8 ms·

Math Intuition Cheatsheet

478 points by jgrodziski 12y ago

fenomas 12y ago
This is terrific. I'm mathy enough to know e to several places, and use it in programs occasionally, but if you'd asked me how it might be intuitively derived I'd have been at a loss. And that's just for something inside my comfort zone - the article on linear algebra is just as illuminating. This looks like it will be my commute reading for the week. But more than that it just makes me happy that it exists.
- kalid 12y ago
  Thanks, glad to hear you're enjoying it. I realized that until I could explain a concept intuitively, I didn't really understand it -- and it became a mission to find some analogy that really clicked for me. Hope it gives you some entertaining reading.
- manualwise 12y ago
  Same here. I always was super interested in mathematics and still am. To have this resource to learn about where it all "came from" and how it actually works is great. Will read it on my commute as well!
akrolsmir 12y ago
I'd read Yudkowsky's popular "An Intuitive Explanation of Bayes' Theorem" as well as a few others, but the article on this site made me feel like I actually understood it for the first time. The examples were clever, short, and simple, while still demonstrating how powerful Bayes' Theorem can be. The concise explanations here seem more helpful than semester-long university courses I've taken. I definitely look forward to exploring his other articles on math and programming.
- kalid 12y ago
  Glad it was helpful! There's a strange tendency in technical explanations to expand, expand, expand. A heuristic I have: Imagine you're writing a letter to yourself, before you started the course. There's no reason to waste your own time, just cover the key difficulties and how to overcome them, in simple language you would have understood. (Oh, this letter also costs $5 a word, so let's make them count!)
- deleted 12y ago
  [deleted]
adamwong246 12y ago
Sure could have used this site a couple years ago when I was in school!
bithive123 12y ago
I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque variable naming schemes, and arbitrary use of jargon where simpler terms would suffice. A friend of mine who is a physicist once complained to me that every time he had to install some scientific software package on his computer, he had to deal with a litany of arbitrary things which seemed to have nothing to do with his task. I countered with my experience learning math and joked that at least programmers are willing to occasionally refresh our idioms and notations to better reflect our mutual understanding.
- kalid 12y ago
  (Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little candle in the darkness (http://betterexplained.com/calculus/lesson-1 http://betterexplained.com/calculus/lesson-1) is to start with intuitive notions (X-Raying a pattern into parts, Time-Lapsing parts into a whole) and then gradually introducing the terms. Eventually, if people are interested, we can get into the theory (which is like getting into the Peano axioms of arithmetic, if you care to go that deep.) I think in math there's a tendency to express things in the lowest-level "machine code" we can. We need more comments and pseudocode outlines :).
  - JacobAldridge 12y ago
    Kalid, I just read your article on Prime numbers (not-quite randomly selected from the homepage). I just wanted to say how well written I found it - even beyond the technical side of things, your writing style, tone, and humility ("if I didn't mess it up") are fabulous.
Sharma 12y ago
Thanks for posting this! I just read a small portion of Integration and trust me, it all make sense now!
Mistral 12y ago
Super, but no induction!
NoPiece 12y ago
I really like this cheatsheet, it is very inspiring actually. You should consider doing a version written for elementary school aged kids. You need to get buy in while they are still young!
- kalid 12y ago
  Thanks! I'd love to do more elementary school topics. I still encounter revelations about things like basic counting (fencepost problem) and arithmetic. My meta goal is to raise people's standards for what it means to understand something. ("If it didn't click, I should keep looking for a better explanation." vs. "It didn't click, I'm no good at this.")
pawannitj 12y ago
Are there other resources like this on Math?
omnibrain 12y ago
Those explanations remind me of Feynman's explanation of elliptical orbits.
curiousDog 12y ago
This is gold. Please do point to more resources like this.
ithinkso 12y ago
I like the idea but I personally really enjoy and prefer 'math english'. At first it may seems odd but after a while you get used to it and it's actually easier, you just scan through epsilons, deltas knowing exactly what they mean. Intuition has edge-case scenarios, double-meanings and almost-truths. I don't know why, but I think intuition (pseudocode math as someone had named it) gives a little bit of a 'illusion' of knowledge. You think you got it but are you really?
- kalid 12y ago
  Great point. I use the word "intuition" but feel it might be overloaded, things with intuitive knowledge, psychics, going with your gut despite the facts, etc. The sense I mean is closer to "grokking" or "clicking", when symbols and definitions actually mean something. You start thinking with the symbols, not despite them. You actually look forward to checking your understanding with real problems, because when it clicks the problems become easy :).
RobinL 12y ago
This September, I'm starting tutoring A level maths to underprivileged kids for a charity after helping them with GCSEs last year. This looks like it will be really helpful. Thanks!
- kalid 12y ago
  That's awesome! I hope you can make use of the site. One goal for the site is to provide "ammo" to teachers who are looking for other ways to present difficult concepts.
gajomi 12y ago
What an excellent collection of articles (well, I only ready a few, but I would like to think my experience extrapolates to the rest)! But I don't think "cheatsheet" is really the right word. Really this is a collection of short expository writings on particular mathematical concepts. And very well done, I think, to the point that calling it a "cheatsheet" (which I associate with a haphazardly conjoined sequence of facts and tricks out of context) is to do it a disservice. Or maybe the name choice has some marketing value?
- kalid 12y ago
  Glad you enjoyed them so far :). "Cheatsheet" might not be the best term for the summary page, perhaps "quick reference". I picked cheatsheet because it's a little more approachable than "reference", which implies something formal (and perhaps offputting).
jgrodziski 12y ago
For me the best one from Kalid is the explanation of the fourier transform: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... This article is simply amazing, moreover Kalid is an humble person and references his inspirations from several people over the web. Thank you very much Kalid for the time you dedicated to share all this wonderful resources.
- kalid 12y ago
  Thanks so much, and for originally submitting the cheatsheet to HN. (I hope it helps people untangle concepts that confused me for literally a decade.) I really appreciate the encouragement and hope to keep cranking as long as I can.
arianvanp 12y ago
I started my 1st year in college doing both math and computer science majors. I really love math but I just hate how (especially calculus and linear algebra courses) they just don't work on growing any intuition at all. in contrast, in my CS major we had a linear algebra class (ugh overlapping classes, huh) as well and we used computer graphics as a case study to learn about the basic concepts which was really fun and enlightening. Eventually I just ragequit math because I didn't have the patience and time to search for the intuition myself, which is a pitty. I'm really happy I followed the classes about mathematical proofs, groups etc because I really enjoyed those but others were just painful. Anyhow recently I discovered this blog too up my math concerning computer science is this: http://jeremykun.com/ http://jeremykun.com/. It's a tad more advanced than this but I'd really recommend it. His blog is freaking amazing and always a joy to read. Math as a sidehobby it is! :)
- j2kun 12y ago
  A lot of mathematics people feel this way about programming, too. Both sides expect one to devote the time to build intuition to grok their subject. I think it's easier to learn programming coming from the mathematics side, because you're going from nobody holding your hand to a compiler giving you error messages. You can't interactively test whether your intuition in mathematics is sound (you can, but you have to play both the role of the compiler and the tester).
factsfinder 12y ago
Seems like there are a lot of things i need to learn in maths
agumonkey 12y ago
I love the content, just wishing for some of their views on explaining electronic circuit design.
FlyingLawnmower 12y ago
Great work. Please keep adding examples!
FlyingLawnmower 12y ago
Great work. Please keep adding examples!

478 points by jgrodziski 12y ago

This is terrific. I'm mathy enough to know e to several places, and use it in programs occasionally, but if you'd asked me how it might be intuitively derived I'd have been at a loss. And that's just for something inside my comfort zone - the article on linear algebra is just as illuminating. This looks like it will be my commute reading for the week. But more than that it just makes me happy that it exists.

kalid 12y ago
Thanks, glad to hear you're enjoying it. I realized that until I could explain a concept intuitively, I didn't really understand it -- and it became a mission to find some analogy that really clicked for me. Hope it gives you some entertaining reading.
manualwise 12y ago
Same here. I always was super interested in mathematics and still am. To have this resource to learn about where it all "came from" and how it actually works is great. Will read it on my commute as well!

akrolsmir 12y ago

I'd read Yudkowsky's popular "An Intuitive Explanation of Bayes' Theorem" as well as a few others, but the article on this site made me feel like I actually understood it for the first time. The examples were clever, short, and simple, while still demonstrating how powerful Bayes' Theorem can be. The concise explanations here seem more helpful than semester-long university courses I've taken. I definitely look forward to exploring his other articles on math and programming.

kalid 12y ago
Glad it was helpful! There's a strange tendency in technical explanations to expand, expand, expand. A heuristic I have: Imagine you're writing a letter to yourself, before you started the course. There's no reason to waste your own time, just cover the key difficulties and how to overcome them, in simple language you would have understood. (Oh, this letter also costs $5 a word, so let's make them count!)
deleted 12y ago
[deleted]

adamwong246 12y ago

Sure could have used this site a couple years ago when I was in school!

bithive123 12y ago

I am often saddened by how much easier it has been for me to acquire computer science versus mathematical literacy. Embarrassingly, years after a poorly-timed calculus course left me thinking I had to be able to prove the central limit theorem in order to use calculus, the biggest barriers have turned out to be things programming as a discipline has learned to avoid; encouraging varied and/or terse notations, opaque variable naming schemes, and arbitrary use of jargon where simpler terms would suffice. A friend of mine who is a physicist once complained to me that every time he had to install some scientific software package on his computer, he had to deal with a litany of arbitrary things which seemed to have nothing to do with his task. I countered with my experience learning math and joked that at least programmers are willing to occasionally refresh our idioms and notations to better reflect our mutual understanding.

kalid 12y ago
(Kalid from BetterExplained here) Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough." My little candle in the darkness (http://betterexplained.com/calculus/lesson-1 http://betterexplained.com/calculus/lesson-1) is to start with intuitive notions (X-Raying a pattern into parts, Time-Lapsing parts into a whole) and then gradually introducing the terms. Eventually, if people are interested, we can get into the theory (which is like getting into the Peano axioms of arithmetic, if you care to go that deep.) I think in math there's a tendency to express things in the lowest-level "machine code" we can. We need more comments and pseudocode outlines :).
- JacobAldridge 12y ago
  Kalid, I just read your article on Prime numbers (not-quite randomly selected from the homepage). I just wanted to say how well written I found it - even beyond the technical side of things, your writing style, tone, and humility ("if I didn't mess it up") are fabulous.

Sharma 12y ago

Thanks for posting this! I just read a small portion of Integration and trust me, it all make sense now!

Mistral 12y ago

Super, but no induction!

NoPiece 12y ago

I really like this cheatsheet, it is very inspiring actually. You should consider doing a version written for elementary school aged kids. You need to get buy in while they are still young!

kalid 12y ago
Thanks! I'd love to do more elementary school topics. I still encounter revelations about things like basic counting (fencepost problem) and arithmetic. My meta goal is to raise people's standards for what it means to understand something. ("If it didn't click, I should keep looking for a better explanation." vs. "It didn't click, I'm no good at this.")

pawannitj 12y ago

Are there other resources like this on Math?

omnibrain 12y ago

Those explanations remind me of Feynman's explanation of elliptical orbits.

curiousDog 12y ago

This is gold. Please do point to more resources like this.

ithinkso 12y ago

I like the idea but I personally really enjoy and prefer 'math english'. At first it may seems odd but after a while you get used to it and it's actually easier, you just scan through epsilons, deltas knowing exactly what they mean. Intuition has edge-case scenarios, double-meanings and almost-truths. I don't know why, but I think intuition (pseudocode math as someone had named it) gives a little bit of a 'illusion' of knowledge. You think you got it but are you really?

kalid 12y ago
Great point. I use the word "intuition" but feel it might be overloaded, things with intuitive knowledge, psychics, going with your gut despite the facts, etc. The sense I mean is closer to "grokking" or "clicking", when symbols and definitions actually mean something. You start thinking with the symbols, not despite them. You actually look forward to checking your understanding with real problems, because when it clicks the problems become easy :).

RobinL 12y ago

This September, I'm starting tutoring A level maths to underprivileged kids for a charity after helping them with GCSEs last year. This looks like it will be really helpful. Thanks!

kalid 12y ago
That's awesome! I hope you can make use of the site. One goal for the site is to provide "ammo" to teachers who are looking for other ways to present difficult concepts.

gajomi 12y ago

What an excellent collection of articles (well, I only ready a few, but I would like to think my experience extrapolates to the rest)! But I don't think "cheatsheet" is really the right word. Really this is a collection of short expository writings on particular mathematical concepts. And very well done, I think, to the point that calling it a "cheatsheet" (which I associate with a haphazardly conjoined sequence of facts and tricks out of context) is to do it a disservice. Or maybe the name choice has some marketing value?

kalid 12y ago
Glad you enjoyed them so far :). "Cheatsheet" might not be the best term for the summary page, perhaps "quick reference". I picked cheatsheet because it's a little more approachable than "reference", which implies something formal (and perhaps offputting).

jgrodziski 12y ago

For me the best one from Kalid is the explanation of the fourier transform: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... This article is simply amazing, moreover Kalid is an humble person and references his inspirations from several people over the web. Thank you very much Kalid for the time you dedicated to share all this wonderful resources.

kalid 12y ago
Thanks so much, and for originally submitting the cheatsheet to HN. (I hope it helps people untangle concepts that confused me for literally a decade.) I really appreciate the encouragement and hope to keep cranking as long as I can.

arianvanp 12y ago

I started my 1st year in college doing both math and computer science majors. I really love math but I just hate how (especially calculus and linear algebra courses) they just don't work on growing any intuition at all. in contrast, in my CS major we had a linear algebra class (ugh overlapping classes, huh) as well and we used computer graphics as a case study to learn about the basic concepts which was really fun and enlightening. Eventually I just ragequit math because I didn't have the patience and time to search for the intuition myself, which is a pitty. I'm really happy I followed the classes about mathematical proofs, groups etc because I really enjoyed those but others were just painful. Anyhow recently I discovered this blog too up my math concerning computer science is this: http://jeremykun.com/ http://jeremykun.com/. It's a tad more advanced than this but I'd really recommend it. His blog is freaking amazing and always a joy to read. Math as a sidehobby it is! :)

j2kun 12y ago
A lot of mathematics people feel this way about programming, too. Both sides expect one to devote the time to build intuition to grok their subject. I think it's easier to learn programming coming from the mathematics side, because you're going from nobody holding your hand to a compiler giving you error messages. You can't interactively test whether your intuition in mathematics is sound (you can, but you have to play both the role of the compiler and the tester).

factsfinder 12y ago

Seems like there are a lot of things i need to learn in maths

agumonkey 12y ago

I love the content, just wishing for some of their views on explaining electronic circuit design.

FlyingLawnmower 12y ago

Great work. Please keep adding examples!

FlyingLawnmower 12y ago

Great work. Please keep adding examples!