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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.

That is clearly a "note to self" that remained in the full text. The following paragraph has a regular definition of group.

Is there a way to get a nicely bound hardcopy ? Doing a single one-off is expensive, I wonder if a hundred people got togehter, would a larger run be more cost effective ? Are there services for those larger runs ?

Qwen3 recommends

Blurb Lulu BookBaby Mixam

For a 1000 page book, it suggest pricing of ≈$120 for single copies, down to $15-25 for a run of 1000.

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 :)

Sheldon Axler's book is the common (now decades old) example of a book doing linear maps first.

I 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 very rusty high-school maths back up to snuff before starting to self-study for a maths degree and it helped a lot. The problems are really excellent and since it's Serge Lang, he treats you like a mathematician right from the beginning even though he really is doing basic stuff.

Thank you for the recommendation, that sounds much more like my level at the moment!

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.

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.

True. There's Morita's a mathematical gift for the same 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.

This is such a common misunderstanding it's worth explaining.

If you get a book in stem called "an introduction to x" it isn't claiming to be short or simple at all. What "introduction" means is that it is intended for a first course in that topic (ie it does not have prerequisites within that topic).

So if I get "an introduction to mechanics" by Kleppner and Kolenkow[1] for example (to pick one off my bookshelf), it is a challenging first course in classical mechanics but it doesn't require you to know any mechanics before reading it.

[1] This is a really good book in my opinion btw.

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.

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.

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!).

“The proof is self-evident, and been left as an exercise for the reader.”

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?

While there are a lot of of textbooks flown around, I'd like to prop up ROB201 textbook, which I came across recently, also aims to cover a lot of ground and is accompanied by videos.

https://grizzle.robotics.umich.edu/education/rob201 - "ROB 201 Calculus for the Modern Engineer"

If I were to write such a text, it would have a lot more about building intuition for advanced mathematical concepts. This intuition is extremely valuable, but missing from almost all advanced-level texts. On the other hand, it’s very difficult to put into words, and probably quite personal.

I wrote one: https://pimbook.org

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."

The “ai” is an expert system yes to calibrate to your ability to answer questions it throws at you. The questions are all human written. I had your initial scepticism as well, I can reassure you that the ai is not an LLM. Also the guy Justin skycak who built it has put a lot of thought into its pedagogy

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.

Try the Princeton Companion.

+1 this is a great reference text

Presumably started before Evan Chen's recent discovery of Typst.

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).

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.

From the book

> Philosophy behind the Napkin approach

> As far as I can tell, higher math for high-school students comes in two flavors:

> • Someone tells you about the hairy ball theorem in the form “you can’t comb the hair on a spherical cat” then doesn’t tell you anything about why it should be true, what it means to actually “comb the hair”, or any of the underlying theory, leaving you with just some vague notion in your head.

> • You take a class and prove every result in full detail, and at some point you stop caring about what the professor is saying.

> Presumably you already know how unsatisfying the first approach is. So the second approach seems to be the default, but I really think there should be some sort of middle ground here. Unlike university, it is not the purpose of this book to train you to solve exercises or write proofs, or prepare you for research in the field. Instead I just want to show you some interesting math. The things that are presented should be memorable and worth caring about. For that reason, proofs that would be included for completeness in any ordinary textbook are often omitted here, unless there is some idea in the proof which I think is worth seeing. In particular, I place a strong emphasis over explaining why a theorem should be true rather than writing down its proof.

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 detail. The proofs get terser as you're expected to be able to fill out the more straightforward details yourself.

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.

I don't quite get how it's supposed to introduce calculus/analysis - the introductory chapters just start talking about metric spaces without even bothering to properly introduce the real numbers or their peoperties. I don't think that's quite sensible. For comparison, mathlib4 of course does it right by starting from topological spaces - and it manages to nicely simplify things throughout, by defining a basic "tends to" notion using set-theoretic filters.

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.

Presumably, GP only worked on the problems in English and someone else would translate it appropriately.

> 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.

If you're going to quote wikipedia:

> "Zéro est le seul nombre qui est à la fois réel, positif, négatif et imaginaire pur."

From: https://fr.wikipedia.org/wiki/Z%C3%A9ro#Propri.C3.A9t.C3.A9s...

It's hard to be more explicit that it is considered both.

================

Added in edit

In speaking with a French colleague, he says that "inférieur" often means "less-than-or-equal-to" rather than "strictly-less-than", so the passage you quote would still imply that 0 is negative (and most likely also positive).

================

Second edit:

> In France, "positive" means "supérieur à 0", and "supérieur à " means "greater than or equal to". Similarly, "négative" means "inférieur à 0", that is "less than or equal to 0".

> (We have the similar reaction towards the anglosaxon world and the introduction of nonnegative…)

-- https://mathstodon.xyz/@antoinechambertloir/1153275891164575...

Well, in any textbook I've read they at least defined ℕ in the beginning and then used e.g ℕ₀ to include 0 or ℕ⁺ to not include it.

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.

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.

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.

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.

I feel like “basic” and “light” might be an overstatement (or should I say understatement). Feels like the audience needs at least a 1 year in a maths tangential uni course

The content is great but static PDFs with minimal hyperlinking is a lost opportunity.

Learning and internalizing higher math is largely about connecting lots of ideas, terms, definitions, named theorems, lemmas, etc. If the book were instead built for the modern web stack with heavy use of tooltips, it would be lots more engaging and fun, supporting a more active learning process.

For many people, learning a heavy topic like mathematics is a lot easier on paper than on a screen.

Def true. I often mark up math papers and books with DIY-hyperlinks. It's very easy for me to skip over an important, foundational clause just because some term isn't immediately familiar, and if that happens frequently in some reading, then I'm mentally checking out.

For the Napkin book, if the underlying metadata were in the latex source, we could have PDF annotations in a sidebar, e.g., ("def: p.123, key application: p.234, ..."), as well as live tooltips for a modern web experience. That would be totally wonderful for this text and its audience.

For another approach at teaching math in an accessible (and self-teaching friendly) approach, I can’t recommend Jay Cummings enough.

I recently tried to go for a math degree in my free time using my countries’ remote learning option, and even though the attempt didn’t last long because the format is hopelessly broken (Mediterranean bureaucracy), I’m still engaging in self learning through his books and they’re an absolute goldmine.

Most basic math books assume no knowledge of the subject but a familiarity with general math that is unreasonable - it’s like saying you don’t need to know what a deadlift is but you need a back that resists 200kg… It’s a borderline fictional audience in practice.

Cummings manages to understand the novice far, far better.

UNED by any chance? Broken indeed

Bingo. I’m guessing you went through the same song and dance?

i will sequeze in real Analysis between complex analysis and measure theory.

It's that Evan Chen. Thanks for teaching me the way of the bary, senpai!

What a fantastic read. I’ve never had higher maths. Having read the first few pages, this perfectly fits my level of knowledge. It makes next paragraphs intuitive by using the remarks and asking me to think. I can’t wait to read more!

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