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Ask HN: Recommend a maths book for a teenager?

297 points by andyjohnson0 ↗ HN
I'm looking for recommendations for a maths book for a bright, self-motivated child in their late teens who is into maths (mainly analysis) at upper high-school / early undergrad level.

It would be a birthday gift, so ideally something that is more than a plain textbook, but which also has depth, and maybe broadens their view of maths beyond analysis. I'm thinking something along the lines of The Princeton Companion to Mathematics, Spivak's Calculus, or Moor & Mertens The Nature of Computation.

What would you have appreciated having been given at that age?

193 comments

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enhdless
The Manga Guide to Linear Algebra was a light, but useful introduction to linear algebra for me during the summer before my freshman year of college.
nbernard
The Pleasures of Counting by T. W. Körner. If you want something more oriented towards analysis, I see he also authored a Calculus for the Ambitious but I have no experience with it.
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giantg2
This isn't bad. I'm surprised it's expensive now.

https://www.amazon.com/No-bullshit-guide-math-physics/dp/099...

ivan_ah
Thx for plug. Indeed it would be a good book for any highschooler interested in more advanced topics.

> I'm surprised it's expensive now.

Yeah amazon pricing is weird. My intent is for the book to be sold ~$30, but if I tell this price to amazon they start selling it for $20 after discounting, and then readers buy it less because they think it is not a complete book, but just some sort of summary notes. Nowadays I set the price to $40 so that after amazon discount the price will end up around $30, but today it is expensive indeed... I might have to bump it down to $35 at some point.

giantg2
I could have sworn that it was between $15 and $20 when I got it a few years ago.
ivan_ah
It's very possible if you purchased it when I had set the price to ~$30 and amazon was discounting it to ~$20.

BTW, I've released several "point" updates and the book is now at v5.3. Please reach out by email if you're interested in having the PDF (I have a free-PDF-with-proof-of-purchase-of-print-version policy, including all updates).

ivan_ah
I forgot to mention, there is an extended PDF preview of the book you can see here: https://minireference.com/static/excerpts/noBSguide_v5_previ...

Also some parts of the best parts of the book are available in full as standalone free tutorials: SymPy = https://minireference.com/static/tutorials/sympy_tutorial.pd... ; mechanics tutorial = https://minireference.com/static/tutorials/mech_in_7_pages.p... ; concept maps = https://minireference.com/static/tutorials/conceptmap.pdf

SamReidHughes
I bought it, but I only read a chapter of it, after seeing it in a bookstore. Nonetheless:

Mrs. Perkins's Electric Quilt: And Other Intriguing Stories of Mathematical Physics by Paul J. Nahin

It sounds like it's at about the right difficulty/knowledge level, and it has interesting stuff, isn't a boring textbook.

Phithagoras
"The Annotated Turing" by Christian Petzold made a huge impression on me around that age. It doesn't discuss analysis but it gives a nice walkthrough of Turing's classic paper where he introduces the Turing machine and uses it to solve the decidability problem of Diophantine equations.

Also, "Street Fighting Mathematics" from the MIT press

rramadass
The author's name is "Charles Petzold" and yes "The Annotated Turing" is a great book.

All of Petzold's books are excellent, in particular; "Code: The Hidden Language of Computer Hardware and Software" should be read by everybody to understand how Computers really work.

rramadass
Some of the books that you mention seem a bit too hard for a teen, so you have to be careful not to demotivate them by expecting too much of them; instead i suggest a simpler approach before tackling the big ones;

* Functions and Graphs by Gelfand et al. - A small but great book to develop intuition.

* Who is Fourier? A Mathematical Adventure - A great "manga type" book to build important concepts from first principles

* Concepts of Modern Mathematics by Ian Stewart - A nice overview in simple language.

* Mathematics: Its Content, Methods and Meaning by Kolmogorov et al. - A broad but concise presentation of a lot of mathematics.

* Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming - A very good applied maths book. All of Hamming's books are recommended.

There are of course plenty more but the above should be good for understanding.

auxym
Just wanted to chime in regarding Concepts of modern mathematics.

Really enjoyed reading it when I was in college. It's not a textbook, just a prose book for enjoyable reading, but it's inspirational and a very interesting overview of the field of mathematics.

jerzyt
Another vote for Ian Stewart's book, or any other of his books. I discovered them at the end of high school and devoured them during the summer before college. Did I just admit I was a nerd? Tough. I'm proud of it.
alimw
I think the Princeton Companion would be a nice gift because it's something they can dip into as they desire. With a more linear book you may appear to confer an obligation to wade through it from beginning to end. (I also really like the Companion and although I've never splashed out on a copy for myself I wish someone else would :) )
bobmaxup
Jan Gullberg - Mathematics: From the Birth of Numbers

https://www.amazon.com/gp/product/039304002X

Amazon.com Review What does mathematics mean? Is it numbers or arithmetic, proofs or equations? Jan Gullberg starts his massive historical overview with some insight into why human beings find it necessary to "reckon," or count, and what math means to us. From there to the last chapter, on differential equations, is a very long, but surprisingly engrossing journey. Mathematics covers how symbolic logic fits into cultures around the world, and gives fascinating biographical tidbits on mathematicians from Archimedes to Wiles. It's a big book, copiously illustrated with goofy little line drawings and cartoon reprints. But the real appeal (at least for math buffs) lies in the scads of problems--with solutions--illustrating the concepts. It really invites readers to sit down with a cup of tea, pencil and paper, and (ahem) a calculator and start solving. Remember the first time you "got it" in math class? With Mathematics you can recapture that bliss, and maybe learn something new, too. Everyone from schoolkids to professors (and maybe even die-hard mathphobes) can find something useful, informative, or entertaining here. --Therese Littleton

linguae
I remember reading this book in 11th grade and I absolutely loved it. It made me appreciate math much more and showed me the beauty of mathematics. It helped me overcome my math anxiety.
logicslave
The classic text on analysis is Principles of Mathematical Analysis by Rudin. Its very difficult and leaves it to the reader to understand the terse proofs. It starts from the beignning, with no math background assumed about the reader. The terse proofs are written in such a way to force the reader to gain deep mathematical intuition. Some of the proofs are elegant and beautiful. I would absolutely recommend it. You can see a pdf here:

https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_math...

bordercases
> It starts from the beignning, with no math background assumed about the reader.

It assumes that you have enough mathematical maturity to deal with proofs left to the reader.

chynaman
IMO, Rudin is difficult not because of its proofs or lack of them (many proofs in discrete math can be no less brutal than anything in Rudin), rather that it's almost completely and utterly devoid of illuminating examples. For example, the definitions of "neighborhood", "limit point", "closed set", "open set", "bounded set", "perfect set", dense set" are crammed into a single definition 2.18 in chapter 2(Topology in Euclidean Spaces) in 3rd edition. The rest of the chapter is made up of theorems and corollaries. No related examples. On the other hand, Raffi Grinberg's analysis book meant to guide one through Rudin's book spends a whole chapter on elaborating on 2.18. And to be honest even that is barely adequate (totally inadequate, actually) if one wishes to become technically proficient in dealing with basic concepts in analysis with ease (that requires exposure to lots and lots of different examples). Although, probably, neither book has the latter as their goal.
seesawtron
Jordan Ellenberg's "How not to be wrong". Recommended even for non teenagers.
mhh__
Visual Complex Analysis. Partly because it's a brilliant book and partly because Complex Analysis is often really really badly taught.

If you haven't read it, it teaches complex analysis in terms of transformations and pictures rather than solely algebra. It's very clever; Also touches on some concepts in physics and vector calculus.

If you like the style 3Blue1Brown uses, he cites VCA as an inspiration for that style.

exmadscientist
A more traditional complex analysis textbook that's really good is Stewart and Tall's Complex Analysis. It's not necessarily a great complement to VCA; I used them both in my course and didn't find myself referring to VCA much, but then I had good lectures in my course and really got on with Stewart and Tall.

The "standard" book was Churchill and Brown and, uh, I'd say that one is best avoided. It's awful enough that it may be responsible for a number of those courses being so badly taught....

jacobolus
If you like pictures, another couple nice books are Nathan Carter’s Visual Group Theory and Marty Weissman’s Illustrated Theory of Numbers, both of which should be accessible to motivated high school students.

http://web.bentley.edu/empl/c/ncarter/vgt/

http://illustratedtheoryofnumbers.com

abhgh
I recently gifted Illustrated Theory of Numbers to a friend, and it is indeed beautifully illustrated.

I would also recommend "Prime Numbers and the Riemann Hypothesis" for its illustrations and exposition [1].

[1] https://www.amazon.com/gp/product/1107499437

ljf
https://en.m.wikipedia.org/wiki/Flatland - Flatland - A romance in many dimensions

It was a great book that helped get my teenage enquiring mind to look at maths, science and thinking in different ways. Not a text book - but well worth a read.

pvg
Flatland also contains some rather, well, Victorian attitudes. Mathematically, one can get 73.8193% of what the book covers from watching Carl Sagan's bit in the relevant Cosmos episode.
airstrike
A bit on the lighter side, I do recommend The Man Who Counted which I read as a kid and absolutely loved

https://www.amazon.com/Man-Who-Counted-Collection-Mathematic...

I read the original in Portuguese but would assume it's just as good in English, given overwhelmingly positive reviews on Amazon

See also https://en.wikipedia.org/wiki/The_Man_Who_Counted

It won't really teach him math per se, but if my experience is any indication, it will get him hooked on developing intuition and he'll find beauty in otherwise mundane topics such as arithmetic. It's an incredibly engaging story aimed at younger readers but fun for people of all ages – think Arabian Nights with a character that loves math.

Come to think of it, I've got to buy it again and re-read it one of these days

npr11
If they might enjoy something on computing, I'd recommend "The Pattern On The Stone: The Simple Ideas That Make Computers Work" by W.Daniel Hillis. It's very clear and well written, is quite short but covers a lot and can be enjoyed cover to cover more like a novel than a textbook.
sgillen
If they have been exposed to diff eq at all I can recommend Strogatz Nonlinear Dynamics and Chaos. It's a very interesting subject, and the text is one of the most approachable I've come across for any subject.
lanstin
Chaos by James Gleick is also a good intro to some chaos maths and a great bio read, with the stories of Feigenbaum wrestling with the new ideas.
JoshTriplett
As a child, I greatly enjoyed "Algebra the Easy Way", "Trigonometry the Easy Way", and "Calculus the Easy Way". They present each of the subjects not as already-invented concepts that you just have to learn, but as things being invented by a fictional kingdom as they need them. I greatly prefer that style over rote memorization; I can remember it better when I know how to recreate it. Even more importantly, it encourages the mindset that all of these things were invented, and that other things can be, too.

(Note: the other books in the "Easy Way" series do not follow the same style, and are just ordinary textbooks.)

Also, in a completely different direction, I haven't seen anyone mention Feynman yet, and that will definitely encourage a broader view of mathematics and science.

Or, to go another angle, you might consider things like "Thinking, Fast and Slow".