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BetterExplained: Math Lessons That Explain Concepts

666 points by rfreytag 11y ago

houzi 11y ago
Is it a data mining site? I'm just entering captcha's for days..
andrewclunn 11y ago
Aw, and here I was hoping for clips from SquareOne TV...
hugs 11y ago
I needed a refresher on trig last year for some robotics simulations I was working on. The trig lesson was extremely helpful: http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/ And fwiw, I finally learned an intuitive understanding of radians by reading the Tau Manifesto (not on BetterExplained). It would be awesome if BetterExplained used tau instead of pi in the lesson on radians, but that's a minor nitpick for a very helpful set of lessons.
- jeffwass 11y ago
  Out of curiosity do you get confused when people talk about 360 degrees, or 90 degree angles, or turning around 180 degrees, etc? Most people I know intuitively understand degrees and could easily understand radians the same. Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto. Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter. (I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).
  - jerf 11y ago
    The anecdata that tau is helpful to learners is, IMHO, pretty strong at this point. Facts beat theory. And since anecdata is all we have, well, it's all we can use to talk about. I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.
    - jeffwass 11y ago
      When someone says 90 degree angle, does that confuse you?
      - jerf 11y ago
        As it happens, I've tutored people in the middle of high school geometry. Radians confuse them. Full stop. If there is something that can make it easier to understand, while at the same time sacrificing absolutely nothing of mathematical consequence, it's a good thing. What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.
        jeffwass 11y ago
        Ok, we're far away from my original point which was only on using tau in one website. My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar. If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers? Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.
- ebbv 11y ago
  No they should not use Tau, because the point of education is to give you knowledge that you can build on later. Nothing else uses Tau, so teaching you Tau does not prepare you for further education. I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.
  - TuringTest 11y ago
    > Nothing else uses Tau, so teaching you Tau does not prepare you for further education. Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it. Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.
- kalid 11y ago
  Hey, thanks for the feedback! Glad to hear the trig article helped (one of my favorite aha moments -- all the trig functions, including stuff like the exotic cosecant, can be drawn together.) I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".
- jgamman 11y ago
  i (mostly) agree. i think thinking about a fraction of a full circle is the easiest way to work with trig. having said that, the mental overload on your average high school student flipping between pi and tau is too much. my personal compromise is to never cancel out the '2' and always work with 2pi. it makes some equations look a bit odd from the textbook but whacks me in the forehead to remember to think about circles...
chairleader 11y ago
Interesting set of articles. "Intuition isn't Optional"[1] could explain why pursuing simple solutions and good naming is important in software. http://betterexplained.com/articles/intuition-isnt-optional/ http://betterexplained.com/articles/intuition-isnt-optional/
- kalid 11y ago
  Nice point. Yes, naming can prime people to understand or be confused by a topic. Renaming the "imaginary numbers" to the "rotated numbers" would make them orders of magnitude easier to learn. "What's so strange about the second dimension?" vs. "How do I understand the square root of -1?".
wiremine 11y ago
I wonder why more secondary teachers don't use these sorts of examples? Any teachers out there that can give insight?
- jerf 11y ago
  The central bureaucracy has so overdetermined education that there is no longer any room for deviation. If it's not in the curriculum, there's no time for it, because there isn't hardly even any time to cover what is in the overdetermined curriculum. If you support innovation in education anymore, your first goal now is to get the central bureaucracy out of it. Until that happens, nothing else can change.
  - criley2 11y ago
    Removing the central bureaucracy is a double edged sword. For every classroom you free to innovate, you also free a teacher to pan science, attack evolution, teach bible and religion in science class, etc. Many of us grew up in school districts prior to the Bush and Obama centralization regimes and quite literally were taught that Evolution is a Satanic lie by public (non-private non-religious) schools.
    - jerf 11y ago
      Innovation is a double-edged sword in general. Doesn't mean we should stop. Besides, are we not discussing exactly a situation in which the central bureaucracy has overspecified an inferior education? Nothing about "panning science, attacking evolution, teaching the bible and religion" has anything to do with whether or not there is a central organization; one stray election and you'll get all those things coming out of the central organization too. The very topic of this discussion is that an inferior education technique is being mandated right now. The casual default presumption that the centrally-mandated education is perfect is already falsified by the evidence.
      - WaxProlix 11y ago
        So are you saying we could fix the mandate and that would be good, or that specifying a minimum basic to which all teaching should be done is wrong in general? Or some third thing?
chris_wot 11y ago
I love this site. This explained logarithms and exponential growth really well.
- kalid 11y ago
  Glad to hear it helped, thanks.
  - chris_wot 11y ago
    Oh! Dude, you are amazing! Seriously, I don't normally say that to people - I can't thank you enough because you honestly got me over a few mathematics hurdles. If you ever have a paid subscription, I'm buying :-)
    - kalid 11y ago
      Thanks Chris, really appreciate it. Hearing when the site clicks with someone is invigorating. No real paid subscriptions yet but it's a great idea =).
tempodox 11y ago
Error 522
alblue 11y ago
Seems like the site has gone down with traffic and CloudFlare is serving up a generic 522 error page.
- pixelkicker 11y ago
  Yep, bummer I was looking forward to checking this out.
  - hugs 11y ago
    Check back later when traffic has died down. The site really is worth a read.
    - kalid 11y ago
      Thanks! Working on getting a fully static version of the site up. Argh, Wordpress with even caching plugins, still folds. (It's on reddit and getting 100x normal traffic). Eventually a postmortem will be in order. Static links to some favorites: • Imaginary numbers: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://webcache.googleusercontent.com/search?q=cache:http://... • Understanding e: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ http://webcache.googleusercontent.com/search?q=cache:http://... • Intuitive Trig: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-trigonometry/ http://webcache.googleusercontent.com/search?q=cache:http://... • Calculus intro: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/calculus/lesson-1 http://webcache.googleusercontent.com/search?q=cache:http://... • Sine waves: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://webcache.googleusercontent.com/search?q=cache:http://... • Euler's Formula: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://webcache.googleusercontent.com/search?q=cache:http://... • Linear Algebra: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/linear-algebra-guide/ http://webcache.googleusercontent.com/search?q=cache:http://...
rfreytag 11y ago
Internet Archive has a copy: https://web.archive.org/web/20151221053341/http://betterexplained.com/ https://web.archive.org/web/20151221053341/http://betterexpl... >>>>> Please donate to the Archive while you are at it! :-) Google cache: http://webcache.googleusercontent.com/search?q=cache:http://www.betterexplained.com/ http://webcache.googleusercontent.com/search?q=cache:http://...
typon 11y ago
This site is a hidden gem on the internet. This guy deserves a lot more credit. I think he's doing something at the level of Khan Academy, but for seemingly simple concepts we all take for-granted. An article I really enjoyed was his explanation of Quake's inverse square root method: https://web.archive.org/web/20150530232103/http://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/ https://web.archive.org/web/20150530232103/http://betterexpl... What I also appreciate is that he admits when he doesn't understand something and doesn't pretend to give an incomplete, vague explanation and just straight up says he does't fully get it (like why the specific 'magic' number in the Quake algorithm). Thankfully he linked the original paper and I read up on it myself. But his initial explanation provided a solid background for the paper.
- morley 11y ago
  This is fascinating! I wanted to know more about the history of that function at id, which lead me to this beyond3d post (which was the topic of the reddit post that BetterExplained listed as a source): https://web.archive.org/web/20070208003255/http://www.beyond3d.com/articles/fastinvsqrt/ https://web.archive.org/web/20070208003255/http://www.beyond...
- kalid 11y ago
  Hey, Kalid from BetterExplained here, thanks! My general philosophy is to be really, really honest with myself if I understood something. It's ok (really!) to admit when we haven't fully understood a concept. Want a fun example? How about percentages. Yeah, that thing we mastered in 4th grade or whatever. That thing we use every day. Well, did you know that a% of b = b% of a? Let's say you want 16% of 25. Ugh. Ok, let's multiply it out... divide by 100... no! How about 25% of 16? Well, that's 4. Easy. But are they the same thing? a% of b = a/100 * b b% of a = b/100 * a Either way, it's ab/100. Now every percentage problem has a 50% chance of being expressed more easily. How did we miss this? (Argh!!!) Math is full of insights like that. Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse). Sine and cosine are unitless numbers, and that's why the can be each other's derivatives (the percentage change of a percentage change...). So many things click! What else have we overlooked? Anyway, really appreciate the note!
robbiemitchell 11y ago
Tangent: with author permission, these lessons are loaded into the recommendation system for Knewton's free learning service: https://www.knewton.com/ https://www.knewton.com/
- shobhitverma 11y ago
  Knewton has done a good job of curating!
- kalid 11y ago
  Awesome! (Kalid from BetterExplained here.) This goes for everyone, all the articles are under a CC license.
kudu 11y ago
Is there anything similar to this for physics? I'm taking an introductory mechanics class and I can't help but think there must be a much more intuitive, logical way to solve even complicated multi-step problems than just falling into algorithmic pattern-based resolution.
- kalid 11y ago
  I haven't done much physics, but Feynman is my teaching inspiration. He has his famous lectures on Physics: http://www.feynmanlectures.caltech.edu http://www.feynmanlectures.caltech.edu (Put online by Michael Hartl, of TauDay fame among others.) I haven't gone through them extensively but Feynman was a master of making the complex simple.
TuringTest 11y ago
I just love this guy's ADEPT method of exposition (Analogy, Diagram, Example, Plain English explanation, and just then Technical definition). [1] That is exactly how I like to be introduced to any new concept, in special when I'm a complete newcomer to the field and can't relate it to previous ideas in it. Judging by the reactions to the BetterExplained site, other people agree with that. [1] http://betterexplained.com/articles/adept-method/ http://betterexplained.com/articles/adept-method/
- kalid 11y ago
  Thanks! I was really hesitant to force an acronym (it actually started as ADE) but then I realized I could work my way up to the full technical definition. Really glad to hear it's resonating. (Sometimes things happen out of order, i.e. you start with a plain English definition. But the idea is to have all 5 parts if you want to truly master a concept.)
- igravious 11y ago
  I've found that I also need to understand the historical context/impetus... I need to be shown and eventually understand why something was developed when it was developed. I need chronology of thought and ideas. For instance, learning about Gödel's theorems without having the Hilbert back-story explained. Learning about Leibniz's and Newton's calculus without learning about infinitesimals. And so on. There's generally a reason _why_ concepts are born _when_ they are born. If you think about your maths classes, sometimes you're instructed to learn a method because it is useful and because it has real-world applications but it I don't think anybody is ever first taught algebraic geometry properly, if I may use that word. I don't think kids are taught the geometry is one thing and algebra is another and that different spaces can have different metrics. Am I making sense here? Do people see what I'm trying to get at? Am I arguing for HADEPT? :) (Historical context, Analogy, Diagram, Example, Plain English explanation, and just then Technical definition)
  - kalid 11y ago
    Great point. I find myself looking at the history of the idea when writing up a post. Did you realize negative numbers were only accepted in the late 1700s? That the Fourier Transform was originally rejected as untrue when first presented, by world-famous mathematicians even? (Yet we require students to internalize it without issue in a single lecture.) Historical context is huge. I think I'm now stuck with this ADEPT name but maybe it fits into the Plain English portion :).
waprin 11y ago
This site is absolutely excellent. There are a lot of things in math that I sort of had a handle of the mechanics of, but less so the intuitions for, which this site fills in well: I really liked the explanation of sine as something that makes things 'circly'. http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-... Does the site take external contributions?
- kalid 11y ago
  I've been considering a patreon page, but the best way is to help share the site with people learning math (students, family friends, etc.). A genuine recommendation is one of the best things I can ask for. Thank you!
- kalid 11y ago
  By popular demand: http://patreon.com/betterexplained http://patreon.com/betterexplained :)
  - waprin 11y ago
    Sorry, I was asking if you accept article contributions.
    - anjanb 11y ago
      as a reader, I think that that should be nice. Also, it would be nice to have a forum where ideas get exchanged. Then Kaled can decide if any of the discussions should make it into the site as an article ?
      - kalid 11y ago
        Thanks for the feedback! I have a community in progress at http://aha.betterexplained.com http://aha.betterexplained.com -- I have some collaboration ideas I'll be announcing early next year :).
    - kalid 11y ago
darkerside 11y ago
From: http://betterexplained.com/static/articles/rethinking-arithmetic-a-visual-guide/ http://betterexplained.com/static/articles/rethinking-arithm... 1 * x^2 = 9 "What transformation (“times x”), when applied twice, will turn 1 into 9?" That's such a great way of thinking about algebra. The "understood" 1 is equivalent to, e.g., an understood "YOU" in the English language.
- kalid 11y ago
  Beautiful way to put it! We aren't always explicit about the subject of the transformation. Similarly, when working with exponents, I think: e^ix is really 1 * e^ix That is, we're starting with 1.0 and doing a transformation (rotation) on it.
nickpsecurity 11y ago
The article on math intuition plus the e article were incredible. I always knew how to work with them and sort of what they meant but never really intuitively. The articles were first time in a while that happened. Reminds me when I first learned the often-hated word problems were the best part of math that explained how to actually map it to real world. Equations took less thought for me so I avoided lots of word problem practice. Was glad I shifted back a bit before calculus or I'd never be able to explain what it was good for. This guy's site should get more attention and probably an award. I'll be experimenting with less-math-inclined people to see how effective it is.
- abc_lisper 11y ago
  Yeah, the e article is awesome. Cleared up a lot of cobwebs in my brain..
- kalid 11y ago
  Hey nick! Thanks for kind words, it's really touching to know when things are clicking. I'd love to see how things go with people that aren't math experts. My philosophy is I found one math up the mountain, it worked for me, but there's other ways too. It's awesome when people take the diagrams/analogies and adapt them for their own style. I saw your other comment too, will reply up there.
elliotec 11y ago
I bought the book a while back, and I still struggled with it. I have very low math skills. My pre algebra teacher was a joke, and everything building on that flew right over my head. I basically talked my counselor into letting me graduate even though I didn't pass basic math graduation requirements in college. So I'm a pretty clean slate, and this was quite difficult for me. I'm a pretty good developer, but telling the computer to do math for you based on looking up what it needs to do is much different from understanding the concepts myself which has been a real difficult thing for me. Kahn academy is the worst, like bringing back old terrible memories of repeating hellish math problems that I don't really get, but still want to move on out of boredom. Such it is.
- j2kun 11y ago
  I would love to hear your thoughts about something I'm working on. Care to shoot me an email? jkun2@uic.edu
- kalid 11y ago
  Thanks for the feedback! The book is more of a "top concepts I wish I knew" but not really a ground-up tutorial. For starting from a clean slate, you might like: http://betterexplained.com/static/articles/rethinking-arithmetic-a-visual-guide/ http://betterexplained.com/static/articles/rethinking-arithm... Once you can visualize the basic operations (add, subtract, multiply, divide), every new math operation becomes a lot easier (complex numbers = rotations, exponents = growth, combining them = orbiting a circle...).
  - lstamour 11y ago
    When you say the opposite can be the multiplication of -2, did you actually mean the multiplication of -1? It's a bit unclear in the example whether the "loss of two" means 1 * -1 = -1 and therefore a relative loss of two, or 1 * -2 = -2 or a relative loss of 3. I always figured opposite meant inverting either the fraction so 2/1 becomes 1/2 or it meant multiplying by -1, effectively toggling the negative sign.
    - 11y ago
akerro 11y ago
>Secondly, if he is not involved in criminal activity, why does Mr Claus use unbreakable encryption? How do they know he encrypts everything?
neilsharma 11y ago
I've been using BetterExplained to review concepts I thought I mastered (based on great test scores and grades) but years later realized I had zero intuition on. It has been, by far, the most useful single source of math content I've found on the internet. The world needs more people like you. Two Questions: 1) What are your thoughts on interactive content like ExplainedVisually? I've been thinking about doing something similar for data structures / algorithm topics. How much of learning math concepts is exploratory vs learn-by-doing? 2) Are you running BetterExplained as a side business, or full-time? And if you're willing to share techniques and numbers for the entrepreneurial HN community, what are some things you've done to market it, monetize, etc and what were the results? Thanks!
- kalid 11y ago
  Really glad to hear the site helps with new insights (especially for someone who has aced the tests/grades). I was in a similar boat, qualified on paper, but not in my heart of hearts. 1) In general I like any efforts to explain things in new ways, and ExplainedVisually is great. My philosophy is that teaching is like humor. You want to make something funny, present it well, but not overexplain it. Let's people enjoy the joke. If you do too much handholding, you ruin the surprise and it's not fun to be told "Ok, the punchline is coming up...". It's not an exact science, but you get a nose for when something is truly illuminating vs. trying to chew your food for you. 2) BetterExplained is a side business. Happy to share numbers, etc. I have some earlier posts about ebook sales and techniques: https://news.ycombinator.com/item?id=870015 https://news.ycombinator.com/item?id=870015 I don't think I should be taken as a model of learning though (I write very infrequently) but thankfully math is evergreen. Many of my most popular articles are 5-7 years old. I do want to dedicate more time to it. I realize I was afraid of ruining my love of learning by turning it into a profession, but I'm slowly coming to grips with it. That's one of the hardest parts for me actually, feeling I'll kill the golden goose by squeezing too hard. I'll probably do a blog post / postmortem on marketing, numbers, etc. so keep an eye out :).
  - neilsharma 11y ago
    1) "Not over explain it" -- that's really good advice. You don't want to take the "aha" moment away from someone. 2) Thanks for the link! Definitely looking forward to your blog post/postmortem (don't let BetterExplained die!). Have you found learning to be less enjoyable by writing about it, or do you end up discovering a dozen other tidbits of math magic you want to share with everyone? Two more questions if you don't mind :) 3) What are your thoughts on word problems? For example, in most linear algebra textbooks, you are given matrix and are asked you to process it. Rarely are you given a word problem and are asked to think through the entire process (data and operations matrices setup --> processing --> meaningful end result). 4) For inspiration, what are you experiences explaining concepts in a cross-disciplinary manner? When I was a student, I never understood why a concept is important. Homework problems were abstracted out of all real-world context to train for mechanical problem solving. Only now, after exploring data and writing algorithms in health, journalism and finance, have I finally been able to answer the question I always had as a kid: "why is this stuff useful?"
romaniv 11y ago
After reading several articles I came to the conclusion that every Wikipedia article on math concepts which are covered on betterexplained should start with the link to betterexplained. I know it's not going to happen, but it would benefit hundreds of thousands of people.
- kalid 11y ago
  Wow, glad to hear the articles had a positive impression :). I'm bouncing ideas for intuitive guides for any Wikipedia topic. After the 10 min intro, you can read the Wikipedia article for more details.
- plaguuuuuu 11y ago
  I feel like it's often easier to learn concepts from equations. The betterexplained articles are so wordy and contain so many analogies (some of which are leaky abstractions) that there's all this noise around the core concepts. Equations cut away all of that noise. And it's not like I'm some huge maths nerd; I've always found language and philosophy far easier to learn than mathematics. Oddly enough, I find betterexplained much more useful when I have already grasped the core concepts, because it does a great job of connecting to other concepts.
  - kalid 11y ago
    Thanks for the feedback! I see the articles as a supplement to the formal description people usually have in hand when they are googling for help :). For the succinct formal definition I definitely recommend Wikipedia or Mathworld.
  - twobits 11y ago
    It may be just me, but I hate learning concepts from equations. IMO, no one ever woke up one day and said, let's have P, Q, Ω, and Φ. (Or whatever they are.) Most probably they started with: Can I win this card game, and I have these cards?
jgamman 11y ago
shoutout to kalid - love your work.
- kalid 11y ago
  Thank you!
outlace 11y ago
This is great, wish someone would do this for more advanced math like topology, or other more abstract/pure math
southclaw 11y ago
I'm so glad this exists going into my next term involving a lot of maths that I just don't fundamentally understand; sure I can remember a set of rules after boring revision of sequences of instructions but I hate that. I asked what a "dot product" _meant_ once and was just told how it could be used to determine these other values. I later learnt what it actually was by reading some renderer library code. I had a similar "aha" moment when I read how you described sine/cosine/etc as percentage values in relation to positions around the circle. I hate how I was taught maths since secondary school (when I was so confused at trig/pythag lessons but learnt the concepts very quickly when I was coding around with positions and angles in a game engine mod to get a position x units in front of a character instead of doing homework) I really think this site will help me in the coming year, thank you!

666 points by rfreytag 11y ago

houzi 11y ago

Is it a data mining site? I'm just entering captcha's for days..

andrewclunn 11y ago

Aw, and here I was hoping for clips from SquareOne TV...

hugs 11y ago

I needed a refresher on trig last year for some robotics simulations I was working on. The trig lesson was extremely helpful: http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/ And fwiw, I finally learned an intuitive understanding of radians by reading the Tau Manifesto (not on BetterExplained). It would be awesome if BetterExplained used tau instead of pi in the lesson on radians, but that's a minor nitpick for a very helpful set of lessons.

jeffwass 11y ago
Out of curiosity do you get confused when people talk about 360 degrees, or 90 degree angles, or turning around 180 degrees, etc? Most people I know intuitively understand degrees and could easily understand radians the same. Why should BetterExplained use tau? Tau is a fairly niche idea that will only confuse way more than explain. That's a bit like saying BetterExplained should be written in Esperanto. Personally while I agree that historically pi would have been better off if defined as 6.28... instead of 3.14... (And even told my students this long before tau manifesto came out) I don't feel that a factor of 2 warrants using up a whole other Greek letter. (I'd be much more supportive of "tau" if instead of the Greek letter Tau Hartl chose a unique new symbol, much like physics uses hbar instead of h as he Planck Constant for radians instead of cycles).
- jerf 11y ago
  The anecdata that tau is helpful to learners is, IMHO, pretty strong at this point. Facts beat theory. And since anecdata is all we have, well, it's all we can use to talk about. I think it's way, way too easy for people on the terminal end of education to forget just how easy it is to get tripped up by very, very little things. Especially if they were themselves "good at math" and really aren't bothered by extra factors of 2 flying around. This is not the normal experience.
  - jeffwass 11y ago
    When someone says 90 degree angle, does that confuse you?
    - jerf 11y ago
      As it happens, I've tutored people in the middle of high school geometry. Radians confuse them. Full stop. If there is something that can make it easier to understand, while at the same time sacrificing absolutely nothing of mathematical consequence, it's a good thing. What confuses me is not the relevant issue. I've got $BOATLOADS of higher ed. I'm not the interesting case.
      - jeffwass 11y ago
        Ok, we're far away from my original point which was only on using tau in one website. My actual only personal argument against Tao as Hartl proposed it is the use of a standard Greek letter. Really would have preferred he picked something that was more "backwards compatible", exactly as physicists did with hbar. If you're for reducing confusion, do you think adding a new fundamental constant to the body of mathematics that is a Greek letter already used for countless other variables in history, will this cause confusion between old and new mathematical science texts and papers? Asking students of science for any texts and papers they read whether tau is 2pi or some other variable, and keeping track across them, seems more confusing to me than just consistently using pi and extra factors of 2.
ebbv 11y ago
No they should not use Tau, because the point of education is to give you knowledge that you can build on later. Nothing else uses Tau, so teaching you Tau does not prepare you for further education. I'm not even gonna touch the Tau vs. Pi debate because it's like a sports team rivalry, and not a rational discussion.
- TuringTest 11y ago
  > Nothing else uses Tau, so teaching you Tau does not prepare you for further education. Nitpicking, but that is a self-defeating argument. If everybody used Tau, there would be no problem in teaching it. Not that I disagree that doing the bootstrapping its usage and reaching there would be a problem, but all by itself is not a reason to not try changing it.
kalid 11y ago
Hey, thanks for the feedback! Glad to hear the trig article helped (one of my favorite aha moments -- all the trig functions, including stuff like the exotic cosecant, can be drawn together.) I'm a fan of Tau as an interesting simplifying concept but don't think mentions of pi should necessary be removed. My intuition is that radians are "distance traveled" as you go along a circle. Pi is the "neutral to max to neutral" distance and Tau is the full cycle (neutral, max, neutral, min, neutral). Ideally we just think about "distance" and context determines whether we want the journey to be "there and back" or just "there".
jgamman 11y ago
i (mostly) agree. i think thinking about a fraction of a full circle is the easiest way to work with trig. having said that, the mental overload on your average high school student flipping between pi and tau is too much. my personal compromise is to never cancel out the '2' and always work with 2pi. it makes some equations look a bit odd from the textbook but whacks me in the forehead to remember to think about circles...

chairleader 11y ago

Interesting set of articles. "Intuition isn't Optional"[1] could explain why pursuing simple solutions and good naming is important in software. http://betterexplained.com/articles/intuition-isnt-optional/ http://betterexplained.com/articles/intuition-isnt-optional/

kalid 11y ago
Nice point. Yes, naming can prime people to understand or be confused by a topic. Renaming the "imaginary numbers" to the "rotated numbers" would make them orders of magnitude easier to learn. "What's so strange about the second dimension?" vs. "How do I understand the square root of -1?".

wiremine 11y ago

I wonder why more secondary teachers don't use these sorts of examples? Any teachers out there that can give insight?

jerf 11y ago
The central bureaucracy has so overdetermined education that there is no longer any room for deviation. If it's not in the curriculum, there's no time for it, because there isn't hardly even any time to cover what is in the overdetermined curriculum. If you support innovation in education anymore, your first goal now is to get the central bureaucracy out of it. Until that happens, nothing else can change.
- criley2 11y ago
  Removing the central bureaucracy is a double edged sword. For every classroom you free to innovate, you also free a teacher to pan science, attack evolution, teach bible and religion in science class, etc. Many of us grew up in school districts prior to the Bush and Obama centralization regimes and quite literally were taught that Evolution is a Satanic lie by public (non-private non-religious) schools.
  - jerf 11y ago
    Innovation is a double-edged sword in general. Doesn't mean we should stop. Besides, are we not discussing exactly a situation in which the central bureaucracy has overspecified an inferior education? Nothing about "panning science, attacking evolution, teaching the bible and religion" has anything to do with whether or not there is a central organization; one stray election and you'll get all those things coming out of the central organization too. The very topic of this discussion is that an inferior education technique is being mandated right now. The casual default presumption that the centrally-mandated education is perfect is already falsified by the evidence.
    - WaxProlix 11y ago
      So are you saying we could fix the mandate and that would be good, or that specifying a minimum basic to which all teaching should be done is wrong in general? Or some third thing?

chris_wot 11y ago

I love this site. This explained logarithms and exponential growth really well.

kalid 11y ago
Glad to hear it helped, thanks.
- chris_wot 11y ago
  Oh! Dude, you are amazing! Seriously, I don't normally say that to people - I can't thank you enough because you honestly got me over a few mathematics hurdles. If you ever have a paid subscription, I'm buying :-)
  - kalid 11y ago
    Thanks Chris, really appreciate it. Hearing when the site clicks with someone is invigorating. No real paid subscriptions yet but it's a great idea =).

tempodox 11y ago

Error 522

alblue 11y ago

Seems like the site has gone down with traffic and CloudFlare is serving up a generic 522 error page.

pixelkicker 11y ago
Yep, bummer I was looking forward to checking this out.
- hugs 11y ago
  Check back later when traffic has died down. The site really is worth a read.
  - kalid 11y ago
    Thanks! Working on getting a fully static version of the site up. Argh, Wordpress with even caching plugins, still folds. (It's on reddit and getting 100x normal traffic). Eventually a postmortem will be in order. Static links to some favorites: • Imaginary numbers: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ http://webcache.googleusercontent.com/search?q=cache:http://... • Understanding e: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ http://webcache.googleusercontent.com/search?q=cache:http://... • Intuitive Trig: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-trigonometry/ http://webcache.googleusercontent.com/search?q=cache:http://... • Calculus intro: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/calculus/lesson-1 http://webcache.googleusercontent.com/search?q=cache:http://... • Sine waves: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://webcache.googleusercontent.com/search?q=cache:http://... • Euler's Formula: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ http://webcache.googleusercontent.com/search?q=cache:http://... • Linear Algebra: http://webcache.googleusercontent.com/search?q=cache:http://betterexplained.com/articles/linear-algebra-guide/ http://webcache.googleusercontent.com/search?q=cache:http://...

rfreytag 11y ago

Internet Archive has a copy: https://web.archive.org/web/20151221053341/http://betterexplained.com/ https://web.archive.org/web/20151221053341/http://betterexpl... >>>>> Please donate to the Archive while you are at it! :-) Google cache: http://webcache.googleusercontent.com/search?q=cache:http://www.betterexplained.com/ http://webcache.googleusercontent.com/search?q=cache:http://...

typon 11y ago

This site is a hidden gem on the internet. This guy deserves a lot more credit. I think he's doing something at the level of Khan Academy, but for seemingly simple concepts we all take for-granted. An article I really enjoyed was his explanation of Quake's inverse square root method: https://web.archive.org/web/20150530232103/http://betterexplained.com/articles/understanding-quakes-fast-inverse-square-root/ https://web.archive.org/web/20150530232103/http://betterexpl... What I also appreciate is that he admits when he doesn't understand something and doesn't pretend to give an incomplete, vague explanation and just straight up says he does't fully get it (like why the specific 'magic' number in the Quake algorithm). Thankfully he linked the original paper and I read up on it myself. But his initial explanation provided a solid background for the paper.

morley 11y ago
This is fascinating! I wanted to know more about the history of that function at id, which lead me to this beyond3d post (which was the topic of the reddit post that BetterExplained listed as a source): https://web.archive.org/web/20070208003255/http://www.beyond3d.com/articles/fastinvsqrt/ https://web.archive.org/web/20070208003255/http://www.beyond...
kalid 11y ago
Hey, Kalid from BetterExplained here, thanks! My general philosophy is to be really, really honest with myself if I understood something. It's ok (really!) to admit when we haven't fully understood a concept. Want a fun example? How about percentages. Yeah, that thing we mastered in 4th grade or whatever. That thing we use every day. Well, did you know that a% of b = b% of a? Let's say you want 16% of 25. Ugh. Ok, let's multiply it out... divide by 100... no! How about 25% of 16? Well, that's 4. Easy. But are they the same thing? a% of b = a/100 * b b% of a = b/100 * a Either way, it's ab/100. Now every percentage problem has a 50% chance of being expressed more easily. How did we miss this? (Argh!!!) Math is full of insights like that. Trig functions (sine, cosine, tan, etc.) are actually themselves percentages. A sine of .95 means you are at 95% of your maximum height (where the max is the hypotenuse). Sine and cosine are unitless numbers, and that's why the can be each other's derivatives (the percentage change of a percentage change...). So many things click! What else have we overlooked? Anyway, really appreciate the note!

robbiemitchell 11y ago

Tangent: with author permission, these lessons are loaded into the recommendation system for Knewton's free learning service: https://www.knewton.com/ https://www.knewton.com/

shobhitverma 11y ago
Knewton has done a good job of curating!
kalid 11y ago
Awesome! (Kalid from BetterExplained here.) This goes for everyone, all the articles are under a CC license.

kudu 11y ago

Is there anything similar to this for physics? I'm taking an introductory mechanics class and I can't help but think there must be a much more intuitive, logical way to solve even complicated multi-step problems than just falling into algorithmic pattern-based resolution.

kalid 11y ago
I haven't done much physics, but Feynman is my teaching inspiration. He has his famous lectures on Physics: http://www.feynmanlectures.caltech.edu http://www.feynmanlectures.caltech.edu (Put online by Michael Hartl, of TauDay fame among others.) I haven't gone through them extensively but Feynman was a master of making the complex simple.

TuringTest 11y ago

I just love this guy's ADEPT method of exposition (Analogy, Diagram, Example, Plain English explanation, and just then Technical definition). [1] That is exactly how I like to be introduced to any new concept, in special when I'm a complete newcomer to the field and can't relate it to previous ideas in it. Judging by the reactions to the BetterExplained site, other people agree with that. [1] http://betterexplained.com/articles/adept-method/ http://betterexplained.com/articles/adept-method/

kalid 11y ago
Thanks! I was really hesitant to force an acronym (it actually started as ADE) but then I realized I could work my way up to the full technical definition. Really glad to hear it's resonating. (Sometimes things happen out of order, i.e. you start with a plain English definition. But the idea is to have all 5 parts if you want to truly master a concept.)
igravious 11y ago
I've found that I also need to understand the historical context/impetus... I need to be shown and eventually understand why something was developed when it was developed. I need chronology of thought and ideas. For instance, learning about Gödel's theorems without having the Hilbert back-story explained. Learning about Leibniz's and Newton's calculus without learning about infinitesimals. And so on. There's generally a reason _why_ concepts are born _when_ they are born. If you think about your maths classes, sometimes you're instructed to learn a method because it is useful and because it has real-world applications but it I don't think anybody is ever first taught algebraic geometry properly, if I may use that word. I don't think kids are taught the geometry is one thing and algebra is another and that different spaces can have different metrics. Am I making sense here? Do people see what I'm trying to get at? Am I arguing for HADEPT? :) (Historical context, Analogy, Diagram, Example, Plain English explanation, and just then Technical definition)
- kalid 11y ago
  Great point. I find myself looking at the history of the idea when writing up a post. Did you realize negative numbers were only accepted in the late 1700s? That the Fourier Transform was originally rejected as untrue when first presented, by world-famous mathematicians even? (Yet we require students to internalize it without issue in a single lecture.) Historical context is huge. I think I'm now stuck with this ADEPT name but maybe it fits into the Plain English portion :).

waprin 11y ago

This site is absolutely excellent. There are a lot of things in math that I sort of had a handle of the mechanics of, but less so the intuitions for, which this site fills in well: I really liked the explanation of sine as something that makes things 'circly'. http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-... Does the site take external contributions?

kalid 11y ago
I've been considering a patreon page, but the best way is to help share the site with people learning math (students, family friends, etc.). A genuine recommendation is one of the best things I can ask for. Thank you!
kalid 11y ago
By popular demand: http://patreon.com/betterexplained http://patreon.com/betterexplained :)
- waprin 11y ago
  Sorry, I was asking if you accept article contributions.
  - anjanb 11y ago
    as a reader, I think that that should be nice. Also, it would be nice to have a forum where ideas get exchanged. Then Kaled can decide if any of the discussions should make it into the site as an article ?
    - kalid 11y ago
      Thanks for the feedback! I have a community in progress at http://aha.betterexplained.com http://aha.betterexplained.com -- I have some collaboration ideas I'll be announcing early next year :).
  - kalid 11y ago

darkerside 11y ago

From: http://betterexplained.com/static/articles/rethinking-arithmetic-a-visual-guide/ http://betterexplained.com/static/articles/rethinking-arithm... 1 * x^2 = 9 "What transformation (“times x”), when applied twice, will turn 1 into 9?" That's such a great way of thinking about algebra. The "understood" 1 is equivalent to, e.g., an understood "YOU" in the English language.

kalid 11y ago
Beautiful way to put it! We aren't always explicit about the subject of the transformation. Similarly, when working with exponents, I think: e^ix is really 1 * e^ix That is, we're starting with 1.0 and doing a transformation (rotation) on it.

nickpsecurity 11y ago

The article on math intuition plus the e article were incredible. I always knew how to work with them and sort of what they meant but never really intuitively. The articles were first time in a while that happened. Reminds me when I first learned the often-hated word problems were the best part of math that explained how to actually map it to real world. Equations took less thought for me so I avoided lots of word problem practice. Was glad I shifted back a bit before calculus or I'd never be able to explain what it was good for. This guy's site should get more attention and probably an award. I'll be experimenting with less-math-inclined people to see how effective it is.

abc_lisper 11y ago
Yeah, the e article is awesome. Cleared up a lot of cobwebs in my brain..
kalid 11y ago
Hey nick! Thanks for kind words, it's really touching to know when things are clicking. I'd love to see how things go with people that aren't math experts. My philosophy is I found one math up the mountain, it worked for me, but there's other ways too. It's awesome when people take the diagrams/analogies and adapt them for their own style. I saw your other comment too, will reply up there.

elliotec 11y ago

I bought the book a while back, and I still struggled with it. I have very low math skills. My pre algebra teacher was a joke, and everything building on that flew right over my head. I basically talked my counselor into letting me graduate even though I didn't pass basic math graduation requirements in college. So I'm a pretty clean slate, and this was quite difficult for me. I'm a pretty good developer, but telling the computer to do math for you based on looking up what it needs to do is much different from understanding the concepts myself which has been a real difficult thing for me. Kahn academy is the worst, like bringing back old terrible memories of repeating hellish math problems that I don't really get, but still want to move on out of boredom. Such it is.

j2kun 11y ago
I would love to hear your thoughts about something I'm working on. Care to shoot me an email? jkun2@uic.edu
kalid 11y ago
Thanks for the feedback! The book is more of a "top concepts I wish I knew" but not really a ground-up tutorial. For starting from a clean slate, you might like: http://betterexplained.com/static/articles/rethinking-arithmetic-a-visual-guide/ http://betterexplained.com/static/articles/rethinking-arithm... Once you can visualize the basic operations (add, subtract, multiply, divide), every new math operation becomes a lot easier (complex numbers = rotations, exponents = growth, combining them = orbiting a circle...).
- lstamour 11y ago
  When you say the opposite can be the multiplication of -2, did you actually mean the multiplication of -1? It's a bit unclear in the example whether the "loss of two" means 1 * -1 = -1 and therefore a relative loss of two, or 1 * -2 = -2 or a relative loss of 3. I always figured opposite meant inverting either the fraction so 2/1 becomes 1/2 or it meant multiplying by -1, effectively toggling the negative sign.
  - 11y ago

akerro 11y ago

>Secondly, if he is not involved in criminal activity, why does Mr Claus use unbreakable encryption? How do they know he encrypts everything?

neilsharma 11y ago

I've been using BetterExplained to review concepts I thought I mastered (based on great test scores and grades) but years later realized I had zero intuition on. It has been, by far, the most useful single source of math content I've found on the internet. The world needs more people like you. Two Questions: 1) What are your thoughts on interactive content like ExplainedVisually? I've been thinking about doing something similar for data structures / algorithm topics. How much of learning math concepts is exploratory vs learn-by-doing? 2) Are you running BetterExplained as a side business, or full-time? And if you're willing to share techniques and numbers for the entrepreneurial HN community, what are some things you've done to market it, monetize, etc and what were the results? Thanks!

kalid 11y ago
Really glad to hear the site helps with new insights (especially for someone who has aced the tests/grades). I was in a similar boat, qualified on paper, but not in my heart of hearts. 1) In general I like any efforts to explain things in new ways, and ExplainedVisually is great. My philosophy is that teaching is like humor. You want to make something funny, present it well, but not overexplain it. Let's people enjoy the joke. If you do too much handholding, you ruin the surprise and it's not fun to be told "Ok, the punchline is coming up...". It's not an exact science, but you get a nose for when something is truly illuminating vs. trying to chew your food for you. 2) BetterExplained is a side business. Happy to share numbers, etc. I have some earlier posts about ebook sales and techniques: https://news.ycombinator.com/item?id=870015 https://news.ycombinator.com/item?id=870015 I don't think I should be taken as a model of learning though (I write very infrequently) but thankfully math is evergreen. Many of my most popular articles are 5-7 years old. I do want to dedicate more time to it. I realize I was afraid of ruining my love of learning by turning it into a profession, but I'm slowly coming to grips with it. That's one of the hardest parts for me actually, feeling I'll kill the golden goose by squeezing too hard. I'll probably do a blog post / postmortem on marketing, numbers, etc. so keep an eye out :).
- neilsharma 11y ago
  1) "Not over explain it" -- that's really good advice. You don't want to take the "aha" moment away from someone. 2) Thanks for the link! Definitely looking forward to your blog post/postmortem (don't let BetterExplained die!). Have you found learning to be less enjoyable by writing about it, or do you end up discovering a dozen other tidbits of math magic you want to share with everyone? Two more questions if you don't mind :) 3) What are your thoughts on word problems? For example, in most linear algebra textbooks, you are given matrix and are asked you to process it. Rarely are you given a word problem and are asked to think through the entire process (data and operations matrices setup --> processing --> meaningful end result). 4) For inspiration, what are you experiences explaining concepts in a cross-disciplinary manner? When I was a student, I never understood why a concept is important. Homework problems were abstracted out of all real-world context to train for mechanical problem solving. Only now, after exploring data and writing algorithms in health, journalism and finance, have I finally been able to answer the question I always had as a kid: "why is this stuff useful?"

romaniv 11y ago

After reading several articles I came to the conclusion that every Wikipedia article on math concepts which are covered on betterexplained should start with the link to betterexplained. I know it's not going to happen, but it would benefit hundreds of thousands of people.

kalid 11y ago
Wow, glad to hear the articles had a positive impression :). I'm bouncing ideas for intuitive guides for any Wikipedia topic. After the 10 min intro, you can read the Wikipedia article for more details.
plaguuuuuu 11y ago
I feel like it's often easier to learn concepts from equations. The betterexplained articles are so wordy and contain so many analogies (some of which are leaky abstractions) that there's all this noise around the core concepts. Equations cut away all of that noise. And it's not like I'm some huge maths nerd; I've always found language and philosophy far easier to learn than mathematics. Oddly enough, I find betterexplained much more useful when I have already grasped the core concepts, because it does a great job of connecting to other concepts.
- kalid 11y ago
  Thanks for the feedback! I see the articles as a supplement to the formal description people usually have in hand when they are googling for help :). For the succinct formal definition I definitely recommend Wikipedia or Mathworld.
- twobits 11y ago
  It may be just me, but I hate learning concepts from equations. IMO, no one ever woke up one day and said, let's have P, Q, Ω, and Φ. (Or whatever they are.) Most probably they started with: Can I win this card game, and I have these cards?

jgamman 11y ago

shoutout to kalid - love your work.

kalid 11y ago
Thank you!

outlace 11y ago

This is great, wish someone would do this for more advanced math like topology, or other more abstract/pure math

southclaw 11y ago

I'm so glad this exists going into my next term involving a lot of maths that I just don't fundamentally understand; sure I can remember a set of rules after boring revision of sequences of instructions but I hate that. I asked what a "dot product" _meant_ once and was just told how it could be used to determine these other values. I later learnt what it actually was by reading some renderer library code. I had a similar "aha" moment when I read how you described sine/cosine/etc as percentage values in relation to positions around the circle. I hate how I was taught maths since secondary school (when I was so confused at trig/pythag lessons but learnt the concepts very quickly when I was coding around with positions and angles in a game engine mod to get a position x units in front of a character instead of doing homework) I really think this site will help me in the coming year, thank you!