged is a property that we just take for granted. It needs to be pointed out how remarkable it is, because there are situations where this doesn't work. Debits and credits in finance are not commutative because of overdraft limits. I'm sure you know all this, so I'm curious what you really mean.
    • MereInterest 3y ago
      I think the argument is more that commutivity of operations is context-dependent, even when the operation is written as multiplication. For example, knowing whether the terms of a multiplication can be commuted requires knowing whether the operands are scalars or matrices.
  • yCombLinks 3y ago
    That particular part glossed over the details. Commutativity is a property of some groups and not others. It is a property of the "normal" numbers, so we all learn it in school. But there are lots of groups that aren't commutative.
  • doubloon 3y ago
    agree. arithmetic in computer languages do not have a theoretical basis that matches the theory of arithmetic in standard algebra. for example, if a = 0.3 b=0.4 and c=5 as f32 in rust, then the following "well known results" from the article are all false: (b) a+(b+c) = (a+b)+c is false (d) a*(b+c) = a*b+a*c is false (f) (ab)^c = a^c b^c is false (g) a^b a^c = a^(b+c) is false (h) (a^b)^c = a^(b*c) is false https://play.rust-lang.org/?version=stable&mode=debug&edition=2021&gist=7a34a46b242f2a12ecb89ae119115660 https://play.rust-lang.org/?version=stable&mode=debug&editio...
  • dboreham 3y ago
    Well there goes the whole CRDT thing...
• tromp 3y ago
  This is the first time I see how the famous equation e^{i*ɸ} = cos ɸ + i * sin ɸ can be discovered from a simple assumption of what raising to the power i could mean. Namely, that for very small x, e^x is close to 1+x, not just for real numbers x, but also for complex numbers x. PS: how do I write superscripts in HN markup?
  • mjburgess 3y ago
    A lot of the issues around complex numbers can be intuitively resolved by drawing a unit circle around 0 on a numberline, and reasoning backwards to a definition of i. Rather than reasoning forwards from sqrt(-1)
    • deleted 3y ago
      [deleted]
    • dboreham 3y ago
      Puzzled expression because that's how I thought complex numbers were taught. (Second generation EE)
      • acuozzo 3y ago
        In many American schools Mathematics is taught as if it were composed solely of magic recipes. Students need only to memorize the patterns described in the red boxes in the textbook to pass.
      • mjburgess 3y ago
        It is often taught by its historical development, so first as an alleged imaginary solution to an equation with no real roots. This has it backwards as far as intuition is concerned. The geometric interpretation is the obvious one, and the use of `i` in finding zeros is a special case.
  • deleted 3y ago
    [deleted]
  • ajkjk 3y ago
• maroonblazer 3y ago
  Every time I read such lucid explanations of math like this I'm filled with resentment for the math instruction I received in junior high and high school in the U.S. There's a real sense of 'play' that these explanations evoke and that make thinking about numbers and their relationships genuinely fun and interesting. That sense of play was entirely absent from my early math education. It was all "These are the rules. Memorize them for the quiz tomorrow." I can't help feeling like my math upbringing was akin to a child being raised by parents who speak their own made-up language. Integrating with the rest of the normal-language-speaking world is anxiety inducing and filled with challenges that may never be completely overcome.
  • lambdasquirrel 3y ago
    I used to think this was just restricted to pre-college math, and to East-Asian parents. After I had erstwhile-parent types infantilize me over my adult hobbies, I realized, you need a creative person in order to facilitate creative discovery. But it takes guts to try to learn something the "right" way. Imagine going through a whole semester, never once learning what was going to be on the AP test. And it takes some stature to teach it too. Imagine doing a writing class, where there absolutely must not be any rules or criteria for grading, where you really must say, I'll know it when I see it, but there really are going to be some yeses and nos. What do you want to produce? People who can come up with new solutions to new problems on their own, or people who do cookie-cutter rules? How do you keep doing that throughout your life, long after school? People say they want the former until they come up against the reality of uncertainty, the possibility that there might not actually be any answers, and then they go right back to memorizing and teaching rules. They go back to justifying things the way they are.
    • wizzwizz4 3y ago
      > Imagine going through a whole semester, never once learning what was going to be on the AP test. This is how I learn mathematics – mostly because I'm empirically incapable of doing otherwise. I will achieve a lower grade than I “could”, and miss out on many opportunities as a result.
    • 3y ago
• pkoird 3y ago
  I remember reading this chapter and being blown away by the "construction" of the entire Mathematics (for me at that time) by Feynman in what seems like a barely over an hour lecture. In particular, the successive "expansions" always seemed so logical and consequential.
• snickmy 3y ago
  is there a calculus version of this ?
  • seanhunter 3y ago
    If you want to learn calculus the way Feynman did, seek out "Calculus for the Practical Man"[1], which is the textbook he used to learn. Fair to say it's nowhere near as good as say Stewart or Spivak, which are the ones I have. https://archive.org/details/j.e.thompsoncalculusforthepracticalman https://archive.org/details/j.e.thompsoncalculusforthepracti...
    • buescher 3y ago
      He learned to differentiate under the integral from Advanced Calculus by Woods, in high school.
• RagnarD 3y ago
  I'm a little surprised that the professor didn't demonstrate how the e^(ix) = cos(x) + isin(x) identity can be used to quickly and intuitively show a more specific famous equation: e^(ipi) = -1 You only have to see that cos(pi)=-1 and sin(pi)=0, giving: -1 + i0 = -1
  • ryan-duve 3y ago
    It doesn't sit with me well to see this being downvoted. I assume downvoters found it dismissive, but I do not read it that way. I agree, at first it *is* surprising to see a Nobel laureate walk from the most-obvious-count-on-my-fingers-elementary-school-math all the way up to Euler's Formula, only to stop there without taking the very short step, done in the comment above, to land on Euler's Equation. After all, that is how I see it done most often. The goal here is not to reproduce famous results. That would be the "we could bring forth this formula in two minutes or so, and be done with it" thing that is deliberately called out at the start. Instead, it is explained > Every so often it is a great pleasure to look back to see what territory has been covered, and what the great map or plan of the whole thing is. Seeing famous relations reduced to one another is probably enjoyable for you, and judging how many authors do it I think you've got a lot of good company. What is done here is different. It starts with things we all know as children and ends with a relationship between algebra and geometry, covering lots of mathematical apparatuses in between. It is notable that this is done without relying on the formality of landing on "famous results" at each step. That approach, combined with the easygoing language, is what I found most enjoyable about the writing.
• tokenadult 3y ago
  Thanks for sharing this particular Feynman lecture. The treatment here, using what are basically the Peano postulates to derive the field properties of the real numbers and then the basic structure of secondary school mathematics, follows the treatment of Landau's Grundlagen der Analysis (Foundations of Analysis), a concise book that Feynman was probably aware of as he presented his lecture. Feynman of course added a sense of excitement and wonder that makes this lecture charming to read. Such treatments of the foundations of secondary school mathematics are fairly commonplace in the better university textbooks of mathematics. I first learned of Landau's book in a discussion of mathematics education in a Usenet newsgroup back in the 1990s, and bought the German edition on the recommendation of Michael Spivak's famous textbook Calculus, which follows a similar approach (but starting from the field properties of real numbers taken as axioms). In those days, I'm pretty sure, the standard calculus textbook at Caltech, where Feynman taught, was Apostol's textbook, which starts a little bit differently but also takes a theorem-proof approach. https://maa.org/press/maa-reviews/calculus-4 https://maa.org/press/maa-reviews/calculus-4
  • haskellandchill 3y ago
    Nice I was going to mention Spivak but you beat me to it. I stumbled on it in a library 20 years ago and was hooked. I bought the Differential Geometry books with the pretty covers and lost my book collection before I could do much with them. I'll have to check out Apostol, I only know the Number Theory book.
  • WalterBright 3y ago
    > the standard calculus textbook at Caltech, where Feynman taught, was Apostol's textbook It remained so in the late 70's when I attended. The prices it still fetches on Amazon shows its timeless worth.
• mbigras 3y ago
  Shoutout to Richard Feynman! I didn’t immediately see his name appear on the page; but once I began reading his words, I glanced at the URL and I see, yes, it is a Feynman lecturer! Lucid and clear; slicing through obscure jargon in a fun and playful—but masterful—way; Feynman totally rocks and is worth reading further!