5 ms·

BetterExplained: Clear, intuitive lessons about mathematics

340 points by parisianka 6y ago

Zhyl 6y ago
I do enjoy Khalid's explanations, and I applaud anyone who is exploring different mediums and methods to help improve education, especially around Mathematics and Computing (my areas of interest). Other favourites: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ https://ncase.me/trust https://ncase.me/trust http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ https://m.youtube.com/user/patrickJMT https://m.youtube.com/user/patrickJMT https://www.3blue1brown.com/ https://www.3blue1brown.com/ https://www.redblobgames.com/pathfinding/a-star/introduction.html https://www.redblobgames.com/pathfinding/a-star/introduction...
- BRedSox 6y ago
  Is there a way to save comments on HN? This is worthy of a save.
  - Zhyl 6y ago
    HN saves a list of your upvoted comments which you can view via your profile. This is private. You can click on the timestamp and it will take you to a page for the comment where you can click 'favourite' which will be public to anyone who visits your profile. Also, thank you!
  - PretzelFisch 6y ago
    you can book mark the comment link, or us HN and favorite the comment.
  - chris_wot 6y ago
    Tap on the date, click on favourite.
  - hyperpallium2 6y ago
    There's a "favorite" (sic) link that appears only when viewing the comment-tree rooted at that particular comment. You can get to that view by clicking the timestamp of the comment (obviously). (Or, in this particular case, clicking the "parent" link when viewing the comment-tree rooted at your comment, which is where you might be reading this, from your "threads" link). These favourites are public and are linked from your user page (https://news.ycombinator.com/user?id=BRedSox https://news.ycombinator.com/user?id=BRedSox), here (https://news.ycombinator.com/favorites?id=BRedSox https://news.ycombinator.com/favorites?id=BRedSox) A surprising number of HNers don't use this feature.
    - branweb 6y ago
      ha ha from the "sic" I can't tell if you object to the verbification of the word or if you're just British. Nice tip though. Had no idea about this feature.
- taliesinb 6y ago
  Second this. I’m working on a set of ideas about how best to enable arbitrary learning trajectories through a knowledge graph — partly in an effort to better capture my own learning in mathematical topics. I liked how Quantum Country integrated spaced repetition into its presentation of quantum computing, but I think you could go further and structure all knowledge you wish to communicate in a graph form. There are so many reasons to do this: it’s a graph in your head anyway, and if a machine can track and manage that graph it can better help you navigate it in a learning-optimal way. Just try reading some math on wikipedia and you’ll see where this could really shine. Currently it’s a frustrating experience because there are many slices of information that you might benefit from when you are looking up eg the article on distributive lattices, but the optimal slice to see depends on what examples you’ve encountered, what other topics are fresh in your mind, and what your short term goals are. Of course a textbook is a particular curated walk through this graph, but people are different, and there is no walk that is optimal for all people. A dedicated expert teacher can curate a path specifically for a learner, but almost no one can afford to employ such a teacher for them. If machines can handle the UX aspects of this graph-walking and graph-building for us (there are many simple ideas that fit into this), I believe we could have a real step change in learning efficiency with only modest investments in building such graphs. And the truth is, such graphs are actually fun to build because they are so interactive! It’s very similar to the semantic web, but catering specificity to humans and their idiosyncratic educational needs rather than to more nebulous things about machine reasoning or scripting.
  - deleted 6y ago
    [deleted]
- an_opabinia 6y ago
  Is there good material that covers advanced mathematics, ie, after multi variable calculus and algebra, but not primarily about proofs?
  - abecedarius 6y ago
    You might check out Penrose's The road to reality. I'm not sure it'll really serve, but it's what comes to mind.
  - deleted 6y ago
    [deleted]
- Debug_Overload 6y ago
  patrickJMT is really great. I remember frequenting his channel when I was taking some math courses in college.
- racl101 6y ago
  So much good stuff in one post. I'm gonna go on a huge tangent bro! I never loved Mathematics in high school, at least not the way we learn a bunch of things but never apply them or have any intuition about them, but I'm interested in it on personal level. This is right up my alley. Edit: forgot to say thank you!
cassepipe 6y ago
I really like it. The article on linear algebra was great and really made me click : https://betterexplained.com/articles/linear-algebra-guide/ https://betterexplained.com/articles/linear-algebra-guide/
wodenokoto 6y ago
What’s a good way to support better explained? I haven’t used the site extensively, but several articles have been instrumental for me understanding a concept, so I am very grateful for the resource and hope for its continuation.
- anjanb 6y ago
  You could buy his books on Amazon and gift them to curious people ?
- kalid 6y ago
  Glad the site was helpful! Helping spread the word with people it could help (educators, students, curious autodidacts) is great. There are BetterExplained books/courses you can get as quick stocking stuffers, though helping support local businesses now may be even better :).
sriram_malhar 6y ago
I really love this site. Ages ago, I was teaching a kid about logs and exponentials, I realised I know about 'e' via the formula, but didn't have a visceral feel for why that formula came to be. Khalid's explanation was the best.
- User23 6y ago
  I got that same result from the book Visual Complex Analysis. In retrospect it seems stupid, but until then I actually believed, to the extent I considered it at all, that it was just a happy coincidence that e^x is its own first derivative.
pdm55 6y ago
I also like https://www.intmath.com/ https://www.intmath.com/ for its short, clear explanations of various mathematical topics.
sizt 6y ago
The problem with math is the english it uses. For instance, there is nothing improper about an improper fraction. This is just idiotic nomenclature. It's not racist. It's not traditional. It's just stupid. Same thing with "numerator" and "denominator" ... or "co-efficient". All ridiculous, idiotic names for simple, basic concepts. And the unimaginative math teachers who apparently cannot speak simple english or consult a dictionary just parrot away what they learned and robotically drill dozing students with this kind of preposterous terminology nonsense. Widespread numerosity will never improve until mathmaticians create new, appropriate, accessible terminology for all the delicate abstractions of mathmatics. Failure to address and change this underlying problem has caused great disparity in scientific understanding. Scientists and educators have no one to blame but themselves for this situation and its consequences, especially in America.
merricksb 6y ago
For the curious, some previous threads: 2015 – https://news.ycombinator.com/item?id=10777638 https://news.ycombinator.com/item?id=10777638 2009 – https://news.ycombinator.com/item?id=950439 https://news.ycombinator.com/item?id=950439 And: 2017 – "Life Lessons After 10 Years of BetterExplained.com" – https://news.ycombinator.com/item?id=13384782 https://news.ycombinator.com/item?id=13384782 Also, many topic pages discussed separately over the years - https://news.ycombinator.com/from?site=betterexplained.com https://news.ycombinator.com/from?site=betterexplained.com
Hnrobert42 6y ago
I read the article on Taylor series. I think the best evidence that it was a good explanation is that at the end, I thought, “Duh. This stuff is all obvious. When does he get to the hard part?” Note, I’d never heard of Taylor series, and I am not a savant. Tangentially, I remember a woman explaining Fibonacci numbers in a youtube video with sunflowers and pineapples about 8 years ago. Her handle was something like “the nerd girl” or “mathchick”. I can’t seem to find it again. Does this ring bells for anyone? Edit: I should have tried harder before asking. https://m.youtube.com/watch?v=ahXIMUkSXX0 https://m.youtube.com/watch?v=ahXIMUkSXX0
- andi999 6y ago
  It is a good article, but actually it skipped the hard part.
  - Shugarl 6y ago
    Well, I guess it's fine. I might be wrong on this, but I always thought that the goal of this site wasn't to make you an expert on a subject, but to provide you a high-level, intuitive understanding of a subject, which then makes understanding the hard parts/the details waaay easier. (Whereas in college, we'd go straight up to the hard parts after a small introduction.)
chris_wot 6y ago
Every time I see this on HN, I always feel the need to thank Khalid. This time is no different - thank you Khalid!
- kalid 6y ago
  Hey Chris, thanks for the support! =)
Shugarl 6y ago
Aside from Zhyl's comment (Thank you by the way), is there a list of Betterexplained-like ressources ? Finding Betterexplained made several things just "click" in my head, and I hope to find something similar with computer science.
savrajsingh 6y ago
He spells his name KALID, just FYI :)
elevenoh 6y ago
I find this a good reference for creating intuitive abstractions/conceptual models. But everyone is different in which abstractions resonate for them, and these don't resonate with my cognitive profile.

340 points by parisianka 6y ago

Zhyl 6y ago

I do enjoy Khalid's explanations, and I applaud anyone who is exploring different mediums and methods to help improve education, especially around Mathematics and Computing (my areas of interest). Other favourites: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ https://ncase.me/trust https://ncase.me/trust http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ https://m.youtube.com/user/patrickJMT https://m.youtube.com/user/patrickJMT https://www.3blue1brown.com/ https://www.3blue1brown.com/ https://www.redblobgames.com/pathfinding/a-star/introduction.html https://www.redblobgames.com/pathfinding/a-star/introduction...

BRedSox 6y ago
Is there a way to save comments on HN? This is worthy of a save.
- Zhyl 6y ago
  HN saves a list of your upvoted comments which you can view via your profile. This is private. You can click on the timestamp and it will take you to a page for the comment where you can click 'favourite' which will be public to anyone who visits your profile. Also, thank you!
- PretzelFisch 6y ago
  you can book mark the comment link, or us HN and favorite the comment.
- chris_wot 6y ago
  Tap on the date, click on favourite.
- hyperpallium2 6y ago
  There's a "favorite" (sic) link that appears only when viewing the comment-tree rooted at that particular comment. You can get to that view by clicking the timestamp of the comment (obviously). (Or, in this particular case, clicking the "parent" link when viewing the comment-tree rooted at your comment, which is where you might be reading this, from your "threads" link). These favourites are public and are linked from your user page (https://news.ycombinator.com/user?id=BRedSox https://news.ycombinator.com/user?id=BRedSox), here (https://news.ycombinator.com/favorites?id=BRedSox https://news.ycombinator.com/favorites?id=BRedSox) A surprising number of HNers don't use this feature.
  - branweb 6y ago
    ha ha from the "sic" I can't tell if you object to the verbification of the word or if you're just British. Nice tip though. Had no idea about this feature.
taliesinb 6y ago
Second this. I’m working on a set of ideas about how best to enable arbitrary learning trajectories through a knowledge graph — partly in an effort to better capture my own learning in mathematical topics. I liked how Quantum Country integrated spaced repetition into its presentation of quantum computing, but I think you could go further and structure all knowledge you wish to communicate in a graph form. There are so many reasons to do this: it’s a graph in your head anyway, and if a machine can track and manage that graph it can better help you navigate it in a learning-optimal way. Just try reading some math on wikipedia and you’ll see where this could really shine. Currently it’s a frustrating experience because there are many slices of information that you might benefit from when you are looking up eg the article on distributive lattices, but the optimal slice to see depends on what examples you’ve encountered, what other topics are fresh in your mind, and what your short term goals are. Of course a textbook is a particular curated walk through this graph, but people are different, and there is no walk that is optimal for all people. A dedicated expert teacher can curate a path specifically for a learner, but almost no one can afford to employ such a teacher for them. If machines can handle the UX aspects of this graph-walking and graph-building for us (there are many simple ideas that fit into this), I believe we could have a real step change in learning efficiency with only modest investments in building such graphs. And the truth is, such graphs are actually fun to build because they are so interactive! It’s very similar to the semantic web, but catering specificity to humans and their idiosyncratic educational needs rather than to more nebulous things about machine reasoning or scripting.
- deleted 6y ago
  [deleted]
an_opabinia 6y ago
Is there good material that covers advanced mathematics, ie, after multi variable calculus and algebra, but not primarily about proofs?
- abecedarius 6y ago
  You might check out Penrose's The road to reality. I'm not sure it'll really serve, but it's what comes to mind.
- deleted 6y ago
  [deleted]
Debug_Overload 6y ago
patrickJMT is really great. I remember frequenting his channel when I was taking some math courses in college.
racl101 6y ago
So much good stuff in one post. I'm gonna go on a huge tangent bro! I never loved Mathematics in high school, at least not the way we learn a bunch of things but never apply them or have any intuition about them, but I'm interested in it on personal level. This is right up my alley. Edit: forgot to say thank you!

cassepipe 6y ago

I really like it. The article on linear algebra was great and really made me click : https://betterexplained.com/articles/linear-algebra-guide/ https://betterexplained.com/articles/linear-algebra-guide/

wodenokoto 6y ago

What’s a good way to support better explained? I haven’t used the site extensively, but several articles have been instrumental for me understanding a concept, so I am very grateful for the resource and hope for its continuation.

anjanb 6y ago
You could buy his books on Amazon and gift them to curious people ?
kalid 6y ago
Glad the site was helpful! Helping spread the word with people it could help (educators, students, curious autodidacts) is great. There are BetterExplained books/courses you can get as quick stocking stuffers, though helping support local businesses now may be even better :).

sriram_malhar 6y ago

I really love this site. Ages ago, I was teaching a kid about logs and exponentials, I realised I know about 'e' via the formula, but didn't have a visceral feel for why that formula came to be. Khalid's explanation was the best.

User23 6y ago
I got that same result from the book Visual Complex Analysis. In retrospect it seems stupid, but until then I actually believed, to the extent I considered it at all, that it was just a happy coincidence that e^x is its own first derivative.

pdm55 6y ago

I also like https://www.intmath.com/ https://www.intmath.com/ for its short, clear explanations of various mathematical topics.

sizt 6y ago

The problem with math is the english it uses. For instance, there is nothing improper about an improper fraction. This is just idiotic nomenclature. It's not racist. It's not traditional. It's just stupid. Same thing with "numerator" and "denominator" ... or "co-efficient". All ridiculous, idiotic names for simple, basic concepts. And the unimaginative math teachers who apparently cannot speak simple english or consult a dictionary just parrot away what they learned and robotically drill dozing students with this kind of preposterous terminology nonsense. Widespread numerosity will never improve until mathmaticians create new, appropriate, accessible terminology for all the delicate abstractions of mathmatics. Failure to address and change this underlying problem has caused great disparity in scientific understanding. Scientists and educators have no one to blame but themselves for this situation and its consequences, especially in America.

merricksb 6y ago

For the curious, some previous threads: 2015 – https://news.ycombinator.com/item?id=10777638 https://news.ycombinator.com/item?id=10777638 2009 – https://news.ycombinator.com/item?id=950439 https://news.ycombinator.com/item?id=950439 And: 2017 – "Life Lessons After 10 Years of BetterExplained.com" – https://news.ycombinator.com/item?id=13384782 https://news.ycombinator.com/item?id=13384782 Also, many topic pages discussed separately over the years - https://news.ycombinator.com/from?site=betterexplained.com https://news.ycombinator.com/from?site=betterexplained.com

Hnrobert42 6y ago

I read the article on Taylor series. I think the best evidence that it was a good explanation is that at the end, I thought, “Duh. This stuff is all obvious. When does he get to the hard part?” Note, I’d never heard of Taylor series, and I am not a savant. Tangentially, I remember a woman explaining Fibonacci numbers in a youtube video with sunflowers and pineapples about 8 years ago. Her handle was something like “the nerd girl” or “mathchick”. I can’t seem to find it again. Does this ring bells for anyone? Edit: I should have tried harder before asking. https://m.youtube.com/watch?v=ahXIMUkSXX0 https://m.youtube.com/watch?v=ahXIMUkSXX0

andi999 6y ago
It is a good article, but actually it skipped the hard part.
- Shugarl 6y ago
  Well, I guess it's fine. I might be wrong on this, but I always thought that the goal of this site wasn't to make you an expert on a subject, but to provide you a high-level, intuitive understanding of a subject, which then makes understanding the hard parts/the details waaay easier. (Whereas in college, we'd go straight up to the hard parts after a small introduction.)

chris_wot 6y ago

Every time I see this on HN, I always feel the need to thank Khalid. This time is no different - thank you Khalid!

kalid 6y ago
Hey Chris, thanks for the support! =)

Shugarl 6y ago

Aside from Zhyl's comment (Thank you by the way), is there a list of Betterexplained-like ressources ? Finding Betterexplained made several things just "click" in my head, and I hope to find something similar with computer science.

savrajsingh 6y ago

He spells his name KALID, just FYI :)

elevenoh 6y ago

I find this a good reference for creating intuitive abstractions/conceptual models. But everyone is different in which abstractions resonate for them, and these don't resonate with my cognitive profile.