4 ms·

Why didn't any one tell me about this book when I was younger! This is so good. :)

partisan 9y ago
As someone who struggled through calculus, this book would have made a huge difference for me. Just reading through the first few pages brought a smile to my face that someone could so plainly explain these critical concepts in such a familiar way. Why didn't my professors do that?
- sideproject 9y ago
  Totally agree with you. I did advanced research in computer science without fully grasping integrals - only if I had this book! But you're right, why can't things be like this? Then I think, there is a certain art to having the ability to explain something so easily.. props to reddit's elif subreddit (https://www.reddit.com/r/explainlikeimfive/ https://www.reddit.com/r/explainlikeimfive/)
- jacquesm 9y ago
  > Why didn't my professors do that? Because to them it's obvious. Most maths teachers are so far ahead of the students they forgot they once were students themselves. I've had 3 different ones in high school and the difference was incredible. All the way from 'only the best students learn anything' to 'everybody earns at least a passing grade'. Maths and physics were the classes where the quality difference between the teachers stood out the most.
  - taejo 9y ago
    > Because to them it's obvious. And to them it's wrong! Much is said in this book which is difficult (but not impossible) to rigourously justify. It took centuries for calculus to be placed on a rigourous mathematical foundation; this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. The presentation here (in particular the idea that dx² is negligible) can be made rigourous through "non-standard analysis", but this 20th century development is less-known (in fact, unknown to many mathematicians) and perhaps even more difficult than real analysis. Rigour is not necessary to understanding, and can even be fatal to understanding, but it's how mathematicians work, and generally mathematics is taught by mathematicians.
    - jacquesm 9y ago
      > in fact, unknown to many mathematicians > and generally mathematics is taught by mathematicians So that leaves the door wide-open to a whole army of mathematicians to who it is unknown that are teachers. And those are the ones for who all of the above does not apply and who still treat their 'entry level problems' as obvious to all their students even when the evidence strongly suggests that to the students it is not at all obvious. And I'm writing this as a high school kid who was pretty good in math but totally lost interest due to such teachers (that, and computers were far more interesting).
    - sundarurfriend 9y ago
      > this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. ... (in particular the idea that dx² is negligible) Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA: > dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ > But if we regard dθ as indefinitely small, then in the limit we may neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes: > dy = 2cosθ × 1/2 dθ This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
      - thanatropism 9y ago
        But this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "nonrigorous elementary algebra", it's a matter of remembering all notations are abbreviations and you have to know how to manipulate them. You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.
    - 9y ago

As someone who struggled through calculus, this book would have made a huge difference for me. Just reading through the first few pages brought a smile to my face that someone could so plainly explain these critical concepts in such a familiar way. Why didn't my professors do that?

sideproject 9y ago
Totally agree with you. I did advanced research in computer science without fully grasping integrals - only if I had this book! But you're right, why can't things be like this? Then I think, there is a certain art to having the ability to explain something so easily.. props to reddit's elif subreddit (https://www.reddit.com/r/explainlikeimfive/ https://www.reddit.com/r/explainlikeimfive/)
jacquesm 9y ago
> Why didn't my professors do that? Because to them it's obvious. Most maths teachers are so far ahead of the students they forgot they once were students themselves. I've had 3 different ones in high school and the difference was incredible. All the way from 'only the best students learn anything' to 'everybody earns at least a passing grade'. Maths and physics were the classes where the quality difference between the teachers stood out the most.
- taejo 9y ago
  > Because to them it's obvious. And to them it's wrong! Much is said in this book which is difficult (but not impossible) to rigourously justify. It took centuries for calculus to be placed on a rigourous mathematical foundation; this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. The presentation here (in particular the idea that dx² is negligible) can be made rigourous through "non-standard analysis", but this 20th century development is less-known (in fact, unknown to many mathematicians) and perhaps even more difficult than real analysis. Rigour is not necessary to understanding, and can even be fatal to understanding, but it's how mathematicians work, and generally mathematics is taught by mathematicians.
  - jacquesm 9y ago
    > in fact, unknown to many mathematicians > and generally mathematics is taught by mathematicians So that leaves the door wide-open to a whole army of mathematicians to who it is unknown that are teachers. And those are the ones for who all of the above does not apply and who still treat their 'entry level problems' as obvious to all their students even when the evidence strongly suggests that to the students it is not at all obvious. And I'm writing this as a high school kid who was pretty good in math but totally lost interest due to such teachers (that, and computers were far more interesting).
  - sundarurfriend 9y ago
    > this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. ... (in particular the idea that dx² is negligible) Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA: > dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ > But if we regard dθ as indefinitely small, then in the limit we may neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes: > dy = 2cosθ × 1/2 dθ This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
    - thanatropism 9y ago
      But this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "nonrigorous elementary algebra", it's a matter of remembering all notations are abbreviations and you have to know how to manipulate them. You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.
  - 9y ago