Basic Math Textbook: The Napkin Project

[diegof79]

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]

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

[aap_]

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]

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

[tocs3]

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?

[barrenko]

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"

[rramadass]

There is no one book which can give you the overall sweep of Mathematics. However, you might find the following (though not textbooks per se) useful.

1) The Princeton Companion to Mathematics by Timothy Gowers et al. and The Princeton Companion to Applied Mathematics by Nicholas Higham et al. - The closest you have to a Modern Encyclopedia of Mathematics. You get unmatched breadth after which you can move on to dedicated books as needed. Well worth the money.

2) Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov et al. - Absolutely brilliant overview of Basic Mathematics. Published by Dover and hence very affordable.

3) Elements of Mathematics: From Euclid to Gödel by John Stillwell - Written as sort of an update to the great Felix Klein's Elementary Mathematics from an Advanced Standpoint books. The Topics are particularly well chosen given modern advances; they include Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.

[eapriv]

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]

I wrote one: https://pimbook.org

[_hao]

Subscription to Math Academy might be more suitable for that.

[BeetleB]

Try the Princeton Companion.

[nxobject]

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]

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.

[stared]

It follows a good tradition of textsbooks in STEM - is it starts with "Introduction to..." it is neither short or simple.

[bonoboTP]

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.

[eapriv]

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.

[sota_pop]

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

[seanhunter]

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]

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

[thibley]

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]

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

[loose-cannon]

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]

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).

[zozbot234]

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.

[golem14]

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.

[golem14]

I can also get a coil binding machine for $50, and can print the book for maybe $15 and spend 1h printing and binding chapters…

Maybe cheap child labour is called for…

[auggierose]

> 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]

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.

[thaumasiotes]

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.

[gjm11]

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]

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.

[sureglymop]

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.

[WillAdams]

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.

[moi2388]

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!

[qwertytyyuu]

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

[kace91]

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]

UNED by any chance? Broken indeed

[mna_]

Try the Open University.

[cubefox]

He uses the word "group" 1297 times. This might be a new record.

[TRiG_Ireland]

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

[jackallis]

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

[qsort]

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