[HN Gopher] Basic Math Textbook: The Napkin Project
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Basic Math Textbook: The Napkin Project
Author : eapriv
Score : 203 points
Date : 2025-10-02 23:46 UTC (3 days ago)
(HTM) web link (web.evanchen.cc)
(TXT) w3m dump (web.evanchen.cc)
| tocs3 wrote:
| 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?
| _hao wrote:
| Subscription to Math Academy might be more suitable for that.
| commandersaki wrote:
| 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)
| ptsneves wrote:
| > 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?
| commandersaki wrote:
| _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.
| delichon wrote:
| 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.
| yorwba wrote:
| 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."
| qwertytyyuu wrote:
| 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
| barrenko wrote:
| 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"
| BeetleB wrote:
| Try the Princeton Companion.
| j2kun wrote:
| +1 this is a great reference text
| eapriv wrote:
| 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.
| j2kun wrote:
| I wrote one: https://pimbook.org
| nxobject wrote:
| 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.
| spankibalt wrote:
| 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.
| schoen wrote:
| 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.
| qsort wrote:
| > 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.
| avdelazeri wrote:
| That's common with mathematics books. Weil's Basic Number
| Theory is enough to give the unsuspecting quite the fright,
| despite the name
| bonoboTP wrote:
| 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.
| avdelazeri wrote:
| True. There's Morita's a mathematical gift for the same
| audience
| stared wrote:
| It follows a good tradition of textsbooks in STEM - is it
| starts with "Introduction to..." it is neither short or simple.
| bonoboTP wrote:
| 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.
| seanhunter wrote:
| 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.
| bonoboTP wrote:
| 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.
| sota_pop wrote:
| "The proof is self-evident, and been left as an exercise for
| the reader."
| eapriv wrote:
| 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.
| dooglius wrote:
| "undergrad math" might be a better phrase to use; "basic" and
| "advanced" mean very different things to people with
| different backgrounds
| schoen wrote:
| 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!).
| auggierose wrote:
| > 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.
| qsort wrote:
| 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.
| ColinWright wrote:
| 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.
| thaumasiotes wrote:
| > 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 negatif_ est un nombre reel qui est inferieur
| a zero, comme -3 ou -p.
|
| ( https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif )
|
| It'd be hard to be more explicit that zero is not a
| negative number.
| ColinWright wrote:
| If you're going to quote wikipedia:
|
| > _" Zero est le seul nombre qui est a la fois reel,
| positif, negatif et imaginaire pur."_
|
| From: https://fr.wikipedia.org/wiki/Z%C3%A9ro#Propri.C3.A
| 9t.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
| "inferieur" 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 "superieur a 0", and
| "superieur a " means "greater than or equal to".
| Similarly, "negative" means "inferieur a 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/1153275891
| 164575...
| dooglius wrote:
| Presumably, GP only worked on the problems in English and
| someone else would translate it appropriately.
| thaumasiotes wrote:
| 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.
| auggierose wrote:
| 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.
| auggierose wrote:
| 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.
| thaumasiotes wrote:
| What do you use for the negative integers?
| gjm11 wrote:
| 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.
| rossant wrote:
| 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.
| qsort wrote:
| It's _that_ Evan Chen. Thanks for teaching me the way of the
| bary, senpai!
| WillAdams wrote:
| 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.
| loose-cannon wrote:
| 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.
| morcus wrote:
| 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).
| loose-cannon wrote:
| 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.
| bonoboTP wrote:
| 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.
| morcus wrote:
| 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.
| schoen wrote:
| 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.
| zozbot234 wrote:
| 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.
| aap_ wrote:
| 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 :)
| j2kun wrote:
| Sheldon Axler's book is the common (now decades old) example of
| a book doing linear maps first.
| diegof79 wrote:
| 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.
| HelloNurse wrote:
| That is clearly a "note to self" that remained in the full
| text. The following paragraph has a regular definition of
| group.
| jackallis wrote:
| i will sequeze in real Analysis between complex analysis and
| measure theory.
| kace91 wrote:
| 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.
| rmonvfer wrote:
| UNED by any chance? Broken indeed
| kace91 wrote:
| Bingo. I'm guessing you went through the same song and dance?
| qwertytyyuu wrote:
| 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
| seanhunter wrote:
| 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.
| dave7 wrote:
| Thank you for the recommendation, that sounds much more like my
| level at the moment!
| moi2388 wrote:
| 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!
| thibley wrote:
| 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.
| BeetleB wrote:
| For many people, learning a heavy topic like mathematics is a
| lot easier on paper than on a screen.
| thibley wrote:
| 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.
| TRiG_Ireland wrote:
| Presumably started before Evan Chen's recent discovery of Typst.
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