I love projects like these. Even when I took algebra and calculus in university, it’s good to refresh and go deeper into the concepts many years later.
However, a small critique to the author: the audience of this book is not clear. It says “basic” math, but then in chapter 1, the group's explanation starts with this sentence: “The additive group of integers (Z,+) and the cyclic group Z/Zm.”
Maybe it was a draft note. To be fair the paragraphs that follow attempt a more basic explanation of groups, but even my “Algebra I” book at the university was friendlier than that.
Really 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 occcur with multivariable calculus, so here I would like to set the record straight"
Math education and textbooks are doing an awesome job obscuring simple ideas by focusing on weird details and bad notation. Always good to see people trying to counter this :)
If you just pick one of those subjects, you'll probably find a textbook just as long as his entire PDF trying to cover 13+ subjects.
Sorry to be negative Nancy over here, but you're going to need more than 54 pages to cover calculus. There is value in organizing the major theorems in the different disciplines. But, to be honest, this doesn't really serve the beginner.
1. I don't think it is at all intended to serve the beginner.
It's geared towards readers wait a reasonable amount of mathematical maturity already (it explicitly says that in the landing page).
2. Many, many of the pages of most introductory calculus textbooks are spent on exercises and on the specifics of computing integrals and derivatives of particular functions - none of this is necessary to understand the concepts themselves.
For example, Baby Rudin (the standard textbook for Analysis for math majors) covers Sequences, Series, Continuity, Differentiation, and the Riemann integral in less than 100 pages (including exercises).
The author’s doing themselves a disservice by using the word “basic” - it doesn’t describe either the mathematics or the description. Perhaps it refers to its focus on the basics of a field.
The actual website never says "Basic Math Textbook", only the submitter typed that in the title here on HN, I guess because "An Infinitely Large Napkin" or "The Napkin Project" would sound ambiguous without a topic context.
"As explained in the preface, the main prerequisite is some amount of mathematical
maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on."
Yeah, that's way beyond what's called basic math instruction, e. g. in schools. A more specific, as in accurate, subtitle (or description) is in order.
The preface has "I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds."
Apparently the author tried to somewhat expand the audience from that, but to me it seems still mostly appropriate for smart high schoolers who have heard some pieces of lore from friends about these topics, but they can't put that puzzle in order in their minds yet.
It's most definitely not aimed at the average student. You need to be highly curious, motivated and find math fun already.
And I think that's a perfectly fine thing. It's great to have books for that kind of audience.
It would make more sense to include the term "higher math" (from the author's own description) in the page title, like "Basic Higher Math Textbook" or "Introductory Higher Math Textbook".
Higher mathematics isn't necessarily very strictly defined anyway, but I guess most people who've heard the term would apply it to branches of math that are developed using formal definitions and at least moderately rigorous proofs, and that usually aim at a level of generality beyond their originally motivating examples.
> that's way beyond what's called basic math instruction, e. g. in schools
I'm not saying you're wrong, I know for a fact that you aren't: unfortunately most high-school students fall extremely short of that bar, but it's not necessarily that way. Many teenagers can and do develop that kind of mathematical maturity.
In this context "basic" means "it doesn't require knowledge in the field", and by and large this book can indeed be followed with no other requirement than the mathematical maturity it talks about. Many classic books self-describe in similar way.
I think it's not just some kind of humblebrag. I know this trope that college students feel like it says it's intro but it's hard so it's not an intro. But you only think this when you don't know the topic well. The "thing itself" is in the journals, at the conferences, and in the professional work of researchers, and (if applicable) the real-world applications of the content in various contexts. Any normal-sized book can really only be an introduction to all that for most topics taught in undergrad or master's level.
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?
> Math Academy is an AI-powered, fully-automated online math-learning platform. Math Academy meets each student where they are via an adaptive diagnostic assessment and introduces and reinforces concepts based on each student’s individual strengths and weaknesses.
What is edutech and why is it unsuitable?
I don't want a computer in the loop when I learn math, plain and simple. My preferred style of learning is instructor led with a mix of Socratic method and hand holding. But bar that, reading texts and using a pen and paper.
As far as I can tell, most of its value comes from having a reasonably thorough dependency tree of math topics and corresponding exercises (which can be solved with pen and paper) and describing it as "AI" is how you get investors to fund a math textbook.
My experience with MathAcademy is very positive. So is my experience using ChatGPT 5 as a math teacher in learning mode. I'm as fed up with AI slop as most people, but for me this is a domain where it excels.
Need to see how this looks on my Kindle Scribe --- I suspect that it will push me over to updating to the newly announced colour model when it becomes available.
> The set ℕ is the set of positive integers, not including 0.
Hell yeah!
I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.
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, ... }
There is no approach that uses "natural language" and is universal, and being aware of this is both frustrating and useful. Whether it is important is up to the individual.
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 is actually also not inconvenient if you don't use static types.
I love projects like these. Even when I took algebra and calculus in university, it’s good to refresh and go deeper into the concepts many years later.
However, a small critique to the author: the audience of this book is not clear. It says “basic” math, but then in chapter 1, the group's explanation starts with this sentence: “The additive group of integers (Z,+) and the cyclic group Z/Zm.” Maybe it was a draft note. To be fair the paragraphs that follow attempt a more basic explanation of groups, but even my “Algebra I” book at the university was friendlier than that.
Really 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 occcur with multivariable calculus, so here I would like to set the record straight"
Math education and textbooks are doing an awesome job obscuring simple ideas by focusing on weird details and bad notation. Always good to see people trying to counter this :)
If you just pick one of those subjects, you'll probably find a textbook just as long as his entire PDF trying to cover 13+ subjects.
Sorry to be negative Nancy over here, but you're going to need more than 54 pages to cover calculus. There is value in organizing the major theorems in the different disciplines. But, to be honest, this doesn't really serve the beginner.
Two thoughts here:
1. I don't think it is at all intended to serve the beginner.
It's geared towards readers wait a reasonable amount of mathematical maturity already (it explicitly says that in the landing page).
2. Many, many of the pages of most introductory calculus textbooks are spent on exercises and on the specifics of computing integrals and derivatives of particular functions - none of this is necessary to understand the concepts themselves.
For example, Baby Rudin (the standard textbook for Analysis for math majors) covers Sequences, Series, Continuity, Differentiation, and the Riemann integral in less than 100 pages (including exercises).
The author’s doing themselves a disservice by using the word “basic” - it doesn’t describe either the mathematics or the description. Perhaps it refers to its focus on the basics of a field.
The actual website never says "Basic Math Textbook", only the submitter typed that in the title here on HN, I guess because "An Infinitely Large Napkin" or "The Napkin Project" would sound ambiguous without a topic context.
From the books advice corner:
"As explained in the preface, the main prerequisite is some amount of mathematical maturity. This means I expect the reader to know how to read and write a proof, follow logical arguments, and so on."
Yeah, that's way beyond what's called basic math instruction, e. g. in schools. A more specific, as in accurate, subtitle (or description) is in order.
The preface has "I initially wrote this book with talented high-school students in mind, particularly those with math-olympiad type backgrounds."
Apparently the author tried to somewhat expand the audience from that, but to me it seems still mostly appropriate for smart high schoolers who have heard some pieces of lore from friends about these topics, but they can't put that puzzle in order in their minds yet.
It's most definitely not aimed at the average student. You need to be highly curious, motivated and find math fun already.
And I think that's a perfectly fine thing. It's great to have books for that kind of audience.
It would make more sense to include the term "higher math" (from the author's own description) in the page title, like "Basic Higher Math Textbook" or "Introductory Higher Math Textbook".
Higher mathematics isn't necessarily very strictly defined anyway, but I guess most people who've heard the term would apply it to branches of math that are developed using formal definitions and at least moderately rigorous proofs, and that usually aim at a level of generality beyond their originally motivating examples.
> that's way beyond what's called basic math instruction, e. g. in schools
I'm not saying you're wrong, I know for a fact that you aren't: unfortunately most high-school students fall extremely short of that bar, but it's not necessarily that way. Many teenagers can and do develop that kind of mathematical maturity.
In this context "basic" means "it doesn't require knowledge in the field", and by and large this book can indeed be followed with no other requirement than the mathematical maturity it talks about. Many classic books self-describe in similar way.
That's common with mathematics books. Weil's Basic Number Theory is enough to give the unsuspecting quite the fright, despite the name
It follows a good tradition of textsbooks in STEM - is it starts with "Introduction to..." it is neither short or simple.
I think it's not just some kind of humblebrag. I know this trope that college students feel like it says it's intro but it's hard so it's not an intro. But you only think this when you don't know the topic well. The "thing itself" is in the journals, at the conferences, and in the professional work of researchers, and (if applicable) the real-world applications of the content in various contexts. Any normal-sized book can really only be an introduction to all that for most topics taught in undergrad or master's level.
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?
Subscription to Math Academy might be more suitable for that.
Red flags of Math Academy:
- Centred around AI
- Seems geared around edutech (which is what I gather from the site)
Green flags for Napkin:
- Covers advanced undergraduate and graduate topics
- Encourages pencil & paper way of learning (took me way too long to learn this is the best appraoch)
> Centred around AI
Where do you see the centered around AI? I have used it a lot and have not touched a single subject around AI.
> - Seems geared around edutech (which is what I gather from the site)
What is edutech and why is it unsuitable?
Finally, have you _used_ MathAcademy at all?
Where do you see the centered around AI?
From https://www.mathacademy.com/how-it-works:
> Math Academy is an AI-powered, fully-automated online math-learning platform. Math Academy meets each student where they are via an adaptive diagnostic assessment and introduces and reinforces concepts based on each student’s individual strengths and weaknesses.
What is edutech and why is it unsuitable?
I don't want a computer in the loop when I learn math, plain and simple. My preferred style of learning is instructor led with a mix of Socratic method and hand holding. But bar that, reading texts and using a pen and paper.
Finally, have you _used_ MathAcademy at all?
Nope, doesn't look like my cup of tea.
As far as I can tell, most of its value comes from having a reasonably thorough dependency tree of math topics and corresponding exercises (which can be solved with pen and paper) and describing it as "AI" is how you get investors to fund a math textbook.
See also How Math Academy Creates its Knowledge Graph https://www.justinmath.com/how-math-academy-creates-its-know... "We do it manually, by hand."
My experience with MathAcademy is very positive. So is my experience using ChatGPT 5 as a math teacher in learning mode. I'm as fed up with AI slop as most people, but for me this is a domain where it excels.
i will sequeze in real Analysis between complex analysis and measure theory.
Previous discussion:
https://news.ycombinator.com/item?id=20168936
Need to see how this looks on my Kindle Scribe --- I suspect that it will push me over to updating to the newly announced colour model when it becomes available.
> The set ℕ is the set of positive integers, not including 0.
Hell yeah!
I've agonised over this quite a lot over the decades. Not including 0 is more intuitive, but including 0 is more convenient. Of course, both approaches are correct. My main reason for not including 0 is that I hate seeing sequences numbered starting with 0.
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, ... }
There is no approach that uses "natural language" and is universal, and being aware of this is both frustrating and useful. Whether it is important is up to the individual.
> I believe that in France (for example) zero is regarded as both positive and negative.
That would cause all kinds of problems, so I'd be pretty surprised if it turned out to be true.
I note that this is the heading of the relevant wikipedia page:
> Un nombre négatif est un nombre réel qui est inférieur à zéro, comme −3 ou −π.
( https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif )
It'd be hard to be more explicit that zéro is not a negative number.
I didn't know that. In French textbooks, I believe ℕ always includes 0. I didn't even know that not including it was another possible convention.
I never write ℕ, for exactly this reason. I write ℤ with a subscript ">0" or ">=0". Doesn't take up much more space, and completely unambiguous.
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 is actually also not inconvenient if you don't use static types.
On the other hand, even for writing a perfectly fine natural number like "10", you need the zero... Maybe it is just ℕ and ℤ after all.
And round we go.
It's that Evan Chen. Thanks for teaching me the way of the bary, senpai!