[HN Gopher] Algebra
___________________________________________________________________
Algebra
Author : luu
Score : 226 points
Date : 2023-10-27 22:48 UTC (1 days ago)
(HTM) web link (www.feynmanlectures.caltech.edu)
(TXT) w3m dump (www.feynmanlectures.caltech.edu)
| turtleyacht wrote:
| From _The Feynman Lectures on Physics:_
|
| https://www.feynmanlectures.caltech.edu/index.html
| lambdasquirrel wrote:
| 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 wrote:
| Commutativity is important for e.g. parallel programming. See
| e.g. <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 wrote:
| 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 wrote:
| In parallel programming, subcomputations may complete out
| of order. The ability to combine those subcomputations in
| any order can be pretty useful!
| BeetleB wrote:
| That, though, is associativity, not commutativity.
|
| In FP arithmetic, operations are commutative, not
| associative.
| phlakaton wrote:
| 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.
| yakubin wrote:
| Both are important. I replied to a comment about
| commutativity, so I focused on that.
| sriram_malhar wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| wffurr wrote:
| None of the common numeric systems in use on computers are
| commutative. IEEE-754 floating point is not. Integer types
| have over- and under-flow.
| sriram_malhar wrote:
| Sure, but the GP alluded to "general purpose programming"
| in a separate comment, by which I take it they meant
| something like conventional python/Java code where one uses
| signed integers and doesn't worry about under/overflow.
| maemre wrote:
| Integer overflows/underflows don't affect commutativity,
| unless you refer to undefined behavior, which
| is...undefined and found only in C and C++ among common
| languages. 2's complement integer arithmetic is commutative
| for both addition and multiplication.
|
| Thanks to BeetleB for pointing out that addition in IEEE
| floats is indeed commutative (I originally claimed "+0 + -0
| = +0 vs. -0 + +0 = -0" which is _incorrect_ ).
|
| As for IEEE-754 floats, addition is commutative as long as
| you don't care about exact bit patterns:
|
| NaN + NaN may return different bit patterns. The result is
| still NaN, so the only way you can tell this apart is by
| bit-casting floats to ints or byte arrays.
|
| Multiplication over floats is also commutative modulo the
| caveat above.
| BeetleB wrote:
| > +0 + -0 = +0 vs. -0 + +0 = -0
|
| These are not exceptions.
|
| First, I will note that your result above depends on the
| rounding mode.
|
| Second, IEEE 754 mandates that +0 and -0 are equal (i.e.
| any equality operator should return True when comparing
| these two). Therefore both expressions are equal.
|
| NaN has several representations in bits, but they are all
| "equal" to one another.[1] If an operation gives you NaN,
| then so will doing it commutatively. It doesn't matter
| that the underlying bits are the same.
|
| [1] Except for the signaling aspect. But I believe that
| is preserved in commutative operations.
| maemre wrote:
| You're right, producing +/- 0 depends on the rounding
| mode and it is commutative, I forgot about that
| completely. I edited my comment to fix that claim.
|
| Also, yes, +0 == -0, but they can produce different
| results when used in the same expression, so the
| distinction does matter (although this doesn't affect
| commutativity, which is the larger point). For example,
| let f(x) = 1 / x. Then f(+0) = +inf, f(-0) = -inf.
|
| I also agree with you about NaN, that's why I mentioned
| having to go outside floating point numbers (bit-
| casting).
| a1369209993 wrote:
| > (I originally claimed "+0 + -0 = +0 vs. -0 + +0 = -0"
| which is _incorrect_ )
|
| Specifically, (-0.0) + (+0.0) = (+0.0) + (-0.0) = +0.0
| (assuming round-to-nearest-or-even). OTOH, (-0.0) +
| (-0.0) = -0.0. This has nothing to do with +0.0 == -0.0
| for comparison, addition just _is_ commutative
| outright[0].
|
| 0: Pedantically, I'm not sure IEEE-754 requires the
| specific choice of which NaN you get when you do
| `some_nan + a_different_nan` versus `a_different_nan +
| some_nan` to be commutative, but it should.
| BeetleB wrote:
| Actually, they all are. They are not _associative_.
|
| If a + b results in an overflow, so will b + a.
| yCombLinks wrote:
| 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 wrote:
| 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&editio...
| dboreham wrote:
| Well there goes the whole CRDT thing...
| tromp wrote:
| This is the first time I see how the famous equation
|
| e^{i*F} = cos F + i * sin F
|
| 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 wrote:
| 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)
| dboreham wrote:
| Puzzled expression because that's how I thought complex
| numbers were taught. (Second generation EE)
| acuozzo wrote:
| 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.
| ajkjk wrote:
| It's a lot easier to see if you imagine i to be a symbol R that
| rotates vectors in the plane, since e^(i F) = cos F + i sin F
| doesn't use the additive property of i at all.
|
| (Well... I guess it does in a sense, because all it means to be
| additive is that it's linear, so you can add up all the
| complex-valued terms in e^(i F) and give their coefficients the
| name sin().)
|
| Figuring out what e^(iF) means requires figuring out "What `i`
| is" and "What e^x means on non-numbers" at the same time. That
| is, it requires you to perform two intellectual jumps at once
| instead of one at a time. No wonder it is so confusing.
|
| If these concepts are made sufficiently simple, I imagine that
| we could live in a world where we also teach e^(a d/dx) f(x) =
| f(x + a) in high school.
| ChainOfFools wrote:
| Pardon my mathematical ignorance here, but I've always been
| curious as to whether negative numbers are the only alternate
| set that extends from the origin of zero, or whether they are
| merely the only set that is diametrically opposed to their
| positive counterparts, and there are in fact an infinite
| number of these lines extending from the origin at zero,
| radiating from it in all directions, and what we think of as
| the positive and negative numbers are simply one pair of rays
| extending from this origin arbitrarily chosen as our base
| units.
| ajkjk wrote:
| I often think that a lot of math would work out more easily
| if we _only_ used polar coordinates and regarded the
| negative numbers as a "separate number line" rather than a
| continuation of the positive numbers.
|
| (In particular, if you know about delta functions... a lot
| of weirdness around x=0 goes away if you write everything
| in terms of r \sgn (r) and take the derivatives of both
| terms. e.g. This gives "for free" the fact that the
| divergence of 1/r^2 is 4 pi delta(r).)
|
| I have heard of systems in which one sticks more lines out
| from 0 than just the positive and negative numbers. At some
| level that's what R^2 is, with four copies of the positive
| number line, but I don't see a strong reason why in
| principle you couldn't have an odd number of lines, which
| would correspond to... uh... R^1.5. But you have to define
| how these lines rotate into each other, and it is gonna be
| weird.
| ithinkso wrote:
| If you want the equation
|
| x + 1 = 0
|
| to have a solution, you need to invent negative numbers,
| now if you want the equation
|
| x^2 + 1 = 0
|
| to have a solution, you need to invent complex numbers and
| 'i'. (Also, complex numbers, turns out, are enough for
| higher powers as well)
|
| The line and plane are just convenient representations of R
| and C but there is nothing inherently profound about them,
| in my opinion
| ajkjk wrote:
| This is one take, and one that has become very popular,
| but it's not necessarily the only take. In particular it
| presupposes that your number-like indeterminates can be
| both (a) multiplied and (b) added to numbers (and,
| implicitly, divided). Naturally the solution has to be a
| division algebra. If instead you asked the question "for
| what values of O would O^2 (v) = -v", or even just O^4
| (v) = v, then you would be more content having the answer
| live in a different space, of operators on vectors rather
| than vectors themselves, instead of in a field extension
| of the present space. Of course they are basically
| isomorphic but I think the alternate interpretations are
| useful to keep in mind so that we don't accidentally
| assume our way into a box of our own making.
|
| edit: I should add, by O^2 I mean O [?] O, so there's no
| definition of "multiplication" on these necessarily, just
| composition.
| maroonblazer wrote:
| 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 wrote:
| 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 wrote:
| > _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.
| anonymouskimmer wrote:
| There are a few colleges/universities that use non-grade
| evaluations. Evergreen State uses a narrative evaluation:
| https://www.evergreen.edu/registration/transcript-format
|
| Really smart people, and sometimes even just moderately smart
| people, coast along as A, B, or C students while learning
| what they want and just doing the minimum of the requirements
| to coast along. And then a bunch of other people have
| academic-like side-interests that they pursue in their free
| time.
| jjtheblunt wrote:
| I think it's very common to have uninspiring math teachers, but
| it's not uniformly the case across the United States. That
| site, for example, is at Caltech. The math teachers I had in
| suburban Chicago in public schools were superlative.
| mmmmpancakes wrote:
| The ability to communicate math this way is honestly rare. It
| comes from a combination of deep understanding, long experience
| in communicating math, and a certain level of "culturing" that
| is specific to the academic experience.
|
| Among the best, Feynman was singular in his ability to
| communicate math and physics.
|
| In other words, don't be so hard on the teachers who were
| disappointing in comparison to the stellar examples you see
| from top mathematical communicators. What you're reading is
| quite rare and, while education quality could certainly
| improve, its not fair to expect this of a 5th grade teacher who
| covers 5 topics in a day. Even for the best, developing this
| type of material takes time and thought that a school teacher
| probably does not have.
| melagonster wrote:
| but government hired professors to design content of
| mathematics textbooks for students. there are some problem...
| mmmmpancakes wrote:
| Disagree. The purpose of a textbook and a lecture is very
| different. A good textbook can be a helpful resource for
| teaching and lecturing, but it is not sufficient to
| guarantee high quality math education. Conversely, a good
| educator who deeply understands the material can deliver
| fantastic education without a good textbook. Claiming that
| profs writing bad textbooks is the cause of poor quality in
| class math instruction is absurd.
| mlyle wrote:
| > Claiming that profs writing bad textbooks is the cause
| of poor quality in class math instruction is absurd.
|
| It certainly doesn't help. The tendency to pile more and
| more into standards, and then to have haphazard treatment
| in the textbooks, with problems that don't make sense...
| isn't great.
|
| Stick a new teacher in the classroom, and they're going
| to run their book's recommended pacing and content. And
| even a veteran is probably going to lean on the book a
| lot in a pinch.
|
| And, your course needs to fit together with 2 other
| teachers who are too likely to be running the absurd
| pacing and content in the courses before and after yours.
| The rushed pace leaves no choice but to devote a huge
| fraction of the time to procedural knowledge.
|
| The net result doesn't serve anyone: the top students are
| left unchallenged and without the context and enrichment
| that could let them really grow. The bottom students are
| in painful struggle. And the middle are perpetually
| slightly confused, learning specific tools that they'll
| immediately forget when the unit completes.
| starcraft2wol wrote:
| Math teaching selects for people who do their homework
| everyday. This is very different from what actual math selects
| for.
| tux3 wrote:
| People who can memorize and apply rules.
|
| I never did any homework, and that worked great until
| undergrad, where the strategy of reading things on my phone
| all day instead of listening to anything in class stopped
| working quite so well.
|
| I'm not very good at proofs, but following arbitrary rules
| provided with no motivation? Decades of experience!
| cafard wrote:
| Doing one's homework is an excellent way of exposing and
| filling gaps in one's understanding. I'm more or less
| adequate at math, but I'd be better if I had done more of my
| homework.
| starcraft2wol wrote:
| No doubt. But ranking math skills, being especially
| consistent in studies/homework is maybe number 5. The
| problem I am pointing out is math teachers often only have
| that 1.
| monadINtop wrote:
| Yes people who follow the rules and do well in "school
| maths" are very likely to also do well and succeed in
| "higher mathematics".
|
| Unfortunately, this selection misses (most?) children who
| may not be well suited to "school maths" - for whatever
| reason. But these children may succeed just as well in
| "higher mathematics".
|
| Two anecdotes: (1) June Huh dropped out of high school and
| stagnated for 6 years in university. In his 6th year, he
| ran into the fields medal Heisuke Hironaka. It was only
| then his "slow thinking" and deep creative insight (perhaps
| the things that hindered him in "school maths" type
| courses?) proved to be fruitful in higher mathematics. June
| Huh now has a fields medal.
|
| (https://www.quantamagazine.org/june-huh-high-school-
| dropout-...)
|
| (2) I was frequently in trouble at school and underachieved
| relative to my predicted grades. I resented the route
| learning and arbitrariness of "school mathematics". Due to
| some miracle I'm currently working towards a PhD in
| theoretical physics, in the mathematics department of a top
| university, and I also spend about 90% of my free time
| working through various advanced maths textbooks for fun.
| Turns out I'm quite suited for thinking about higher
| mathematics, despite not being particularly well disposed
| for school. If my school experience was different, I
| probably would have done a PhD in pure mathematics instead.
| starcraft2wol wrote:
| > people who follow the rules and do well in "school
| maths" are very likely to also do well and succeed in
| "higher mathematics".
|
| I somewhat disagree. There were plenty of students who
| start to hit higher classes and just don't have the
| aptitude for it. They really didn't know it wasn't their
| thing until junior year of undergrad, despite always
| being told they were "good at math" as a kid.
| JackFr wrote:
| Agree. Saw that as a math major undergrad, there were
| some people with more of an engineering bent who just
| crushed multivariable calc, differential equations stats
| and numerical methods, but then just got stuck at
| abstract algebra and point set topology proofs and stuff
| because there was no concrete application or "real world"
| anchor for the work.
| Der_Einzige wrote:
| Boy topology is useful in AI/ML. Just look at T-SNE or
| UMAP
| analog31 wrote:
| Junior year in college is when they start doing proofs.
| This is a crime.
|
| "Back in my day," my school district adopted a math
| curriculum that introduced sets in first grade, and eased
| us into proofs. We were not unfamiliar with proofs when
| we hit high school geometry, which was almost entirely
| proofs. Also, by doing proofs we could recognize that the
| manipulations we were doing in the regular problem sets
| could be seen as mini-proofs, rather than just guessing
| the right algorithm and grinding through it without
| knowing why.
|
| When my kids took math, no proofs. Even geometry was all
| problems and no proofs. Moreover, kids are all aware of
| the conventional wisdom that "you just need math to get
| through school, you will never use it after you
| graduate."
|
| For me, proofs were what made math come alive, and I
| started college as a math major. Today, despite my
| theoretical bent, I'm one of the few people at my
| workplace who is willing to solve practical math problems
| that don't have a canned solution in a software package.
| photon_lines wrote:
| For anyone interested in reading a bit more about Huh's
| story and some key lessons from it - I made a bit of a
| write up about him and his teacher which you can find
| here: https://photonlines.substack.com/p/lessons-from-
| june-huh-fro...
| krsrhe wrote:
| Usually when people say this, it's because learning something
| for the first time is harder than learning something you
| already know.
| jodrellblank wrote:
| > " _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._"
|
| You could wonder why you feel this way. If there is a sense of
| play which makes it interesting, and you are reading it and
| enjoying it today, why isn't that enough? What is the
| resentment about - you haven't missed out on the interesting
| math explanation - it's right here, you're reading it and
| enjoying it.
|
| (This is the work of cognitive behavioural therapy - "I read a
| math thing which I found interesting, but instead of feeling
| elevated, happy, awed in the presence of brilliance, greatful
| that I stumbled upon such a thing when I could have gone my
| whole life not knowing about it, I instead jumped to feelings
| of resentment about things which happened many years ago,
| leaving me in a bad mood. I wonder what in my head made that
| connection and why?" Ref: a different person I was replying to
| in a different thread who was claiming that CBT is about
| colouring-in while handwaving problems away or waiting for
| acceptance that life sucks to bestow itself upon you).
| plonk wrote:
| This isn't about enjoying the explanation now, it's about the
| academic success and open doors one could have had if their
| teachers had been better communicators. Math doesn't get
| studied in a vacuum, it's the biggest selection criterion for
| STEM education.
| jodrellblank wrote:
| You could be right, that's one possible explanation, but
| that comment didn't read like "I resent my highschool math
| teacher because I don't earn enough in a STEM job" to me.
| harry8 wrote:
| Upset because of all that was missed out on. The
| realisation of what might have been.
|
| Could _I_ have been a decent mathematician if the
| opportunity was not missed while I was totally unaware of
| its existence? Obviously there can be no definitive
| answer to that question.
|
| And it's also ok to be a little resentful of having been
| tortured for many hours with something that clearly could
| have been a lot of fun.
| jwells89 wrote:
| This was an issue with my high school math classes too, but the
| other thing that made for something of a stumbling block for me
| was the total absence of and refusal to produce practical
| examples of the math being taught in action.
|
| The way my mind works, in order to grasp a concept well enough
| to be able to actively use it, I need to see it in action by
| way of a non-contrived, realistic example. The way any math
| past basic arithmetic tends to be taught in abstract
| dramatically slows acquisition.
|
| It's a quality that cuts both ways. It doesn't work well with a
| lot of traditional academics, but it enables me to self-teach
| highly applied subjects like programming with little friction.
| jibe wrote:
| Push the language analogy further. In a sense, a child really I
| s raised by parents speaking a made-up language. It just
| happens to be a locally shared made-up language.
|
| Like math, you brute force language learning. Imitation and
| memorization. Only after years, or decades, can you go from
| basic language understanding to appreciating the beauty of
| words, poetry, literature. I don't think there is a short cut
| in language or math. You have to go through the unfun
| multiplication table, spelling test phase to build the
| foundation for higher level appreciation l.
| User23 wrote:
| I think you'd really enjoy the book Visual Complex Analysis.
| devchix wrote:
| I feel the same way. My college years math was a clueless grind
| (to me) and I don't think I made the connection to the real
| world until long after school was behind me. There's a very
| good site called BetterExplained, where I go to when I need to
| review some concept. I wish I had that when I was in school.
| photochemsyn wrote:
| I think this paragraph in particular should be explained to all
| introductory students in any STEM-related field. It sets the
| stage for a basic yet profound understanding of what
| mathematics is all about:
|
| > "To discuss this subject we start in the middle. We suppose
| that we already know what integers are, what zero is, and what
| it means to increase a number by one unit. You may say, "That
| is not in the middle!" But it is the middle from a mathematical
| standpoint, because we could go even further back and describe
| the theory of sets in order to derive some of these properties
| of integers. But we are not going in that direction, the
| direction of mathematical philosophy and mathematical logic,
| but rather in the other direction, from the assumption that we
| know what integers are and we know how to count."
|
| Beyond that, an understanding of algebraic concepts in terms of
| how equations can be manipulated from one form to another (and
| of the rationale for why one can) might benefit from the
| addition of the concepts of distribution and commutation, which
| are not included in this lecture (although the important basic
| idea of a successor is).
| pkoird wrote:
| 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 wrote:
| is there a calculus version of this ?
| seanhunter wrote:
| 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.thompsoncalculusforthepracti...
| buescher wrote:
| He learned to differentiate under the integral from Advanced
| Calculus by Woods, in high school.
| RagnarD wrote:
| I'm a little surprised that the professor didn't demonstrate how
| the e^(i _x) = cos(x) + i_ sin(x) identity can be used to quickly
| and intuitively show a more specific famous equation:
|
| e^(i _pi) = -1
|
| You only have to see that cos(pi)=-1 and sin(pi)=0, giving:
|
| -1 + i_0 = -1
| ryan-duve wrote:
| 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 wrote:
| 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
| haskellandchill wrote:
| 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 wrote:
| > 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 wrote:
| 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!
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