[HN Gopher] Ask HN: I suck at math, where to start?
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Ask HN: I suck at math, where to start?
As a self taught programmer I feel like I have a big gap on my math
knowledge, where can I start learning math concepts that will help
me in my career?
Author : novakor
Score : 150 points
Date : 2022-01-12 12:42 UTC (10 hours ago)
| sesm wrote:
| Try 'Mathematics can be fun' by Perelman.
|
| I was a 'math prodigy' in school. To me mathematics was always
| about solving problems, learning any theory/apparatus makes much
| more sense when you understand what problems it helps you solve.
| As a programmer you probably have a problem-solution mindset too.
| If you just have fun solving problems and slowly build your
| mathematical foundation from there, you'll probably discover that
| you don't suck at all, just the way you were taught mathematics
| at school was too dogmatic.
| progre wrote:
| Are you sure you suck at math? I used to think I sucked at math
| (now I think I'm merly bad at it). Turns out I suck at _numbers_.
| I can 't get a feel for them. Manipulating equations and such
| though I found quite easy back when all that was fresh in school.
|
| As for career advise, depends on what you want to do: There is
| not much use in learning calculus if you end up in astatistics-
| heavy field like data science. I'd say figure out what kind of
| computing you want to work with and see if there is a specific
| part of math that are useful there, if there even is one.
|
| I've done work for finance and never had to go much deeper than
| simple multiplication.
| kasey_junk wrote:
| I thought I sucked at math for a long time, reinforced because
| my math teachers in school believed it too.
|
| Turns out I suck at spatial thinking. I can't manipulate things
| like plots and curves in my head and they don't help me
| understand mathematical principles. Algebra and logic though
| come easy to me. So learning math became an exercise in
| translation.
|
| That is to say, please don't think you suck at math and
| definitely look at different ways to learn the same concepts if
| something is not clicking.
| lief79 wrote:
| Agreed, I've used trig once since I left college for an image
| manipulation problem. Discrete mathematics, Algebra,
| statistics, and depending on the roles ... linear algebra are
| far more likely to be useful.
| k3liutZu wrote:
| I strongly suggest Khan Academy --
| https://www.khanacademy.org/math
|
| Sal Khan is a great teacher
|
| ---
|
| I am doing now a uni course (economy/informatics) and had to
| brush up on calculus and other math areas. Khan Academy helped me
| understand a lot of required concepts.
| jcadam wrote:
| I managed to get through a BSCS w/ a minor in mathematics. And 20
| years later I still count on my fingers when working out
| problems. Or counting change. Or working out the final price on a
| 30% off sale while shopping.
|
| The odd look I got from from calc professors while taking exams
| was kinda funny.
| orionblastar wrote:
| http://khanacademy.org
| bennysomething wrote:
| This is a big coincidence for me, a few hours ago I posted a
| reply to someone else saying I've decided this year to plough
| through khan academy at least until I can do some calculus.
|
| I'm in the same boat as you I think.
|
| I'm 40, work as a programmer and I have a fear of being found out
| for not know enough maths.
|
| I bought "math for programmers"
| https://www.manning.com/books/math-for-programmers But I realised
| I really need to get better at algebra first.
|
| I'm treating this as how I treated learning the guitar on my own
| when I was 12, sit in my room at night and practice.
|
| I'd be interested to hear what people think of my khan academy
| plan :)
| [deleted]
| tatsean wrote:
| https://www.khanacademy.org/
| BFay wrote:
| I really like Jeremy Kun's writing: https://jeremykun.com/ and
| recommend his book, A Programmer's Introduction to Mathematics:
| https://pimbook.org/.
|
| So far I've only read the first few chapters of the book, and the
| exercises often feel too difficult to me. But I think he does a
| great job of easing into mathematical notation, pausing to
| reflect on what a seasoned mathematician might be thinking when
| they come across that notation. He also makes a lot of analogies
| to programming, and has example programs that are easy to follow.
| It's helpful to have that angle to understand things from.
| silent_cal wrote:
| I find it helpful to start with "quantitative reasoning" instead
| of "math" if you're not math person... Math people speak "math
| language" which can be frustrating if you're not cut from that
| cloth.
|
| Here's an example of a good quantitative reasoning textbook that
| I looked at once, it is pretty well received by non-math people:
| https://www.amazon.com/Using-Understanding-Mathematics-Quant...
| schleck8 wrote:
| If you happen to be german speaking
|
| Maths for high school: https://studyflix.de/mathematik-schueler
|
| Maths for university: https://studyflix.de/mathematik
| gspr wrote:
| Do you struggle with mathematical reasoning in itself, or with
| different aspects of mathematics (aspects that are built on, and
| work based on, said reasoning)?
| [deleted]
| xtiansimon wrote:
| Same situation. It actually helped me to read about the history
| of math. I found a love for some ideas in math and dispelled some
| myths I had about the "Queen of the Sciences".
| mixedmath wrote:
| If you have a math-savvy person in your circles, it might be a
| good idea to set up a chat with them. I think a good beginning
| point would be to identify a specific problem or concept that
| you've encountered that gave you some difficulty, and then talk
| it through with them.
|
| In my experience as a mathematician, it is not always easy to
| identify the source of frustration in a problem. Having a
| different perspective can be really helpful. (Also in my
| experience as a math educator, a shockingly large proportion of
| problems stem from fractions, exponentials, and logarithms --- or
| not knowing what a function is.)
| warrenm wrote:
| Check out this Farnam Street post -
| https://fs.blog/mathematicians-lament-2
| sjg007 wrote:
| I agree with this. I think that the way we teach math is a
| disservice to the kids. When you watch a talented
| mathematician's lecture you always see how relatable and
| intrinsically motivating it is.
|
| On the flip side, I think that when people write about math in
| scientific reports for example, they don't explain the math
| they use either.
| dboreham wrote:
| https://www.youtube.com/c/3blue1brown
| yboris wrote:
| I second this recommendation. It's a lovely way to explore some
| of the beauty of mathematics.
|
| I have a BA in math and I think the most-useful mathematics for
| everyone is _probability_ (math theory and basic calculations).
| This is because you 'll be better able to deal with (quantify)
| uncertainty. The concept to aim for is (learn enough so you can
| understand and use) _expected value calculations_ - which are,
| I think, the foundation of rational decision making under
| uncertainty.
| godwhoa wrote:
| +1 I really enjoyed the "Essence of ..." playlists from
| 2blue1brown.
| [deleted]
| sillysaurusx wrote:
| > where can I start learning math concepts that will help me in
| my career?
|
| I wouldn't worry about it. The amount of math that a working
| programmer needs is minuscule.
| bigbillheck wrote:
| Depends on the job.
| ur-whale wrote:
| IMO "I suck at subject X ..." simply means the amount of
| work/time you're going to need to invest to master subject X is
| larger (sometimes much larger) than folks who have a knack for
| it.
|
| It rarely means you're completely incapable of mastering subject
| X.
|
| The real question is: is the investment worth it? If it takes you
| 5 years to master basic linear algebra because your brain truly
| isn't wired for it, how much is the 5 years you are going to
| spend going to pay back in the long run?
|
| Oh, and: actually enjoying doing X is usually a _tremendous_ help
| towards mastering it.
| ianai wrote:
| Time. Be honest about how much time you're willing to invest into
| understanding math. You can understand math if you're willing to
| give it the time, but without that time you're likely setup for
| frustration.
|
| Where to start?
|
| Math is huge. I suspect discrete math may be the most useful to a
| programmer who's looking to just be a more theoretical
| programmer. Proof by induction stands out as a core, helpful
| concept too.
|
| Otherwise I suspect it's about problem domain. Geometry has its
| uses as does algebra. It all kinda branches from there.
| saityi wrote:
| _Introduction to Graph Theory_ - Richard J. Trudeau
|
| _A History of Pi_ - Petr Beckmann
|
| _Journey through Genius: The Great Theorems of Mathematics_ -
| William Dunham
|
| _How to Bake Pi_ - Eugenia Chang
|
| These are all sort of 'pop-math' books -- that is, they're more
| intended to spark a joy & love for math than teach rigorous
| mathematics. _Great Theorems_ and _A History of Pi_ include a lot
| of history (edit to add: in addition to covering the math
| involved!) -- did you know some mathematicians in history would
| duel over their theorems? That theorems were a carefully guarded
| secret instead of something you shared?
|
| _Introduction to Graph Theory_ is specifically intended as an
| introduction to mathematics for 'the mathematically
| traumatized'.
|
| In my opinion, after reading these, if you've sparked a joy for
| the puzzles and fun of mathematics, then I would then suggest
| branching out into more formal presentations of them relevant to
| your interests... it's much easier to slog through a book on
| abstract mathematics when you receive from enjoyment from the
| puzzles presented.
| ultra_nick wrote:
| There's nothing better for filling math gaps than going through
| Kahnacademy.
|
| Start skimming at the 1st grade level and then slow down and
| focus when you reach material that isn't easy.
| motohagiography wrote:
| Out of curiosity, what does it look like to not-suck at math?
|
| Do you wake up in the morning and are reminded of constants and
| theorems in everyday things and laugh empathetically with donuts
| as you recognize your shared absurd topological homology, or go
| to the office and automatically recognize what's possible and not
| from its implied complexity class in office conversations? Are
| you checking Bayesian priors in your interpretation of the news
| or other risk?
|
| This is only half kidding, as I also identify as useless at math,
| but I think its words and concepts can be very funny, and I
| wondered what someone who actually knew this stuff might think
| about. It begs the question of what one is actually doing when
| they are doing math as well. Are you reasoning with abstract and
| quantitative models, are you writing papers and proofs of new
| ideas, or are you encoding a narrative dynamic into a
| symbolically defined logical relationship? Maybe you're just
| being recognized as a peer by the community of people who
| recognize each other as good at it?
|
| When I learned music again later in life, I found I appreciated
| the same things, but with better intuition and words for
| articulating and reproducing them. Working musicians have a trade
| where it's expected they can show up and perform in a group based
| on sight reading of the notation, same as a programmer.
|
| All this is to say, I see a lot of these "I can't math / how do I
| math" threads and am always interested in them, but maybe we
| should start with, "how does anyone math" first. So, how do you
| math?
| jscholes wrote:
| > Out of curiosity, what does it look like to not-suck at math?
|
| To me, it looks like the rest of your comment, wherein you
| casually mention constants, theorems, topological homology and
| other concepts that don't even come to mind for many people who
| self-identify as being bad at maths (as I do).
| motohagiography wrote:
| Thank you, but I am mathematically illiterate. I can't
| identify greek letters after sigma without looking them up, I
| ignore equations in papers and other texts unless they are
| supporting something that offends me, I don't have a working
| understanding of why certain operations are meaningful or
| their uses other than knowing there is probably a python
| library that does anything I could need.
|
| However, things have shapes and relationships, most of which
| we can't physically see, but we can reason about and compare
| them, and that's about as deep as I get. e.g. do things at a
| certain level of abstraction have a similar shape, and what
| are the words that describe that? Does this tell us more
| about the thing itself, or the limits of our ability to
| percieve it, and it's just an artifact of our lens? It's like
| having an ear for music, where you hear fragments of other
| pieces in everything, and interpret forms and symmetries and
| respond to intuitions, expectations, and resolutions - but
| not being able to play it.
|
| I think we're entering into an era where we can finally have
| punk math, where some idiots get on stage and the people
| watching them go, "omg, this is terrible but fun, and I could
| do this better," and a thousand bands get launched. When I
| write stuff like that, it's because I don't mind acting as
| one of those idiots. I figure if most of my favourite bands
| can't read music, there's interesting math to be done by
| people who aren't proving anything or making progress, but
| intentionally or not, like all idiots, they exist as an
| example to inspire others. :)
| paulpauper wrote:
| Knowing how to solve problems in a mathematically precise way,
| such as writing down the formula for the expected value of a
| path-dependent investing strategy an hour after conceptualizing
| it. Like being a handyman but the formulas are your tools.
| dleslie wrote:
| I'm no mathematics savant, but math has always been easy for
| me. I never recall struggling to understand concepts, even
| through my university degree.
|
| What I vividly recall is my fascination with algebra and
| geometry as a child. While on hikes I would imagine the
| equations necessary to approximate the flora around me, to
| calculate the parts of the rays of light, and so forth.
|
| And while I never applied myself strongly to homework, I did
| pay with math as I've would pay an instrument.
|
| That is to say, this should sound like how musicians describe
| hearing the music of nature, and their fascination with sounds
| and musical theory.
|
| Math is an instrument that one can play to make beautiful
| music.
| ur-whale wrote:
| > Out of curiosity, what does it look like to not-suck at math?
|
| The speed at which you an understand, internalize and can put
| to use a new concept when it is presented to you.
|
| This is usually combined with the practice of the trade
| providing you with actual pleasure, which is, of course, a
| self-reinforcing loop.
|
| Some folks can take a single semester of linear algebra and
| absorb the majority of the thing, including learning far more
| than the course required because they truly liked it.
|
| Others can take the first half of their career to finally
| understand the ideas and apply them properly.
| javajosh wrote:
| A related question: which part of math can be used to model
| object-oriented programming? AFAIK set theory deals purely with
| associations of objects, not how you make new ones. HS algebra is
| concerned with simplifying and inverting numeric equations.
| Meanwhile abstract algebra (and number theory) is concerned with
| finding talking about broad categories of numbers, relating those
| categories to each other, and making new, presumably interesting
| (read: surprising) statements about categories of numbers. Math
| is a study in "going meta" since each component of each activity
| gets a name, which are further grouped and named into other
| categories, limited only by what the thinker can stomach/finds
| useful.
|
| But I am hard-pressed to think of any mathematical construct that
| reflects, even a little, the reality of OOP. This is evidence
| that OOP is an engineering concern, not a math concern. That is,
| OOP is one method to help humans deal with the complexity of a
| large amount of shared, mutable state (SMS), by partitioning it
| into smaller units of SMS. But math itself doesn't care about the
| scale of anything, and will happily encode any state into a
| single, very large integer, if you let it.
|
| Some parts of programming are better grounded in math, like
| functional programming and relational algebra. Some distributed
| programming problems have some nice, ad hoc mathy treatment (e.g.
| Paxos), but don't really have a clear correspondence to anything.
|
| Interesting the field that is closest to programming in real
| life, IMHO, is statistical thermodynamics. This is usually taught
| as part of the physics curriculum, and is pretty math intensive,
| and the field's remarkable job is to generally model microscopic
| behavior and then predict macroscopic behavior of huge
| aggregates. Programs always deal with huge numbers of tiny
| things, each having unique degrees of freedom, (alternatively,
| which have unique constraints), so there is some connection
| there. ST is also the field most closely related to certain
| "quant" jobs in the finance field, AFAIK, since the same tools
| let you model individuals in an economy and from that predict
| markets.
| GuB-42 wrote:
| What kind of math can help a programmer in their career?
|
| There are some fields that can make use of math, like computer
| graphics, machine learning and data science. Most of it being
| linear algebra. But although they interest me, I don't work in
| these fields. For actual, paid, work, I don't remember using math
| beyond middle school level (ex: solving linear equations), and
| even that is uncommon.
|
| If you want to learn some math, maybe try playing with shaders
| (see: https://www.shadertoy.com/ ) or more generally, 3D
| graphics. There is lots of math in here, but that's awesome
| looking math, and you can actually see the results.
| josefrichter wrote:
| It also helps not to tell yourself "I suck at math", that's a
| very common self-fulfilling prophecy.
| the_only_law wrote:
| I am beginning to think "I suck at math" less and more of "I'll
| never be good at it", from the comments on this thread.
|
| I absolutely despise grinding and drilling uncontexualized and
| contrived problems and have never been able to maintain
| something stuff like that for very long. Which of course
| becomes problematic at I find myself interested in problems
| that require a decent grasp on some math.
| Jtsummers wrote:
| The grinding is, by analogy, equivalent to the "reader" books
| in many languages. "See Spot. See Spot run. Run Spot, run!"
| level and then up. It's very hard (for most people) to jump
| from "I now know the English alphabet" to "I can read any
| article in a literary magazine." Many people forget that they
| went through that process because they did it so young, but
| also because language (written and spoken) is far more
| pervasive than mathematics. This pervasiveness means that you
| have less "grinding" to do, because you get to learn just by
| being in the environment the language is used in (if it's
| your native language, or if you can afford or are otherwise
| forced to move into an environment where it's the local
| native language).
|
| Unless you're like me as a kid, you don't look around and
| start counting, adding, and multiplying based on the objects
| or scenes that you see, and "thinking in numbers". Thus the
| need for grinding, you need some math fluency (which takes
| practice to develop and maintain, like any other domain)
| before you can move on to more advanced topics (at least
| easily). If you lack fluency, even at a low level, then you
| have to do a refresher every time you try and get to
| something moderately advanced. Which is frustrating, at
| least, if not demotivating to the point of causing most
| people to stop. The more fluent you are, the easier it is to
| pick up an arbitrary advanced topic (at least to read, if not
| to apply or extend).
| ianai wrote:
| Agree! I've seen many a mathematics professor struggle with
| math. It's not innate to any of us. But certain of us have
| spent more time with it and seen more of it. The time and
| familiarity eventually goes a long way - to a phd, even.
| ur-whale wrote:
| > It also helps not to tell yourself "I suck at math", that's a
| very common self-fulfilling prophecy.
|
| Wholeheartedly agree, and this applies much more broadly than
| math.
|
| Once you've built the mental fence, breaking it down is a giant
| PITA.
| qsmi wrote:
| Definitely.
|
| I think when people say they suck at math what they mean is; I
| find math homework extremely boring and unrewarding. Which
| perpetuates sucking at math and only compounds. Then things get
| worse as modern society works better for the individual when
| one does not totally suck a math.
|
| Kinda like, I suck at piano because I find practicing the piano
| boring and unrewarding, but without the ramifications of
| sucking at math.
| kypro wrote:
| Agreed, although it's good to recognise what your strengths and
| weaknesses are. I would argue it's often better to focus on
| being great at the things you're good at than trying to be
| decent at stuff you really struggle with. If OP struggles with
| math they may be better off focusing on something else, at
| least if they're optimising for career success. If you're
| optimising for happiness and enjoy math problems then ability
| doesn't really matter.
| qsmi wrote:
| I suppose depending on how extreme we're going with
| unfocusing on math.
|
| But one doesn't really get to ignore math in life because of
| money. You'll still need to figure some things out. I can say
| I'm a poor painter so I'm just going to ignore that and
| things will be fine. The cost of ignoring math can be quite
| high.
| upofadown wrote:
| Chances are you actually suck at algebra. I went through this
| issue when I started electrical engineering as part of an overly
| elaborate midlife crisis. I got the book in the bookstore that
| was supposed to bring you up to high school level and just did
| all of the exercises. Then I did a whole lot of exercises for the
| introductory university math courses.
|
| Basically it is something you have to grind at. Once you do
| enough problems all algebra will seem easy and you are done.
|
| I am not sure that you actually _need_ math for programming. Code
| is its own algebra.
| Tutanota1 wrote:
| This is so true. I sucked at arithmetic but was quite good at
| Higher secondary maths.
| amelius wrote:
| > I went through this issue when I started electrical
| engineering as part of an overly elaborate midlife crisis.
|
| Did it ease your midlife crisis? Would you do it again?
| sho_hn wrote:
| As someone with a "Watch Later" queue recently filled up with
| oscilloscope and variable power bench PSU reviews - I'd also
| love to know.
| randmeerkat wrote:
| I'm curious, was getting the EE degree as part of a midlife
| crisis worth it? Or would you have done things differently
| looking back?
| bigbillheck wrote:
| I used to teach college calculus, and can confirm: if you're
| not comfortable with algebra, you're going to have a bad time
| of it.
| ransom1538 wrote:
| IF!! You have kids learning math is _right there_ with a full
| program. I forgot all my math so I keep up with my kids and
| help at least 15 minutes per day. I have been tripped up with
| 5th grade word problems - it is fun. Follow along with your
| kids math - it will move quicker than you think.
|
| I remember my last math class: Number theory. I got through the
| final, shook the professors hand and told him, "this is the end
| of intellectual ability". I have never seen a professor laugh
| that hard. Once you get past abstract math classes thin out
| fast and it just gets weird.
| dls2016 wrote:
| I second the grind. There's not really a shortcut... like
| lifting weights or practicing scales/rudiments.
|
| I returned to grad school at 27 for math and had trouble
| passing qualifying exams. I ended up using spaced-repetition to
| memorize large sections of books and old questions. I had
| previously shunned memorization... but coupled with focused
| practice it is a powerful tool. Memorizing forces you to
| distill the material and patterns to their essence and allows
| you to recognize the patterns in new situations.
| HumanReadable wrote:
| I agree there are no shortcuts, but just like the scales will
| come much more easily to someone naturally gifted at music,
| so will algebraic operations to someone gifted at math.
|
| We can all improve our abilities, but those not gifted has a
| much steeper hill to climb.
| 300bps wrote:
| _There 's not really a shortcut... like lifting weights_
|
| And just like lifting weights, you will lose it when you stop
| doing it.
|
| I remember doing the CFA program which is math and formula
| heavy. I had hundreds of formulas memorized. I couldn't tell
| you the formula to calculate the standard deviation of a
| portfolio of three assets now if my life depended on it.
| solaxun wrote:
| This is my main qualm with testing as a means of validating
| understanding. If you truly understand a topic, you will
| probably pass a test on it. However passing a test most
| definitely does not guarantee understanding.
|
| I remember very, very little from the CFA exams (passed L3
| about 10 yrs ago), because it tested surface level
| knowledge of an incredibly broad array of subjects. Far too
| broad to allow a deep-dive in any one thing. At the time, I
| could fit a bunch of formulas in my head and regurgitate
| them on command, but it was like filling up a leaky bucket.
| I had to load that sucker up and run into the exam center
| before it poured out.
|
| IMO testing well is it's own skill, and has little to do
| with understanding a topic. Understanding is when you've
| forgotten all the stuff you've memorized, but you can work
| your way back to a solution from first principles. I think
| the only way to get to that point is genuine interest and
| sustained study.
| Py-o7 wrote:
| I am a big fan of the working out analogy. In all cases you
| need to have a decent plan then execute-- basically this
| means showing up and putting in the work. If you do it right
| it should be a bit painful at times.
|
| And just like working out some people will progress more
| rapidly than you and have an easier time at it. So what? As
| long as your goal isn't to 'place' at the elite level, this
| is irrelevant. You really just need to show up and embrace
| the struggle. (For elite level tasks including grad school,
| its a bit more involved.)
| commandlinefan wrote:
| > I did a whole lot of exercises
|
| Same happened to me. I completed a BS in CS and learned just
| enough algebra, trig & calculus to pass the classes and then
| forgot almost everything. Many years after graduation, I came
| across an interesting post in comp.lang.java - somebody was
| asking how to create a java applet that could draw a 3D block
| arrow that could orient itself in any direction. I'm pretty
| sure somebody was asking for help with his homework, but I
| thought it was an interesting problem so I started looking at
| it and the first issue I ran into was drawing a line
| perpendicular to another. I vaguely remembered that there was a
| formula for computing the slope of a line perpendicular to
| another and when I looked it up - in context to a problem I was
| trying to solve - it just kind of "clicked" and I found it
| interesting. From there I fell down the rabbit hole of re-
| learning everything up to differential equations that I was
| _supposed_ to be learning when I was an undergraduate.
| jordanpg wrote:
| Agree with this and would also add: in addition to being able
| to do algebra fast and have it be second-nature, at higher
| levels it helps if you are able to do it 100% accurately. 5
| nines isn't enough.
|
| I didn't appreciate this until I was a graduate physics student
| and was frequently frustrated by minor algebra errors in long,
| complex problems.
| sdenton4 wrote:
| What's important isn't being able to do it right quickly the
| first time. It's having a strong habit of checking your work
| as you go, both at a low level and a high level. (Kinda like
| unit testing and integration testing...)
|
| Fast arithmetic and algebra are fairly particular skills. As
| one progresses into new world of math, the calculations and
| operations may be completely different from what's been
| practiced, making the fast math less useful. But a good habit
| for careful progress is completely general!
| bigbillheck wrote:
| I've known a lot of mathematicians, some of whom were really
| good, and five nines (one-in-one hundred thousand) is an
| implausibly high standard for humans to reach. I'm certainly
| nowhere near there.
|
| What's worked for me is to develop an intuition about how
| things 'should' look, so that if I do make a mistake I'll get
| a 'hey, that's not right' feeling a couple steps later.
| qsmi wrote:
| Just to expand on this, when I worked as a grader in
| college I noted on engineering test keys the professor's
| answers were always roughly three short lines of
| handwritten equation. But the student's answer section were
| often totally darken with pencil they wrote so much. The
| heuristic was definitely the longer the answer the more
| wrong it was.
|
| This is something I've taken to heart professionally too.
| If the equations are getting out of hand it's either wrong
| or I need some simplifying assumptions (like that cow needs
| to be spherical). Otherwise one just can't keep track of it
| and reason about it.
|
| If I need a more precise answer, that's what computer
| numerics is for.
| melenaboija wrote:
| 3Blue1Brown series on linear algebra is an amazing resource as
| a starting point.
|
| https://youtube.com/playlist?list=PL0-GT3co4r2y2YErbmuJw2L5t...
| inetknght wrote:
| I was going to add my own post recommending 3Blue1Brown. I'm
| glad you've already recommended it :)
| animal_spirits wrote:
| Linear algebra is _much_ more complicated than normal algebra
| threatofrain wrote:
| 3B1B has stated that the series is mostly useful for
| summarizing or augmenting existing knowledge _after_ taking
| the class; it 's not for _learning_ Linear Algebra fresh. If
| you sum all the minutes in the 3B1B videos it 's basically
| the length of one video from a university course.
| melenaboija wrote:
| Agree, I said starting point assuming that he has some
| basic knowledge as he is struggling so he must know
| something.
|
| For more advanced linear algebra I would recommend the
| class from Prof. Gilbert at MIT.
|
| https://youtube.com/playlist?list=PLE7DDD91010BC51F8
|
| https://ocw.mit.edu/courses/mathematics/18-06-linear-
| algebra...
|
| What is your recommendation?
| threatofrain wrote:
| I'm not sure about recommending video courses because...
| there are so few people who have taken up the burden of
| producing a full Linear Algebra video course. It's sad.
| But I would at least say that whether you're following
| Gilbert Strang or someone else, you should be following
| with a book as well.
|
| There's another guy, Jim Hefferon of St Michael's
| College, who has written a full book for his own class
| and produced a full video course on Linear Algebra (all
| free).
|
| https://hefferon.net/linearalgebra/
|
| https://www.youtube.com/watch?v=JnTa9XtvmfI
|
| https://www.youtube.com/watch?v=DJ6YwBN7Ya8
| melenaboija wrote:
| I totally agree with you and that is why I have also
| added to the post the link to the open course MIT site.
|
| I also believe that is quite naive thinking that you can
| learn math in some depth just watching videos (at least
| with my capacity). For me it is a way to keep some pace
| and order when learning and, as you say, to find some
| bibliography and material that is being used by the
| professor.
| jeffreyrogers wrote:
| It's good for concepts but you can watch all his videos,
| think you know what linear algebra is about, and still not
| know how to calculate anything. You have to do exercises if
| you want to actually learn math.
| phatfish wrote:
| I think the sort of visualizations that are around on Youtube
| are great for those who don't "get" maths easily and need
| another angle to help them understand. Although 3Blue1Brown
| is not for beginners there are other beginner resources.
|
| For example, this goes right back to basic algebra but is a
| very well explained with the afore mentioned visualizations.
|
| Veritasium - How Imaginary Numbers Were Invented
| https://www.youtube.com/watch?v=cUzklzVXJwo
| RegBarclay wrote:
| I hadn't thought about it that way. I was good at algebra, but
| can't get through calculus to save my life. I've mostly found
| that coding is more about logic than math. When math has come
| up in my work, it's generally a formula or calculation that's
| already defined and I just have to code it into a system, which
| has never been a problem for me.
| ThinkBeat wrote:
| Algebra is a wide topic.
|
| My second year of university is where i fell off math.
|
| Several people have told me that once they reached post
| graduate level at university that math made sense to them. They
| saw the perspective of unified math. How it all fits together.
|
| Sadly, all I did know I have forgotten.
| redman25 wrote:
| I've enjoyed revisiting topics I learned in school through the
| Kahn Academy[1]. You can pick any topic you want to start with
| and follow it through as far as you want.
|
| [1] https://www.khanacademy.org
| web007 wrote:
| Seconding this.
|
| I went through precalc in school, then promptly forgot most
| of it once I started working. All of the bits I had learned
| felt rushed, so a lot of them fell out of my brain even while
| in school. I only had a basic understanding of most of what I
| "learned".
|
| About a year ago I decided to go back to basics, and start
| over from zero on Khan Academy. I've been v.e.r.y slow, maybe
| a lesson a month on average, but have been grinding through
| the lowest level math to really understand and remember it.
| Down to addition, subtraction and multiplication even - I've
| relearned how to multiply small numbers in my head, and
| gotten much better at quick mental addition and subtraction.
|
| I have scheduled and dedicated more time for it this year, so
| I hope to catch up to my original math basis. This time,
| hopefully, remembering all of it instead of skimming. That
| should set me up well for linear algebra, and then calculus
| next year and beyond.
|
| One thing that may help regardless of how you choose to
| learn: pick a problem you want to tackle that needs Fancy
| Maths. It doesn't need to be anything useful, or anything
| difficult like "prove this unproven theorem". It will
| motivate you as long as it keeps your brain engaged. For me
| it's factoring and primality proving, where I can't even read
| the equations / algorithms because I don't have the basic
| tools. As I get further along I keep looking back to those
| problems and seeing how the bits I'm learning apply around
| the edges or to simpler factoring algorithms. It keeps me
| motivated to learn more to be able to dive deeper into the
| juicy parts.
| munchbunny wrote:
| 100% agree this is something you grind at. Calculus, even
| multivariate, and even the stuff beyond that, still grind. I
| used to be really good at it, definitely not the best but among
| the higher scores in my college math classes. But 10+ years
| later, when I go and look at stuff that would have barely fazed
| me then, I go cross eyed now. I remember the concepts, vaguely,
| but following the math and linking together the various
| theorems and transformations feels like wading through syrup.
| It's a problem for me as I'm trying to dig into data science
| stuff, and it feels like getting back into marathon shape after
| 10 years of not running.
|
| There's absolutely a component of learning the concepts, but a
| huge part of being good at the math is just building up the
| mental muscle memory for symbolic manipulation. Sure there's
| innately talented people out there, but the consistent
| application of incremental effort will get you very far
| regardless of talent.
| badactor wrote:
| The problem for me is identifying where exactly in my
| mathematical fundamentals things fall apart (like with prime
| numbers, some basic arithmetic, geometry) that makes higher-
| level math a struggle conceptually. It's always felt like there
| were holes in my fundamentals. I suspect the only way to figure
| this out is to go back and do everything again either through
| Khan Academy or books. Does anyone else have this issue?
|
| When it comes to math, I try to avoid self-diagnosing with
| dyscalculia or with poor working memory. Instead, I think I've
| gone so long with math being an anxiety-inducing subject that
| any time I need to solve math under the tiniest amounts of
| pressure, I fall apart.
|
| Mind naming what book you used?
| graycat wrote:
| Part I
|
| How to get good with math?
|
| Okay. Been there. Done that. Learned a lot of it. Got a Ph.D.
| in it. Taught it. Applied it. Published peer-reviewed
| original research in it. Had a good career applying it. Am
| using it as an advantage in the core of my startup.
|
| Broadly, for a career in computing, at times math can be an
| advantage, one that might be significant, e.g., get you
| founder's stock in a startup that becomes successful.
|
| Math and computing can be a career _one-two_ punch: With some
| math you might find an application, maybe a valuable one, and
| then with some computing you get to do the associated
| programming. Maybe then you can show up at work one morning,
| maybe after doing an _all-nighter_ , and show the final,
| useful, maybe quite valuable results -- done deal, no
| waiting, meetings, project approvals, etc.
|
| This is a great time for both math and computing, no doubt
| unique in all of history. We are awash in what is in
| historical terms just astounding computing, and part of that
| is that a lot of math is just a few clicks away at Wikipedia,
| YouTube, in PDF files from word processing with TeX, etc.
|
| The first thing in math is arithmetic. Of course, current
| computing eats arithmetic problems much faster than Godzilla
| eats fish.
|
| You should know basic arithmetic for whole numbers and
| fractions.
|
| Then you should know the basics of ratios, proportions,
| percentages, square roots and exponents, logarithms, compound
| interest, areas, and volumes. E.g., on my instance of Windows
| 10 Home Edition (that I have as a result of a sad situation,
| long story), the key in the upper right corner of the
| keyboard runs ( _opens, launches_ -- maybe computing will
| think of more silly synonyms) a version of an old scientific-
| engineering pocket calculator that has a lot of such
| arithmetic and math.
|
| Uh, that software is harder to learn to use than the math it
| does! If you can find out how to use such software in less
| than a few hours of clicking guesses, you can also learn the
| associated math!
|
| Then on to _algebra_ : That subject is just doing arithmetic
| with symbols instead of specific values, and that should be
| really easy for anyone who can write math expressions in a
| computer language.
|
| Then on to plane geometry: The most important idea there is
| triangles, especially ones with one angle 90 degrees -- right
| triangles. Then, sure, the biggie result is the Pythagorean
| theorem -- it gets applied throughout our economy and has
| surprisingly far reaching generalizations. For a proof, take
| 4 of the right triangles and arrange them so that they form a
| square where each side of the square is the longest side of
| one of the triangles and all the triangles are inside the
| square. Then will also see a square in the middle. Then write
| out the area of the squares and, presto, bingo, get the
| theorem. There are also 149 or so other proofs.
|
| For a while, I taught trigonometry (about triangles) at
| Indiana University. The best student in the class was a
| pretty girl, and later I dated and married her -- see, math
| can be useful!
|
| Then there is second year algebra where learn some more,
| e.g., about, say,
|
| (x + y)^n
|
| for numbers x and y and a positive integer n. From that can
| learn a lot about how many HEADS might get if flip a fair
| coin 1000 times and can understand the math shown in the
| baseball movie _Moneyball_. Also that way can start to
| understand the _bell curve_ of Gauss and the powerful _law of
| large numbers_.
|
| Might study solid geometry, that is, planes, lines
| perpendicular to planes, spheres, circles on spheres, etc.
|
| Next up, calculus: As you already know, in a car the
| speedometer is the rate of change of the odometer. The rate
| of change of the speedometer is _acceleration_. From Newton
| 's law of motion F = ma, that is, force is mass times
| acceleration, in a car you feel the force as you are pressed
| back in your seat when your Tesla does 0 to 60 MPH in less
| than 4 seconds! Going around in a circle is also
| acceleration, and that's why when you make a fast left turn
| the sack of groceries slides to the right. So, rate of change
| -- that is the first half of calculus.
|
| Given all the speedometer readings, should be able to
| reconstruct the odometer readings, and you can: That is the
| second half of calculus and also is the way both to define
| and to find the lengths of curved lines (e.g., that the Webb
| telescope is following), areas and volumes of spheres,
| cylinders, etc.
|
| How to learn calculus? Long story short, I was not permitted
| to take calculus yet so got a good calculus book and dug in.
| Went to a better school and started on their second year
| calculus and did fine. So, I never took first year calculus
| -- learned it, taught it, applied it, published research in
| it, learned math _analysis_ (that calculus is part of) far
| beyond calculus, but never took a course in it.
|
| How to learn calculus: Get a good book. At each section, (1)
| study the text and examples and (2) work at least half the
| exercises, especially the more difficult ones, and check your
| work with the answers in the back of the book. Don't go for
| _pre-calculus_ , high school calculus, or high school
| _advanced placement_ calculus. Instead, just get a good book
| in CALCULUS. Or get several such books. Then get a quiet
| place, good light, big chair, clipboard with a sharp, soft
| mechanical pencil, big, soft eraser and dig in. Since
| calculus has not changed much in 50+ years, you don 't need a
| recent book. Instead just do an Internet search of used book
| sites.
|
| I learned mostly from
|
| Richard E. Johnson and Fred L. Kiokemeister, _Calculus with
| Analytic Geometry_.
|
| It is VERY well written, even _polished_ , and with an
| unusually good collection of exercises. When I used it, it
| was also used at Harvard. You may be able to get a used copy
| in very good condition for less than $10.
|
| For on-line sources, my opinion is that nearly none of them
| are good. I've seen a lot of the on-line video sources, and I
| never saw a good one. E.g., last time I looked at Khan
| Academy, I concluded that they didn't understand calculus
| very well.
|
| To learn calculus, or nearly anything in math, whether you
| are in a course or not, essentially you still need to study
| as I have outlined. Learning math is not a spectator sport.
|
| If you have taught yourself to be good at C++ and Win32, then
| you should have NO trouble learning calculus QUITE WELL!
|
| Of COURSE you can teach yourself calculus and nearly anything
| in math: To keep up, that is what college professors and
| anyone applying math as a professional do.
| graycat wrote:
| Part II
|
| If you do much with computer graphics you will encounter
| matrix theory. That takes you into _linear algebra_ ; next
| to calculus it is likely the most useful math. Evidence:
| There are a lot of downloads of LINPACK.
|
| Can start a course in linear algebra by considering solving
| several equations in several unknowns. The standard
| technique is _Gauss elimination_ , and can program that in
| about one page of code. Linear algebra is a good start on
| curve fitting in statistics and the math of quantum
| mechanics.
|
| If you want to understand more about cryptography and error
| correcting codes, you should study _abstract algebra_. Here
| I would suggest that you actually take a course (a) to help
| you get through that quite different world of thought and
| (b) especially to learn how to write proofs. And for (b),
| take a course where the prof is really good and also
| carefully reads and comments on your proofs. Abstract
| algebra is the easy place to learn to write proofs.
|
| Can get more guidance on how to learn math at
|
| https://news.ycombinator.com/item?id=28215105
|
| Somehow long, maybe still, knowledge of both math and
| computing can be welcome and lucrative in parts of US
| national security. That was the case early in my career
| when my annual salary was 6+ times the cost of a new high
| end Camaro.
|
| Soon FedEx had what their founder, COB, CEO called their
| "most important problem" -- fleet scheduling. The BoD was
| concerned, and crucial funding was at risk. I typed
| furiously, wrote some software, the output "solved" the
| problem, enabled the funding, and saved FedEx. There, sure,
| needed to calculate great circle distances so used the law
| of cosines for spherical triangles -- solid geometry can be
| good stuff! Also had to handle wind vectors -- linear
| algebra can be powerful stuff. Then I went off to do much
| more, _integer linear programming set covering_ where can
| discover much of the motivation for currently the most
| important problem in computer science, P versus NP.
|
| Later the BoD wanted some revenue projections. I did a
| little with some calculus and got a nice answer. Long story
| short, that work saved FedEx a second time.
|
| For another long story -- I needed to be better at office
| politics -- I just missed out on some FedEx stock that
| should be worth ~$500 million now.
|
| The US Navy was collecting ocean wave data at sea, and I
| was in a software house bidding on writing some software to
| analyze the data. One customer engineer wanted (a) to know
| the _power spectrum_ of the ocean waves (that is, what
| _frequencies_ have the power) and, then, (b) to generate
| _synthetic_ , _random_ ocean waves with that power
| spectrum. I quickly read a book by Blackman and Tukey,
| typed in some software, showed the engineer the results on
| how to find the power spectra (with an important point
| about handling low frequencies) and how to generate the
| synthetic waves, and our company got "sole source" on the
| software work.
|
| Later at IBM's Watson research lab, we were doing AI for
| monitoring of server farms and networks. I thought of
| another way, for some of the monitoring much more powerful
| than the AI, based on some original math, and published the
| results.
|
| Net, some math, especially through calculus and linear
| algebra, can at times be an important career advantage. For
| more, get good with probability theory, if you can, the
| version based on the subject _measure theory_. Then learn
| some about _stochastic processes_. E.g., once the US Navy
| wanted an evaluation of the survivability of the US SSBN
| (missile firing submarines) fleet under a special scenario
| of global nuclear war limited to sea -- in two weeks. From
| some old work by B. Koopman, I saw a continuous time,
| discrete state space Markov process subordinated to a
| Poisson process, wrote some software, and was done on time.
| My work got reviewed by a famous mathematician, and he
| questioned how my software could "fathom the enormous
| state space". I answered, at each time, the number of SSBNs
| surviving is a real valued _random variable_. It is
| positive and not greater than the number of submarines to
| begin with so is bounded and has an expectation and a
| finite variance. Then the law of large numbers applies. So,
| generate 500 independent sample paths, average them, and
| get the expectation "within a gnat's ass nearly all the
| time". He agreed. I passed the review!
|
| If you go for a Ph.D., then understand that, in the US,
| academic positions at the better universities are about
| three things, research, research, and research, especially
| because that leads to grant money. The _operational_
| definition of _research_ is that it got published in a
| peer-reviewed journal. If you publish, say, 3 papers a
| year, then likely people will stay off your case and you
| will likely make progress to tenure. People making the
| promotion and /or funding decisions will rarely look at the
| papers and, instead, just count them. Papers that result in
| prizes are usually quite powerful for a career. Generally,
| though, academics is not very promising for providing a
| good standard of living and good financial security for you
| and your family and these days can't hope to compete with
| what is available in computing, the Internet, etc.
|
| Then the math? It can be an advantage. The "advantage" can
| have you push ahead, maybe by a little or a lot, useful
| technology, economic productivity, and civilization. Such
| progress happens, actually fairly regularly, but is rarely
| easy. So, if want to push civilization ahead, (a) don't
| expect that the work will be easy but (b) math can be one
| of the most powerful advantages.
|
| Now you know some of what I wish I'd known at the beginning
| of my career. I want a _do-over_ -- where can I apply?
| Kye wrote:
| https://www.amazon.com/Schaums-Outline-Elementary-
| Mathematic...
|
| Schaum's Outlines are like that "Learn X in Y Minutes" site
| for school subjects. They give the clearest, most bare bones
| explanation so you can quickly identify gaps. I had the
| college algebra book for the required math class in technical
| school. Between that and YouTube, I was able to pass the
| class.
| openknot wrote:
| For books on strong fundamentals, you can try the Art of
| Problem Solving series [0]. They suggest a curriculum to
| start with prealgebra, move to algebra, then counting &
| probability, then geometry, then precalculus, and finally to
| calculus (though a regular calculus book like Thomas
| Calculus/Stewart Calculus/even Spivak/Apostol would work
| fine).
|
| The main advantage of these books are its focus on building
| intuition by visualizing shapes or immediately rephrasing
| notation (e.g. 4/2 is better understood as 4*(1/2), which
| better explains why you should avoid cases where you divide
| by zero; I also found their exponent rules easier to
| understand, because it encourages visualization instead of
| just memorizing the rules).
|
| The downside is that they're time-consuming due to a large
| number of exercises (I'm currently still trying to slowly
| work through them when I can, but if you need higher-level
| math in the short-term, it's probably better to start there).
| They're also not a free resource.
|
| For free lecture videos, I've found Professor Leonard's
| lectures to be excellent, and equivalent to lectures at a
| university classroom [1].
|
| [0] https://artofproblemsolving.com/store/recommendations
|
| [1] https://www.youtube.com/c/ProfessorLeonard/playlists
| ambrozk wrote:
| "Mathematical fundamentals" is a sliding scale. You have to
| define your end goal. What do you want to understand, that
| you are currently unable to understand due to your incapacity
| with mathematics?
|
| If you want to understand computational complexity theory,
| for instance, you need a different set of "fundamentals" than
| you do if you want to understand high-school physics. It'll
| be a different thing if you want to read econ papers, and a
| different thing if you want to study machine learning.
|
| Within the intersection of all these fundamentals is probably
| basic arithmetic and algebra. Those tools are basic
| requirements for everything else. Beyond that, you need to
| define your goal before you decide what math to learn.
| pthread_t wrote:
| Try Khan Academy. The exercises (especially with how they
| focus on mastery), coupled with the videos will help you root
| out your weak points and even master them. Do not be ashamed
| in starting with the kindergarten math topics and moving
| upwards from there. If you are able to power through it, good
| on you. If you get stuck on something, even better -- now is
| your chance to master it.
|
| KA helped me go from hating math in high school to double
| majoring in Math & CS in college, and graduating with honors.
| I donate to KA now.
| corysama wrote:
| Anyone who sucks at algebra and has a low grind tolerance
| should try https://dragonbox.com/products/algebra-12 It's very
| well made. And, it's designed for 12 year olds. So, adults can
| usually manage to keep up.
| znpy wrote:
| I'm no math genius however I just want to add this tiny bits:
|
| - you don't really "learn" math, you just get used to it
|
| - math is really mental gymnastics, and "learning math"
| actually is doing a lot of math exercises
| maxgiraldo wrote:
| If anyone is looking to brush up on their Algebra fundamentals,
| I built an app for that and it's free to use:
| https://apps.apple.com/us/app/pensend/id1571322730. I had a
| similar problem where I didn't understand where my fundamental
| knowledge gaps were so I ended up starting from the beginning
| (pre-algebra). I hope this is helpful for people!
|
| (sorry for the shameless plug)
| thih9 wrote:
| At what moments do you feel that your math knowledge limits your
| programming skills?
|
| Also: what kind of programming are you doing? E.g. working with
| 3d games relies on different math skills than dealing with AB
| testing.
| jesperlang wrote:
| I'm currently going through "Math for programmers" (Manning
| Publ.) and I enjoy it immensely! A lot of math books can feel too
| abstract and dry but this book was perfect for me coming mostly
| from a programming background. Start learning learning linear
| algebra and you will soon find yourself in graphics programming
| and game engines. That's where I am at the moment and I feel I
| regained my passion for programming after years in webdev..
| the_hob_code wrote:
| I completely agree about a lot of math books seeming too
| abstract. I hated math until I took an interest in physics and
| realized I needed math to understand the ideas being presented.
| Suddenly it clicked that the math was actually a way to better
| represent the ideas, not just a trick for getting answers.
| Seeing how math applies to create and represent knowledge gave
| me a huge appreciation for it.
| donenas wrote:
| Learn the concepts and start practicing
| nerbert wrote:
| The brilliant app is not bad at this. It takes you through a good
| learning curve.
| arisbe__ wrote:
| First Khan Academy, then if you want to go further:
|
| Bill Shillito | Introduction to Higher Mathematics (YouTube
| lecture course) -
| https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz...
|
| Richard Hammack | Book of Proof (pdf book) -
| https://www.people.vcu.edu/~rhammack/BookOfProof/
|
| Taylor Dupuy | Fundamentals of Mathematics (YouTube lecture
| course) -
| https://www.youtube.com/playlist?list=PLJmfLfPx1OedcIUn5nSCZ...
|
| Silvanus P Thompson | Calculus Made Easy (html book) -
| https://calculusmadeeasy.org/ (This shouldn't be your only
| exposure to Calculus. It is more for building intuition.)
|
| Dana Mosely | Understanding Basic Statistics (YouTube lecture
| course, no calculus) -
| https://www.youtube.com/playlist?list=PL9Wxhr5qVFN0WY2CXB4tR...
|
| Gilbert Strang | Highlights of Calculus (YouTube lecture course)
| - https://www.youtube.com/playlist?list=PLBE9407EA64E2C318
|
| Josh Starmer | StatQuest (Short various statistics videos) -
| https://www.youtube.com/c/joshstarmer/playlists
|
| Bob Franzosa | Introduction to Topology (single public lecture) -
| https://www.youtube.com/watch?v=zsN_guq__Ac
|
| Socratica | Abstract Algebra (short videos) -
| https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...
|
| MIT Calculus Revisited (Single Variable Calculus):
| https://www.youtube.com/playlist?list=PL3B08AE665AB9002A
|
| MIT Calculus Revisited (Multivariable Calculus):
| https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B
|
| MIT Calculus Revisited (Complex Variables, Differential
| Equations, Linear Algebra):
| https://www.youtube.com/playlist?list=PLD971E94905A70448
|
| Matthew Macauley | Visual Group Theory, Differential Equations,
| _Discrete Mathematical Structures_ , Advanced Linear Algebra, and
| Advanced Engineering Mathematics (YouTube lecture courses) -
| https://www.youtube.com/channel/UCH1cV4RtgI_N97M8jepiUzw/pla...
|
| _The Discrete Mathematics course above is probably the most
| important for your work. In fact I would look for more Discrete
| Mathematics courses if I were you as it is far more important
| than anything else here._
|
| Open University (BBC) | Geometric Topology (YouTube lecture
| course) -
| https://www.youtube.com/playlist?list=PLKB3Q5Oyy_RNBrS3V2WbO...
|
| Joel David Hamkins | Philosophy of Mathematics (YouTube lecture
| course) -
| https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIn...
|
| Marco Taboga | Probability and Statistics & Matrix Algebra (html
| book, need calculus) - https://www.statlect.com/
|
| On YouTube you can literally watch a good lecture course for just
| about any typical undergraduate course. You just need to know
| where to look. Also there are even some really good master's
| degree courses out there.
|
| _Of course the only way to really learn the mathematics deeply
| is to "learn by doing", aka problems and proofs._
|
| Other than the usual big American universities another good
| source from India is NPTEL (https://nptel.ac.in/course.html).
|
| For somewhat more entertaining short lectures try:
|
| Grant Sanderson | 3Blue1Brown -
| https://www.youtube.com/c/3blue1brown
|
| Brady Haran | Numberphile -
| https://www.youtube.com/c/numberphile/
|
| Tai-Danae Bradley, Gabe Perez-Giz, and Kelsey Houston-Edwards |
| PBS Infinite Series -
| https://www.youtube.com/c/pbsinfiniteseries/
|
| Raymond Flood (YouTube public lectures at Gresham College) |
| History of Mathematics -
| https://www.youtube.com/playlist?list=PL_jwwOG0kPgPPiX0pcbzL...
|
| There are a ton of channels starting to pop up like Grant's 3B1B
| (I find like a new one every week). He had a contest recently so
| maybe look at some of the winners.
|
| Lastly this is pretty useful if you get into higher mathematics:
|
| Math Vault | The Definitive Glossary of Higher Mathematical
| Jargon - https://mathvault.ca/math-glossary/
| RamblingCTO wrote:
| Saving this for later by commenting it. Many thanks for
| curating this!
| jesperlang wrote:
| You can click on the timestamp of the comment and then
| "favorite" to save it :)
| hiyer wrote:
| This is an incredibly detailed set of recommendations! Thank
| you for this :-)
| irchans wrote:
| Wow, Great, Thanks!
| [deleted]
| secretsatan wrote:
| I wouldn't say I'm great at math, I scraped an A-Level in it over
| twenty years ago, didn't use it much for years, but then I got
| interested in graphics again and there are some severe gaps in my
| knowledge.
|
| I remember at the time, I just did not understand matrices, I now
| use them a hell of a lot and I still suck at them. Linear algebra
| is another one.
|
| Most other stuff day to day stuff I can reasonably understand, or
| at least sit down with pen and paper and work out what I need to
| do but the above 2 frequently get my head stuck in knots, I often
| think about maybe doing a course on these 2
| silicaroach wrote:
| Identify the concept you don't get and read! Specifically read a
| _variety_ of explanations on every concept you struggle with.
| Eventually you will find or form a point of view that will make
| it clear. There's no easy way, no one source.
| gspr wrote:
| And while reading: do! And when doing (solving a problem):
| don't read (the answer)!
| bradleyankrom wrote:
| I was asking a similar question a few years ago and found Ivan
| Savov's "No Bullshit Guide to Math & Physics" to be really
| helpful.
| vmilner wrote:
| I really like the "placement exam" in this, that tells you
| where your weaknesses are and what to focus on. I feel like
| this is the kind of tool I want when someone says "I suck at
| X".
| jackyinger wrote:
| The thing about math is that it just boils down to simple rules
| that have wide ranging consequences and interrelations.
|
| To keep from being overwhelmed, pick a problem and try to focus
| on only the information relevant to that problem.
|
| I sucked at math as a small child because the social message
| teachers give that math is hard did not really make sense in
| light of boring arithmetic so I thought I was missing something.
| Turns out I wasn't.
|
| Go into it driven by curiosity. Focus on building strong
| fundamentals; algebra and trigonometry are very useful. Then look
| into calculus or linear algebra.
|
| Make sure to solve actual problems for practice. But spend plenty
| of time researching. Take notes. Write out all the steps in your
| problem solving so that you can debug any mistakes.
|
| Lastly, have fun! There's a lot of neat math out there, treat
| yourself to some research into whatever is interesting to you
| when you get sick of grinding on the fundamentals.
| ianai wrote:
| Yes. I'd encourage people to not be afraid to check out random
| math topics from wolfram's site. I killed a lot of hours in
| undergrad that way and it's helpful in non-obvious ways.
| ur-whale wrote:
| There are a number of facets to this problem.
|
| My own personal experience has been that learning existing math
| is not very hard and quite fun (YMMV of course and it also
| depends on how much pre-requisite knowledge is required to even
| approach the topic)
|
| Once you've acquired the tools, applying them to solve actual
| engineering problems, also relatively easy (depending on the
| problem of course) and _very_ fun.
|
| However, solving math problems is a completely different game,
| and this is where (again for me), the discipline is the most
| frustrating.
|
| Solving a math problem is like finding a path out of a dense
| forest, and some people seem to have a "natural compass" guiding
| them towards it.
|
| For me (born w/o much of a compass), it's always felt like I have
| to recursively try all possible paths until I find the one that
| gets me there. Needless to say, if the forest is dense and thick
| enough, that's a completely hopeless endeavor.
|
| For example, reading the proof to a theorem, assuming it uses
| tools, concepts and facts you're familiar enough with and does
| not take giant leaps (the infamous "from here it obviously follow
| that ...") is easy and can be fun.
|
| But when I get to the QED, I'm always left wondering how the guy
| who first proved it effing found the path in the first place.
|
| It's borderline disheartening.
| rzarate wrote:
| I recently faced the same realisation. I'm a software engineer
| with several years of experience and somehow felt like I didn't
| know enough about formal computer science/math to tackle the
| problems I found most interesting (usually very abstract,
| foundational stuff).
|
| One day I decided to go to a physical bookstore and buy a bunch
| of books from the Math and Computer Science sections and started
| from there. It was probably not the best way to start but A START
| nonetheless. Given that I really enjoyed reading about these
| topics, I decided to enrol into an online university to pursue a
| degree in Mathematics.
|
| The book I found the most useful was "How to Prove It"[0]. From
| my point of view, it was a great starting point for two reasons:
| * It is approachable (specially for Software Engineers) without
| being boring. You start building intuitions and it really ignites
| your curiosity. * It is also sufficient for understanding proves
| and mathematical notation/language. This is an important building
| block that will allow you to start tackling the branches in Math
| you are interested in.
|
| [0] https://www.goodreads.com/book/show/739735.How_to_Prove_It
| danielvaughn wrote:
| Nice, I'd been looking for a good book that explains the
| context behind the conventions and thought processes. So many
| articles/websites/books claim to start out at the basics, but
| then they toss a bunch of confusing shit at you with barely an
| explanation. It's very demoralizing. I'll pick this up, thanks
| for the recommendation.
| slingnow wrote:
| Have you tried the internet?
|
| Why has there been such a proliferation of these Reddit-style
| obvious questions upvoted? The front page is full of "Ask HN:
| what can I do about <insert obvious problem>?"
| WHA8m wrote:
| Maybe we could link to https://hn.algolia.com more often...
|
| But to be fair, HN is not aiming to be a wiki. It's a vivid
| forum that discusses topics that are relevant to its
| participants. Looking at the amount of comments and upvotes on
| this thread, it's clear that a lot of people are willing to
| help or engage in some way. Also, people often not only discuss
| the initial topic (or question) but something else that arose
| from it. As I said, it's a vivid place. And that's for good.
| brg wrote:
| Two pieces of advice.
|
| First, pick a topic that you believe you can be fully engaged
| with. Here are some examples. But you can look at the threads of
| these from middle school through first year graduate school.
|
| * Geometry (From Euclid to Topology to tensor analysis)
|
| * Linear Algebra (From vectors to Convex Optimization)
|
| * Calculus (Trig to PDEs')
|
| * Algebra (From Groups to Number Theory)
|
| Second, is do the work. With math is easy to trick yourself, in
| the moment, that you know the solution and understand the
| concept. But that mistake acrues, you get to a point where
| everything is opaque and there isn't a starting point without a
| hint. This feeling of self-assurance needs to be challenged. You
| need to do the work, rewrite the proofs, do the exercises
| completely, and explore the concept on your own a bit.
| gaoshan wrote:
| I'm the same. Self taught, brain is not mathematically oriented
| and it feels like a gap in my abilities.
|
| People will often say something like, "You don't really need
| math" for programming but that is missing the point, in my
| opinion. The point is that it feels to me (after many years of
| experience and working with many great programmers) that people
| with a mathematically oriented mind tend to find certain common
| programming realms easier to grasp. It makes them faster and more
| productive as they almost intuitively "get" things that are
| oriented in the same manner as their brain already works.
|
| For others (me) I have much more trouble in these areas and have
| to really pound my head on the problem to even get close to
| understanding it as fluently as they do. They probably do not
| realize this but it's a real struggle (I've had a few occasions
| where the other person seemed genuinely confused that I was not
| really "getting it").
| danielvaughn wrote:
| Same. I definitely don't need math in my day-to-day. But I've
| been working on this side project for years, and haven't made
| much progress because I've hit a ceiling that requires math to
| transcend.
| akomtu wrote:
| If I were you, I'd study only linear algebra. Other subjects,
| including calculus, are of little use to programners. The
| quickest way to learn linear algebra is to pick a high level
| topic, say... spectral decomposition of a matrix, look it up on
| wikipedia, see what theorems this topic is made of, and try to
| repeat the proofs. You'll quickly see that those proofs rely on
| lower level theorems, so you'll need to look them up too, and so
| on, until you descend to the definitions of numbers and sets.
| jbot27 wrote:
| I would say I felt the same way. I am also self taught and just
| have high school level math. For the most part of my career
| though it hasn't really been an issue.
|
| But I want to learn ML so digging in. I feel like it is not as
| bad as I thought it would be, I think the problem with math
| information is assumes a lot of things. There are tons of
| notation that is really dense.
|
| My recommendation would be to pick a project or an area, because
| math is huge. Try to find resources for that. So for ML it is
| linear algebra and Calculus. Try to find a bunch of resources and
| get different ways of explaining it.
|
| I highly recommend:
|
| Math for Programmers by Paul Orland
|
| https://betterexplained.com/
| distalx wrote:
| I'd recommend going through https://www.mathsisfun.com/ website.
| I'm currently using it for brushing up my algebra and finding it
| really well phrased.
| hashtones wrote:
| One interesting fact that you may not know is that the often
| assumed symbolic notation of algebra (and the rest of
| mathematics) is fairly recent: that is, the likes of Euler and
| Fermat were known to write out the entirety of the mathematical
| logic as a "word problem".
|
| Why I bring this up, is because often I've thought that the
| innovators of mathematics probably benefited from this action:
| probably it is what enabled them to solve and derive problems we
| still today have difficulty resolving.
|
| I bring this up to highlight a point about the philosophical
| underpinnings of mathematics--that as necessary as it is to
| understand the syntax and grammar of mathematics today, it is
| just as necessary to wrestle with the ideas in a form more
| palpable to your mind: language.
|
| So what I'm saying, really, is that if you find yourself having
| difficulty with mathematics, as much as it is a matter of "doing
| the work" (solving the problem, crunching the number) as it is
| with any other skill, it is as equally important (and maybe even
| "more" helpful) to approach and take on the logical reasoning as
| a function of what you can put into words... At least, doing so,
| I think and hope it would help you render yourself more capable
| of tackling mathematics.
|
| A good book to start you off in this way, is Bertrand Russel's
| Introduction to the Mathematical Philosophy. If you have to read
| it several times, it's been shown rewatching something as higher
| playerback speed is more effective than just reading it once, so
| don't be afraid to reread sections (or even in math) as many
| times as it takes for the knowledge to become explicit to you.
|
| Oh and Khan Academy is a great resource.
|
| Finally, if you have some money you can definitely find a math
| tutor--if you can find one who you can relate to / who speaks to
| you, it'll make a radical difference too.
|
| Hope this helps!
|
| Afterward: if you want a problem that'll stump any mathematician,
| take a look at the Collatz Conjecture: very simple, but
| understanding it might help you understand how to approach
| problems in mathematics (although this one has still yet to be
| proven, and as Paul Erdos said, mathematics is still not yet
| equiped to prove it, despite how simple it is).
| jyriand wrote:
| I would go with Art of Problem Solving series. Especially volume
| 1.
| hdjjhhvvhga wrote:
| Personal advice: unless you have a very specific reason to do so,
| completely pass on advanced calculus, real/complex analysis etc.,
| and continue with discrete maths instead - you will find it
| immediately useful. Only later, when you find you need it, choose
| other topics, carefully selected. We only have so much time in
| life, and the range of fields to study is enormous, so choose
| wisely. Don't blindly follow the advice of people who will tell
| you to study everything, you will soon realize it's impossible
| and it will only leave you sad.
| sjg007 wrote:
| What do you want to learn? And what area do you want to work in?
|
| From a CS view math is:
|
| Discrete math and combinatorics? Really useful for proving
| algorithm properties etc.. graph theory is useful as well.
|
| Statistics? Extremely useful. Linear algebra? Extremely useful as
| applied to statistics and deep learning.
|
| Theorem proving? Useful for determining program correctness,
| important in some industries.
|
| Of these, I think stats and linear algebra are the most
| fundamental. You can use these to build models of things and
| estimate parameters and create predictions.
|
| I think the critical piece is to learn how to apply these
| tools/concepts correctly to solve problems, determine when they
| are valid / what the limits are / and how to intellectually debug
| them.
|
| Otherwise learning about algorithmic complexity and how to solve
| CS type problems with algorithms is more likely to help your
| career.
| vasili111 wrote:
| I would suggest math books from "for Dummies" series if it is
| hard for you to understand other math books.
| jostylr wrote:
| I recommend making sure you are good with Guesstimation to start
| (The book of that name by Weinstein and Adams). Be sure to create
| your own questions and attempt to answer them. Watch some TED
| talks and try to use mathematical skepticism to criticize them.
| Doing all this should make you comfortable using mathematics as a
| tool. Once you are in that frame of mind, you can explore and
| have fun with the other topics as ably listed in the other
| comments on this page. Think of math as a faithful toolkit to
| explore the world. Once you start to try to describe the world in
| mathematical language, the more proper tools of mathematics will
| make much more sense.
|
| Also, keep in mind that pretending things are lines is really
| useful. See the secant method.
| user_235711 wrote:
| Any edition of "Mathematical Ideas"[0] might be a good place to
| start. While it may seem too basic for some, I feel that it
| covers a lot of concepts and material that are very useful when
| programming - problem solving, set theory, logic, number theory,
| basic algebra, etcetera - and does so in a way that is gradually
| cumulative and not so daunting.
|
| Maybe this book will help you to realize you don't really suck at
| math, you just had some terrible teachers or whatever. It's also
| a great introduction to many different mathematical subfields so
| you can see which ones are most interesting/useful to you for
| future study.
|
| [0] https://www.amazon.com/Mathematical-Ideas-14th-Charles-
| Mille...
| bjourne wrote:
| Khan Academy
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