I have been looking for a general all around math text since last century (as an amateur / recreational mathematician). I m starting to look at this. It seems to cover lots of ground. Any observations?
I wrote one: https://pimbook.org
Basic Math Textbook: The Napkin Project
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Re: Basic Math Textbook: The Napkin Project
#62Re: Basic Math Textbook: The Napkin Project
#63Really cool! This is the sorta thing that, just yesterday, I wished existed. And it's already on the HN frontpage. It's hard to see the forest for the trees in many math books, a bird's eye view is a really valuable perspective. I highly appreciate this approach: "As i have ranted about before, linear algebra is done wrong by the extensive use of matrices to obscure the structure of a linear map. Similar problems occ…
Sheldon Axler's book is the common (now decades old) example of a book doing linear maps first.
Re: Basic Math Textbook: The Napkin Project
#64Earlier quoted context omitted.
From a technical perspective you frequently need 0 in there. From a pure convenience perspective, it doesn't make sense to assign ℕ to the positive integers when they're already called ℤ⁺. Now you have two convenient names for the smaller set and none for the larger set.
By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there. Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers. Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It i…
What do you use for the negative integers?
Re: Basic Math Textbook: The Napkin Project
#65I would strongly recommend getting, and working through Serge Lang's book "Basic Mathematics" for people who want to self-study what is normally considered "basic maths" (ie the stuff you might have covered in high school plus some of what in the US is called "college algebra" (in the UK and Europe that is just covered in high school and "algebra" at university generally means abstract algebra. I did it to get my ver…
Thank you for the recommendation, that sounds much more like my level at the moment!
Re: Basic Math Textbook: The Napkin Project
#66Earlier quoted context omitted.
I used to write and review problems for math competitions. This is why we avoided saying "natural numbers". We used "nonnegative integers" or "positive integers" instead.
You need to be careful about this ... I believe that in France (for example) zero is regarded as both positive and negative. So in France: Non-negative integers: 1, 2, 3, 4, 5, ... Positive integers: 0, 1, 2, 3, 4, 5, ... Similarly, for some countries "Whole Numbers" is equivalent to all the integers, while in other countries it's the set { 0, 1, 2, 3, 4, ... } while in still other countries it's { 1, 2, 3, 4, ... }…
Presumably, GP only worked on the problems in English and someone else would translate it appropriately.
Re: Basic Math Textbook: The Napkin Project
#67Earlier quoted context omitted.
So this is aimed at somebody who has mathematical maturity but prefers... less content and detail? The point is that you are losing something in a shortened presentation. You're not just losing "unnecessary exercises" as you put it.
As I said, intro calculus books will spend a large amount of time teaching you the mechanics of finding closed form solutions for integrals and derivatives of various kinds of functions. Look at https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/pa... for an example. Most of that content is not that important to understand the concepts. And yes, with more mathematical maturity you definitely don't need as much…
My first calculus class in high school was about 10% "conceptual explanation of limits, derivatives, and integrals", 30% "techniques for evaluating derivatives", 50% "techniques for evaluating integrals", and maybe another 10% (or less) "justifications of the correctness of those techniques". (I guess I'm putting the Fundamental Theorem of Calculus in the the last 10% here.)
The style of this textbook does seem to primarily skip the "techniques for evaluating" stuff, on the basis that you just wanted to understand what each branch of mathematics is about and what kinds of theorems it has that might relate to the larger edifice of mathematics.
Re: Basic Math Textbook: The Napkin Project
#68Presumably started before Evan Chen's recent discovery of Typst.
Re: Basic Math Textbook: The Napkin Project
#69Earlier quoted context omitted.
I submitted it, and the word “basic” is mine, because the author doesn’t really go deep into what I would consider “advanced” mathematics. It can be a good prerequisite for advanced things, though.
"undergrad math" might be a better phrase to use; "basic" and "advanced" mean very different things to people with different backgrounds
As elsewhere in the thread, I'd advocate for "basic higher mathematics" or "introductory higher mathematics" (which would make clear that it's for people actively studying math as a subject and not as a standard part of primary or secondary education, or a prerequisite in an engineering major or something).
The author says that this is largely aimed at high school students who are doing self-study, which is a realistic audience but not a context where a lot of people would naturally apply the word "basic". But this material is basic for mathematicians, I guess (although even a lot of mathematicians may not have quite as broad a knowledge of mathematics as the author does!).
Re: Basic Math Textbook: The Napkin Project
#70Earlier quoted context omitted.
By convenience I mean "convenient from a technical perspective", and yes, you often need 0 in there. Your other argument doesn't make much sense. I learnt both in school and at university ℕ, ℕ₀, and ℤ as THE symbols for the natural numbers, the natural numbers including 0, and the whole numbers. Fuck convenience. ℕ, ℕ₀, and ℤ it is :-) It is just so much prettier (ℤ⁺ is a really ugly symbol for such a nice set). It i…
What do you use for the negative integers?
I very rarely use just the negative integers, so I don't need a symbol for it.