7 ms·

Learn difficult concepts with the ADEPT method (2014)

270 points by adamzerner 6y ago

cjhveal 6y ago
This is actually really good advice. I've tended to try a formal description of something first, in case the other party knows the jargon, but it's been surprising to me how quickly you can shut someone down by throwing words at them they don't understand.
jakub_g 6y ago
Never heard about the method before but I've noticed long time ago this is the way that works for me. When someone starts with definition I get bored. I want to start from example and use case and then I will built a mental formalization on my own. Starting from formalization I just can't be bothered. You can see the diff e.g. how Wikipedia and Investopedia articles are written. Each Investopedia article shows an example and repeats some key things multiple times. Whereas trying to learn any economics or maths stuff from Wikipedia is almost impossible. It can only be a reference if you already know the thing and just need to double-check the formula.
- punnerud 6y ago
  This is why I like Jeremy Howard’s approach for teaching machine learning. Begin with simple code and examples, then break apart and explain. https://course.fast.ai/ https://course.fast.ai/
- hikarudo 6y ago
  Same here. I've heard somewhere, in a video about language acquisition, that we have a pattern recognition algorithm in our brains, so all we need is to see examples, and the algorithm does the rest (builds a model by itself).
  - nullsense 6y ago
    That's basically how I learned Japanese grammar. All I needed were 3 to 5 example sentences in an Anki and I could get an intuition for it after that.
  - epiecs 6y ago
    I would love to see that video. I have been thinking about this a lot since I don't easily learn new languages. If you do find it please paste it here or pm me :)
  - cyphar 6y ago
    This is the idea of the "language acquisition device" pioneered by Chomsky in the 60s. Research from the 60s up to the current day has pretty conclusively shown that language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that deliberate learning of grammar and vocabulary has very little to no impact on your language skills. Basically, our brains all learn language in the same manner no matter what language and no matter what age. Children don't learn their native language through reading books on grammar, they acquire it through listening and reading stories which they can comprehend. Adults are exactly the same. (The theories of the existence of a physical module in the brain or innate universal grammar are not widely accepted science today, but it is inarguable that the above second-order effects have been observed regardless of the mechanism of action.)
    - tasogare 6y ago
      > (1) language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that (2) deliberate learning of grammar and vocabulary has very little to no impact on your language skills (1) is somewhat true while (2) is utter garbage. In fact, the point of (2) is to reach (1). In his book, Nation (2001) explains very well, with data from numerous studies, that cramming vocabulary is in fact an extremely optimized productive use of time and effort. And given how Nation details the dimension of vocabulary learners, he is above suspicion in respect to being biased towards rote learning.
      - deleted 6y ago
        [deleted]
      - cyphar 6y ago
        Yeah that part of my comment was a bit sloppy (I didn't mention vocabulary originally and then foolishly added it in a later edit). Cramming vocabulary does help make input more comprehensible, but it should be noted that cramming vocabulary doesn't (by itself) cause you to acquire that vocabulary. Graded readers have been shown to also be very effective, despite a lack of vocabulary cramming (meaning that it's not necessary -- which is the point I was unsuccessfully trying to make). I do agree that it is a productive use of time (at least in the early stages of learning a language) and in my own language learning, I am cramming vocabulary. However, consciously learning grammar definitely does not have significant enough returns to be worth the effort (it can help somewhat in certain cases, but comprehensible input has a much larger effect).
- snarkypixel 6y ago
  Same, give me an example, and if I don't get it I'll dig in.
- shahbaby 6y ago
  I've also discovered that this type of bottom up approach is faster than the top down approach. Starting with an example gives a more concrete frame of reference. I don't think this is just a matter of personal preference either. I believe that the way you learn a new concept is the same way as how you learn to navigate a new physical environment. We don't have a GPS in our brain for mapping, we learn locations by reference to other locations.
- jiggawatts 6y ago
  This is a disease of Mathematics in general. Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class. However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems. Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely. It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes! An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as: □²A = J This is total gibberish to anyone who doesn't already know volumes about Geometric Algebra. It literally states that: Square squared letter A is letter J. What!? The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc... Pasting that equation into an article can't possibly cover all of that...
  - nubbins 6y ago
    Out of all resources I have found Wikipedia to be almost uniquely useless for teaching yourself a new concept, its like "explanation by committee". On the other hand if I can understand a Wikipedia article I know I am at least intermediate in a area.
    - chrisweekly 6y ago
      Yep. Highest recommendation for https://betterexplained.com https://betterexplained.com which is at the other pole.
  - penguin_booze 6y ago
abalaji 6y ago
This website's got some great explanations for topics that a lot of people get hung up on. Here's one of their articles on Flux. [1] I was curious how this mental model compares to other ones I had previously encountered and found this article that compares each. [2] [1] https://betterexplained.com/articles/flux/ https://betterexplained.com/articles/flux/ [2] https://tomroth.com.au/learning/ https://tomroth.com.au/learning/
- alexnastase 6y ago
  I've read Tom Roth's article and have to say it's very well written and insightful. Thanks for sharing!
deleted 6y ago
[deleted]
QueensGambit 6y ago
ADEPT method is like using "Uber for X" to explain an idea. Here, I used ADEPT method to explain the ADEPT method.
- kaycebasques 6y ago
  You used the A part but didn't cover the DEPT part.
  - QueensGambit 6y ago
    That's the idea of analogy, isn't it? It invokes the listener to ask for more information, so that you can explain the DEPT part aka "the pitch deck".
blennon 6y ago
This is tangential but does anyone have a "better explanation" for topology?
deleted 6y ago
[deleted]
carls 6y ago
This is the way that I wish more teachers taught. I love finding analogies or visual diagrams for CS concepts to share with my students. Request: can others share links to books, courses, videos, articles or other resources that employ incredibly thoughtful pedagogy in explaining, rather than diving into the nitty-gritty details immediately? I'd love to start a collection of examples to share with others.
- moallabbad 6y ago
  This is a book on Linear Algebra I am working on: https://moallabbad.github.io/Intuitive-Guide-to-Linear-Algebra https://moallabbad.github.io/Intuitive-Guide-to-Linear-Algeb... I am going through somewhat lengthy explanations and using interactive graphs in 2D and 3D. My goal is to introduce things gradually and provide different perspectives to increase the chance of at least one clicking. It’s in no way complete. In fact, it currently only has one chapter. But I am actively working on it. Hope you like it.
  - cnasc 6y ago
    This looks very cool! Is there a way to subscribe to updates?
kaycebasques 6y ago
I've been a full-time technical writer for ~8 years. These seem like solid guidelines to me. One trick-of-the-trade that I've learned about diagrams is to avoid hardcoding English text into the diagram if you're working on an internationalized doc site. Instead, insert numbers like "(1)" and "(2)" and then provide a numbered list after the diagram explaining each number from the diagram. This makes the content easier to translate. E.g. if you hardcode English into the diagrams, you've got to translate that diagram for every language or (more realistically) your non-English docs are going to have English text hardcoded into their diagrams. Hat tip to David Friedman for teaching me that trick. I've sometimes had technical people take issue with analogies precisely because they're not completely technically accurate. When that happens I compromise with them by adding an explicit sentence right before or after the analogy along the lines of "This is just an analogy to help you develop an intuition for the topic. It's not 100% technically correct. See X for technical details." I think the importance and helpfulness of examples is majorly underrated across docs sites.
- sn41 6y ago
  It may hark back to the terse crypticness of Unix manpages. On the other hand, php has a better balance with user contributed examples at the end of the official doc, with exceptional comments and examples eventually becoming part of the official documentation.
dang 6y ago
If curious see also 2016 https://news.ycombinator.com/item?id=12031893 https://news.ycombinator.com/item?id=12031893
unnouinceput 6y ago
In my experience the most difficult for us techies to lay down to somebody outside is the analogy. Is the make or break of entire explanation. And usually I fail 75% of the time right there at the start. Some recover later when I go with examples, but most of them are lost from beginning. Yeah, I know, I'd make a terrible teacher.
- nubbins 6y ago
  I think the key here is the analogy is just a warm up for the actual full detailed explanation. Where I find analogies always fail is in journalistic tech books that just use analogies to explain concepts but never get down to low level details and leave the reader with a vague feeling of understanding something, but if you asked them about any of the implications of the model explained they can't state anything.
sn41 6y ago
Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example, [1]) The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact. [1] https://nrich.maths.org/5961 https://nrich.maths.org/5961
- thunderbong 6y ago
  The 'negative' on a number line is just a visualization. If you instead, think of the 'negation' as direction (as in a vector), it becomes easier to understand why multiplying two negative numbers makes the answer positive (You've basically turned around twice). I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
  - sn41 6y ago
    Thanks, this was helpful.
    - mkl 6y ago
      To better understand this, and see how complex numbers can be viewed the same way, go through this article, especially up to the first set of slides: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
- deleted 6y ago
  [deleted]
- cyphar 6y ago
  > The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact. Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative: a*0 = 0 [Assumed but you can prove this too.] a*(b-b) = 0 [Definition of 0.] a*b + a*(-b) = 0 [Distributive law.] For any X, only its inverse -X has the property that X + -X = 0. Therefore a×(-b) is the inverse of a×b [which is -(a×b)]. Then apply the same idea but with a negative a: (-a)*0 = 0 [Assumed but you can prove this too.] (-a)*(b-b) = 0 [Definition of 0] (-a)*b + (-a)*(-b) = 0 [Distributive law.] -(a*b) + (-a)*(-b) = 0 [From above.] (-a)*(-b) + -(a*b) = 0 [Commutative addition to rearrange to X + -X = 0 form.] And as before, only the inverse has this property so -(a×b) must be the inverse of (-a)×(-b) -- which is a×b! So a negative number times a negative number is positive. QED. In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1]. The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal). [1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive_identity_annihilates_ring_elements https://en.wikipedia.org/wiki/Additive_identity#The_additive...
zaptheimpaler 6y ago
Books exist. A well-written book usually condenses years and years of knowledge and experience into a few hundred pages. Honestly the modern trend of having to make all educational content always engaging, with immediate rewards at all times seems wrong to me. Sometimes, its better to just chug through a book and boring formalization for 50 hours until you get to a point of clarity, because getting to the same knowledge via dozens of half-complete blogs and online sources takes MUCH more time in the long run. Books assume you will read the whole thing, so they build up the mental framework "breadth-first", so it seems less useful until you get to a sufficiently high depth. Once you get over the hump though, its great. With quick online learning, they give you a few good "depth-first" dives, but leave the foundational "knowledge/concept tree" underdeveloped. Books aren't distracting, they aren't internet enabled and make it easy to just sit down and go through page by page. Online stuff is usually a great supplement if you miss some intuition but it hasn't quite replaced books. Everyone is looking for some way to make the fundamental act of paying attention and learning easier.. but paying attention IS difficult. That is why its valuable. </oldmanrant>
- sn41 6y ago
  Agree++. I really prefer the disconnected mode of a book. The feeling of insight dripping in after being steeped in one thing alone.
- klibertp 6y ago
  No. Books were the most cost-effective medium for transfering knowledge for a long time, but that's in the past. They were never the most efficient - a few hours of one on one chat with domain expert, possibly with a whiteboard (or, failing that, some sand and a stick) at hand, could easily be an equivalent of ten times as much time of reading a book. Now that there's negligible difference in distribution cost between a flat text and a full-blown multimedia presentation, sticking with text is simply a waste of time.
  - zaptheimpaler 6y ago
    I agree that learning directly from an expert is much more efficient. A multimedia presentation can be just as good as a book or better. Its just that as of today, I haven't seen many websites that match the depth and completeness of books yet. 3blue1brown is awesome, many people are writing linear algebra ebooks, quantum country looks great for quantum computation but as of _now_, none are complete or comprehensive the way a textbook is.
qwerty456127 6y ago
What are some more sources where I can find things explained in this manner?
qwerty456127 6y ago
I understood nearly zero of all mathematics (beyond the elementary school stuff) before I've found betterexplained and some relevant YouTube videos. I could learn formulae and some typical solutions by heart to substitute values and solve some exam problems but hardly ever understood them. I felt school/college math pure waste of time as I didn't actually understand anything. Things like betterexplained were a blast that opened an entire new beautiful exciting world to me instantly.
- penguin_booze 6y ago
  Back when I was taught matrix multiplication, I was told matrix is "just a table of numbers", and in order to "multiply" matrix to another, you walk from left to right, and on the other matrix, you go top to bottom. And th... (muffled voices). I thought it was game invented to pracitce more integer multiplication and improve concentation. Later, during college, I see matrices again! I went, "hang on, I thought... that wasn't a game after all?!". Decades later I discover 3b1b and Pavel Grinfeld [1] videos. My fist emotion was anger: "This is what you told me 'just a table of numbers'?". What a cruel joke! [1] https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1..., https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMRULWJYthculb2QWEiZOkwTSU https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMR..., https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5 https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMR...
  - qwerty456127 6y ago
    Thank you very much for the links. Please share any other cool educational materials you know.

270 points by adamzerner 6y ago

This is actually really good advice. I've tended to try a formal description of something first, in case the other party knows the jargon, but it's been surprising to me how quickly you can shut someone down by throwing words at them they don't understand.

jakub_g 6y ago

Never heard about the method before but I've noticed long time ago this is the way that works for me. When someone starts with definition I get bored. I want to start from example and use case and then I will built a mental formalization on my own. Starting from formalization I just can't be bothered. You can see the diff e.g. how Wikipedia and Investopedia articles are written. Each Investopedia article shows an example and repeats some key things multiple times. Whereas trying to learn any economics or maths stuff from Wikipedia is almost impossible. It can only be a reference if you already know the thing and just need to double-check the formula.

punnerud 6y ago
This is why I like Jeremy Howard’s approach for teaching machine learning. Begin with simple code and examples, then break apart and explain. https://course.fast.ai/ https://course.fast.ai/
hikarudo 6y ago
Same here. I've heard somewhere, in a video about language acquisition, that we have a pattern recognition algorithm in our brains, so all we need is to see examples, and the algorithm does the rest (builds a model by itself).
- nullsense 6y ago
  That's basically how I learned Japanese grammar. All I needed were 3 to 5 example sentences in an Anki and I could get an intuition for it after that.
- epiecs 6y ago
  I would love to see that video. I have been thinking about this a lot since I don't easily learn new languages. If you do find it please paste it here or pm me :)
- cyphar 6y ago
  This is the idea of the "language acquisition device" pioneered by Chomsky in the 60s. Research from the 60s up to the current day has pretty conclusively shown that language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that deliberate learning of grammar and vocabulary has very little to no impact on your language skills. Basically, our brains all learn language in the same manner no matter what language and no matter what age. Children don't learn their native language through reading books on grammar, they acquire it through listening and reading stories which they can comprehend. Adults are exactly the same. (The theories of the existence of a physical module in the brain or innate universal grammar are not widely accepted science today, but it is inarguable that the above second-order effects have been observed regardless of the mechanism of action.)
  - tasogare 6y ago
    > (1) language acquisition only occurs through comprehensible input (this is known as the "Input Hypothesis" popularised by Stephen Krashen) and that (2) deliberate learning of grammar and vocabulary has very little to no impact on your language skills (1) is somewhat true while (2) is utter garbage. In fact, the point of (2) is to reach (1). In his book, Nation (2001) explains very well, with data from numerous studies, that cramming vocabulary is in fact an extremely optimized productive use of time and effort. And given how Nation details the dimension of vocabulary learners, he is above suspicion in respect to being biased towards rote learning.
    - deleted 6y ago
      [deleted]
    - cyphar 6y ago
      Yeah that part of my comment was a bit sloppy (I didn't mention vocabulary originally and then foolishly added it in a later edit). Cramming vocabulary does help make input more comprehensible, but it should be noted that cramming vocabulary doesn't (by itself) cause you to acquire that vocabulary. Graded readers have been shown to also be very effective, despite a lack of vocabulary cramming (meaning that it's not necessary -- which is the point I was unsuccessfully trying to make). I do agree that it is a productive use of time (at least in the early stages of learning a language) and in my own language learning, I am cramming vocabulary. However, consciously learning grammar definitely does not have significant enough returns to be worth the effort (it can help somewhat in certain cases, but comprehensible input has a much larger effect).
snarkypixel 6y ago
Same, give me an example, and if I don't get it I'll dig in.
shahbaby 6y ago
I've also discovered that this type of bottom up approach is faster than the top down approach. Starting with an example gives a more concrete frame of reference. I don't think this is just a matter of personal preference either. I believe that the way you learn a new concept is the same way as how you learn to navigate a new physical environment. We don't have a GPS in our brain for mapping, we learn locations by reference to other locations.
jiggawatts 6y ago
This is a disease of Mathematics in general. Actual mathematics teaching, as seen in typical University classes typically introduces a new abstract definition, and then immediately follows that up with a number of "worked examples", either in the lecture itself or a follow-up tutorial class. However, the reference material will typically only have the abstract definitions, but at least there will be a sequence of them building up the formalities piece by piece. In principle, a gifted mathematician could follow the trail and get to the "aha!" moment without even having to do the homework problems. Encyclopedias like Wikipedia or Mathworld tend to just mention the most abstract, "end product" of a long chain of mathematics. Terse formulas by themselves are utterly useless to anyone who hasn't gone through the step-by-step learning process and already understands the topic completely. It's like a historian being given a handful of Egyptian hieroglyphic characters before finding the Rosetta Stone. It's just squiggles and shapes! An example I came across personally was the simplification of Maxwell's Equations using Geometric Algebra. In GA, it is possible to write the equations of the EM fields incredibly tersely, down to as little as: □²A = J This is total gibberish to anyone who doesn't already know volumes about Geometric Algebra. It literally states that: Square squared letter A is letter J. What!? The 5 characters in that equation pack in an awful lot: The Geometric Product. Graded vector spaces. The four dimensional spacetime of special relativity. Vector calculus. Natural physical units. Etc... Pasting that equation into an article can't possibly cover all of that...
- nubbins 6y ago
  Out of all resources I have found Wikipedia to be almost uniquely useless for teaching yourself a new concept, its like "explanation by committee". On the other hand if I can understand a Wikipedia article I know I am at least intermediate in a area.
  - chrisweekly 6y ago
    Yep. Highest recommendation for https://betterexplained.com https://betterexplained.com which is at the other pole.
- penguin_booze 6y ago

abalaji 6y ago

This website's got some great explanations for topics that a lot of people get hung up on. Here's one of their articles on Flux. [1] I was curious how this mental model compares to other ones I had previously encountered and found this article that compares each. [2] [1] https://betterexplained.com/articles/flux/ https://betterexplained.com/articles/flux/ [2] https://tomroth.com.au/learning/ https://tomroth.com.au/learning/

alexnastase 6y ago
I've read Tom Roth's article and have to say it's very well written and insightful. Thanks for sharing!

deleted 6y ago

[deleted]

QueensGambit 6y ago

ADEPT method is like using "Uber for X" to explain an idea. Here, I used ADEPT method to explain the ADEPT method.

kaycebasques 6y ago
You used the A part but didn't cover the DEPT part.
- QueensGambit 6y ago
  That's the idea of analogy, isn't it? It invokes the listener to ask for more information, so that you can explain the DEPT part aka "the pitch deck".

blennon 6y ago

This is tangential but does anyone have a "better explanation" for topology?

deleted 6y ago

[deleted]

carls 6y ago

This is the way that I wish more teachers taught. I love finding analogies or visual diagrams for CS concepts to share with my students. Request: can others share links to books, courses, videos, articles or other resources that employ incredibly thoughtful pedagogy in explaining, rather than diving into the nitty-gritty details immediately? I'd love to start a collection of examples to share with others.

moallabbad 6y ago
This is a book on Linear Algebra I am working on: https://moallabbad.github.io/Intuitive-Guide-to-Linear-Algebra https://moallabbad.github.io/Intuitive-Guide-to-Linear-Algeb... I am going through somewhat lengthy explanations and using interactive graphs in 2D and 3D. My goal is to introduce things gradually and provide different perspectives to increase the chance of at least one clicking. It’s in no way complete. In fact, it currently only has one chapter. But I am actively working on it. Hope you like it.
- cnasc 6y ago
  This looks very cool! Is there a way to subscribe to updates?

kaycebasques 6y ago

I've been a full-time technical writer for ~8 years. These seem like solid guidelines to me. One trick-of-the-trade that I've learned about diagrams is to avoid hardcoding English text into the diagram if you're working on an internationalized doc site. Instead, insert numbers like "(1)" and "(2)" and then provide a numbered list after the diagram explaining each number from the diagram. This makes the content easier to translate. E.g. if you hardcode English into the diagrams, you've got to translate that diagram for every language or (more realistically) your non-English docs are going to have English text hardcoded into their diagrams. Hat tip to David Friedman for teaching me that trick. I've sometimes had technical people take issue with analogies precisely because they're not completely technically accurate. When that happens I compromise with them by adding an explicit sentence right before or after the analogy along the lines of "This is just an analogy to help you develop an intuition for the topic. It's not 100% technically correct. See X for technical details." I think the importance and helpfulness of examples is majorly underrated across docs sites.

sn41 6y ago
It may hark back to the terse crypticness of Unix manpages. On the other hand, php has a better balance with user contributed examples at the end of the official doc, with exceptional comments and examples eventually becoming part of the official documentation.

dang 6y ago

If curious see also 2016 https://news.ycombinator.com/item?id=12031893 https://news.ycombinator.com/item?id=12031893

unnouinceput 6y ago

In my experience the most difficult for us techies to lay down to somebody outside is the analogy. Is the make or break of entire explanation. And usually I fail 75% of the time right there at the start. Some recover later when I go with examples, but most of them are lost from beginning. Yeah, I know, I'd make a terrible teacher.

nubbins 6y ago
I think the key here is the analogy is just a warm up for the actual full detailed explanation. Where I find analogies always fail is in journalistic tech books that just use analogies to explain concepts but never get down to low level details and leave the reader with a vague feeling of understanding something, but if you asked them about any of the implications of the model explained they can't state anything.

sn41 6y ago

Negative numbers were hardly distrusted in Indian mathematics - Brahmagupta in the 6th Century just asked called a negative number as "debt". (see for example, [1]) The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact. [1] https://nrich.maths.org/5961 https://nrich.maths.org/5961

thunderbong 6y ago
The 'negative' on a number line is just a visualization. If you instead, think of the 'negation' as direction (as in a vector), it becomes easier to understand why multiplying two negative numbers makes the answer positive (You've basically turned around twice). I actually prefer this visualization rather than the one with a number line with numbers going to the left of 0.
- sn41 6y ago
  Thanks, this was helpful.
  - mkl 6y ago
    To better understand this, and see how complex numbers can be viewed the same way, go through this article, especially up to the first set of slides: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/
deleted 6y ago
[deleted]
cyphar 6y ago
> The real question is why negative times negative is positive - I had this doubt in school. I am a bit ashamed to admit that I still don't know any "from first principle explanation" for this fact. Don't sweat it, here's a rough proof. First a proof that a negative times a positive is negative: a*0 = 0 [Assumed but you can prove this too.] a*(b-b) = 0 [Definition of 0.] a*b + a*(-b) = 0 [Distributive law.] For any X, only its inverse -X has the property that X + -X = 0. Therefore a×(-b) is the inverse of a×b [which is -(a×b)]. Then apply the same idea but with a negative a: (-a)*0 = 0 [Assumed but you can prove this too.] (-a)*(b-b) = 0 [Definition of 0] (-a)*b + (-a)*(-b) = 0 [Distributive law.] -(a*b) + (-a)*(-b) = 0 [From above.] (-a)*(-b) + -(a*b) = 0 [Commutative addition to rearrange to X + -X = 0 form.] And as before, only the inverse has this property so -(a×b) must be the inverse of (-a)×(-b) -- which is a×b! So a negative number times a negative number is positive. QED. In case you're interested, the proof that the additive element is a multiplicative absorbing element (a×0 = 0, for all a) is also fairly trivial[1]. The above does somewhat implicitly assume that 0 and -0 are equal (or at least that (-a)×0 = (a)×0 = 0, which implies that 0 and -0 are equal). [1]: https://en.wikipedia.org/wiki/Additive_identity#The_additive_identity_annihilates_ring_elements https://en.wikipedia.org/wiki/Additive_identity#The_additive...

zaptheimpaler 6y ago

Books exist. A well-written book usually condenses years and years of knowledge and experience into a few hundred pages. Honestly the modern trend of having to make all educational content always engaging, with immediate rewards at all times seems wrong to me. Sometimes, its better to just chug through a book and boring formalization for 50 hours until you get to a point of clarity, because getting to the same knowledge via dozens of half-complete blogs and online sources takes MUCH more time in the long run. Books assume you will read the whole thing, so they build up the mental framework "breadth-first", so it seems less useful until you get to a sufficiently high depth. Once you get over the hump though, its great. With quick online learning, they give you a few good "depth-first" dives, but leave the foundational "knowledge/concept tree" underdeveloped. Books aren't distracting, they aren't internet enabled and make it easy to just sit down and go through page by page. Online stuff is usually a great supplement if you miss some intuition but it hasn't quite replaced books. Everyone is looking for some way to make the fundamental act of paying attention and learning easier.. but paying attention IS difficult. That is why its valuable. </oldmanrant>

sn41 6y ago
Agree++. I really prefer the disconnected mode of a book. The feeling of insight dripping in after being steeped in one thing alone.
klibertp 6y ago
No. Books were the most cost-effective medium for transfering knowledge for a long time, but that's in the past. They were never the most efficient - a few hours of one on one chat with domain expert, possibly with a whiteboard (or, failing that, some sand and a stick) at hand, could easily be an equivalent of ten times as much time of reading a book. Now that there's negligible difference in distribution cost between a flat text and a full-blown multimedia presentation, sticking with text is simply a waste of time.
- zaptheimpaler 6y ago
  I agree that learning directly from an expert is much more efficient. A multimedia presentation can be just as good as a book or better. Its just that as of today, I haven't seen many websites that match the depth and completeness of books yet. 3blue1brown is awesome, many people are writing linear algebra ebooks, quantum country looks great for quantum computation but as of _now_, none are complete or comprehensive the way a textbook is.

qwerty456127 6y ago

What are some more sources where I can find things explained in this manner?

qwerty456127 6y ago

I understood nearly zero of all mathematics (beyond the elementary school stuff) before I've found betterexplained and some relevant YouTube videos. I could learn formulae and some typical solutions by heart to substitute values and solve some exam problems but hardly ever understood them. I felt school/college math pure waste of time as I didn't actually understand anything. Things like betterexplained were a blast that opened an entire new beautiful exciting world to me instantly.

penguin_booze 6y ago
Back when I was taught matrix multiplication, I was told matrix is "just a table of numbers", and in order to "multiply" matrix to another, you walk from left to right, and on the other matrix, you go top to bottom. And th... (muffled voices). I thought it was game invented to pracitce more integer multiplication and improve concentation. Later, during college, I see matrices again! I went, "hang on, I thought... that wasn't a game after all?!". Decades later I discover 3b1b and Pavel Grinfeld [1] videos. My fist emotion was anger: "This is what you told me 'just a table of numbers'?". What a cruel joke! [1] https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS14F4NxAZudSmv https://www.youtube.com/playlist?list=PLlXfTHzgMRUKXD88IdzS1..., https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMRULWJYthculb2QWEiZOkwTSU https://www.youtube.com/watch?v=I04GHpKc-WQ&list=PLlXfTHzgMR..., https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5 https://www.youtube.com/watch?v=odV3oJOpE8s&list=PLlXfTHzgMR...
- qwerty456127 6y ago
  Thank you very much for the links. Please share any other cool educational materials you know.