Geometry Revisited by Coxeter and Greitzer and/or Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger if the kid is into plane geometry. It's an idyllic subject, great for independent exploration, and the books shouldn't take long to read. Not very deep, though (at least Honsberger).

Anything by Tom Körner, just because of the writing. Seriously, he can make the axiomatic construction of the real number system read like a novel; open https://web.archive.org/web/20190813160507/https://www.dpmms... on any page and you will see.

Proofs from the BOOK by Aigner and Ziegler is a cross-section of some of the nicest proofs in reasonably elementary (read: undergrad-comprehensible) maths. Might be a bit too advanced, though (the writing is terse and a lot of ground is covered).

Problems from the BOOK by Andreescu and Dospinescu (a play on the previous title, which itself is a play on an Erdös quote) is an olympiad problem book; it might be one of the best in its genre.

Oystein Ore has some nice introductory books on number theory (Number Theory and its History) and on graphs (Graphs and their uses); they should be cheap now due to their age, but haven't gotten any less readable.

Kvant Selecta by Serge Tabachnikov is a 3(?)-volume series of articles from the Kvant journal translated into English. These are short expositions of elementary mathematical topics written for talented (and experienced) high-schoolers.

I wouldn't do Princeton Companion; it's a panorama shot from high orbit, not a book you can really read and learn from.

In general transformation geometry is drastically underemphasized in American (and possibly other countries’) secondary and early undergraduate math education.

It is a really interesting book as it covers multiple topics from mathematics and being that you are a young reader you might benefit from that exposure to topic.

From there on you can fine the topics that interest and dive deeper and learn the nitty gritty!

Thats how I personally found my passion in topology.

* 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.

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.

I own this book, and it's a favorite of mine.

https://www.amazon.com/Concrete-Mathematics-Foundation-Compu...

It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the creation of groups of students, teachers, and mathematicians called "Mathematical Circles". The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport-without necessarily being competitive. This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of the school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be "extracurricular mathematics".

You are right that a book aimed at well prepared Russian 12-year-olds in an extracurricular math circle might be fine for 16-year-old average American students.

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

Hope someone sees this and shares the book with a high schooler—it’s also available for free online!

https://archive.org/details/MIRZeldovichYAndYaglomIHigherMat...

Zeldovich's book with Myshkis on applied mathematics is also excellent: https://archive.org/details/ZeldovichMyskisElementsOfApplied...

Subjectively, I prefer typesetting of the latter, but that is because it is closer to the original Russian edition. Zeldovich was a physicist, so these take an engineer's/physicist's approach, which is, in my opinion, the right entry point to analysis. The reader effectively has to follow the historic development of the subject, starting with some intuitive observations, and eventually developing quite delicate insights.

I read HMFB when I was about 17, and it was great. I remember making up questions of the sort "What level of soda in a can makes it the most stable", and the like, inspired by the book.

All these concepts are central to higher level mathematics, and are not covered in high school (at least not the Danish one).

I'm was very thankful for that introduction, hopefully they would be as well :)

I'm curious: Do you feel this way because it isn't a math textbook?

https://journal.infinitenegativeutility.com/why-i-dont-love-...

It's a negative review, and I agree with all its points. In addition, I don't like Hofstadter's proof of Gödel's theorem - it's so LONG that it's hard to keep track of the parts. (The best proof of Gödel's theorem I've come across was given verbally by John Conway, and he had the opposite strategy - make the proof as brief as possible so that you can easily see an overview of it.)

But despite these criticisms, I think it's a wonderful book, and I agree with the idea of giving it to a teenager as an introduction to mathematics.

I first read it when I was 14 and went on to do a lot of mathematics and philosophy of mathematics.

https://en.wikipedia.org/wiki/What_Is_Mathematics%3F

I got it, then put it on a shelf for 20 years. When I picked it back up, it somehow had become delightful! Perfect subway reading.

Review: http://www.ams.org/notices/200111/rev-blank.pdf

https://artofproblemsolving.com/

[0] https://www.maa.org/press/maa-reviews/the-cauchy-schwarz-mas...

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.

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

http://illustratedtheoryofnumbers.com

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

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

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....

(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".

From the blurb:

"...includes landmark discoveries spanning 2500 years and representing the work of mathematicians such as Euclid, Georg Cantor, Kurt Godel, Augustin Cauchy, Bernard Riemann and Alan Turing. Each chapter begins with a biography of the featured mathematician, clearly explaining the significance of the result, followed by the full proof of the work, reproduced from the original publication, many in new translations."

What's great about this book for a teenager is that they get to read original sources for the stuff they've already learned! And indeed, as they learn more they can keep coming back for more original sources. Personally, reading Descartes original words in Geometry was awe-inspiring, not because every word was so perfect, but because he comes across as just so damn human, the ideas he presents are subtle and profound, and yet presented with an interesting combination of humility and pride that is instantly recognizable. I truly wish I'd had something like that book before embarking on my own journey through math - we stand on the shoulders of giants, but we so rarely look down to see their faces.

By one of my early mathematics tutors in San Diego math circle

Then buy something like: Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) (Featured Titles for Transition to Advanced Mathematics) https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_api_i_K4...

https://www.amazon.com/Differential-Equations-Scientists-Eng...

Might be a little advanced for most teenagers (I was 19 that summer), but I love the book and still refer to it from time to time. I did have experience with ordinary differential equations at the time, but I haven't found an ODE book that's quite the same.

https://artofproblemsolving.com/store/list/aops-curriculum

I've got a PhD in bioengineering but I'm currently going through Introduction to Counting and Probability and I'm really enjoying it.

Some others (not AOPS series):

Nelsen - Proofs Without Words

Polya - How to Solve it

Strogatz - Nonlinear Dynamics and Chaos

Plus, it's a free PDF on the internet! Doesn't get better than that.

[1] https://web.evanchen.cc/napkin.html

For a kind of "cabinet of curiosities", I endorse "Wonders of Numbers" by Clifford Pickover. This book was pivotal in my relationship with mathematics, containing as it does brief excursions into all manner of fascinating topics like cellular automata, and the Collatz Conjecture, as well as a host of more obscure oddities. It's a perfect book to have around when learning programing as well, since it has a nearly bottomless well of interesting things to code. Nor is it dry, thanks to Pickover's whimsical style.

Anything by Ian Stewart would also be good, 'Letters to a Young Mathematician' springs to mind.

Given that you use the word 'maths' with an s I'm guessing you're not American. If you're British like me, I would recommend avoiding American books for high schoolers because they will assume quite different prerequisites.

Stillwell is a magnificent writer -- he loves to go on digressions, and to talk about the history of the subject. My impression is that his books are a bit rambling for traditional use as textbooks, but perfect for self-motivated reading for exactly the same reason. He makes the subject fun.

(Disclaimer: I haven't read this book in any sort of depth, but I have read another of Stillwell's books cover to cover.)

Concerning your other recommendations: The Princeton Companion to Mathematics is magnificent, but in practice it's something he'd be more likely proudly own and display on his bookshelf than to read; it's quite dense. Spivak's Calculus, from what I've heard, is magnificent. Probably best in the context of a freshman honors class, but I can imagine that someone disciplined could love it for self-study. Don't know Moor and Mertens.

If I'd read this book as a teenager, maybe I would've passed Calc I on my first try as opposed to my third. With a C-.

The Four Pillars of Geometry approaches geometry in four different ways, spending two chapters on each. This makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic. Not only does each approach offer a different view; the combination of viewpoints yields insights not available in most books at this level. For example, it is shown how algebra emerges from projective geometry, and how the hyperbolic plane emerges from the real projective line.

John Stillwell is a great mathematician and writer. You won’t regret reading this.

PS. John Stillwell’s Mathematics and Its History is also worth reading. In fact, the book doesn’t aim to tell the story of mathematics. The book connects and the parts of mathematics with a historical perspective.

The point being, of course, that it may take a few different expositions before something 'clicks'. I think this observation is particularly important for self study.

So, in answer to your question: maybe more than one book?

Problems in general physics by IE Irodov [1] was one of those "bang your head on the wall, but when you get it it's ecstasy" kind of books for me.

I am not even sure if I would recommend it to every one. Maybe masochists. But, looking back on it, I have some really fond memories of locking myself in a room for 2 days to get a problem that I felt oh-so-close to solving. Eventually getting it is intensely rewarding.

It it right at the grade 10-12 level.

https://smile.amazon.com/Problems-General-Physics-I-Irodov/d...

https://www.amazon.com/Surreal-Numbers-Donald-Knuth/dp/02010...

That's education wise. Story wise I like "love and math" despite the corny title.

Puzzle/mystery wise "the Scottish book" would have seemed like alien speak to me in HS, aspirational but probably too tough.

Inside interesting integrals is cool if you want to go on a computation spree.

My fave academic book from HS was General Chemistry by Pauling.

IMO the best calculus/real analysis book is by Benedetto & Czaja. But HS age much better is Advanced Calculus by Fitzpatrick.

Introduction to statistical learning is very readable at that age.

CS wise I think Skienas algorithm design manual is the best.

Discrete Math With Ducks[0] (and the professor that taught from it) is the reason I focused on the discrete side of things. It doesn't take itself too seriously, and it introduces a range of topics in the area. Plus the mindset is different from analysis. It's an interesting shift

[0] https://www.maa.org/press/maa-reviews/discrete-mathematics-w...

Tim Gowers' Mathematics: A Very Short Introduction is a popular book on doing mathematics. Not a textbook that teaches you mathematics, so wouldn't give this as the only book, but popular "what's the field like" books could be very interesting to a high schooler.

Also, not suitable for the only book, Penrose's The Road to Reality. It gets very advanced and probably can't be fully tackled without additional mathematical education, but it tries to be an honest exposition of the math needed for modern physics from the ground up without explicitly resorting to external knowledge. I would have loved a "this will teach you all of the math if you can get through it" book like this even if I never did manage to get through it.

I remember getting God Created The Integers when I was a teenager and... not finishing it. I also got a copy of Brown & Churchill's Complex Variables and Applications and spent hundreds of hours on it. As a teenager, I preferred textbooks with problem sets to popularizations. (I still do.) Of course, this was [complex] analysis, so it doesn't qualify.

One book which is fully technical but also entertaining by way of the subject matter, and which was inspiring to me around 14-15, was Kenneth Falconer's Fractal Geometry:

https://www.amazon.com/Fractal-Geometry-Mathematical-Foundat...

Of course, at that age, I didn't understand what Falconer meant by describing the Cantor set as "uncountable", or what a "topological dimension" was, but I was able to grasp the gist of many of the arguments in the book because it is very well illustrated and does not rely too much on abstruse algebra techniques. Some people don't enjoy reading a book if they don't fully understand it, but I liked that kind of thing. As I got older and learned more, I started to be able to understand the technical arguments in the book as well.

https://books.google.com/books/about/Mathematics_A_Discrete_...

I bought this book when I was ~16 because I wanted to learn some discrete maths, but it actually touches many different interesting topics that you don't see in secondary school (including some cryptography!).

or Calculus Made Easy: https://www.math.wisc.edu/~keisler/keislercalc-09-04-19.pdf

[1] https://www.maa.org/external_archive/devlin/LockhartsLament....

> 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?

Common Sense Mathematics by Ethan D. Bolker and Maura B. Mast

My friend was assigned this book for a quantitative reasoning class in college and I was so impressed by how approachable it was. It's got sections on things like climate change and Red Sox ticket prices.

Excerpt from preface:

""" One of the most important questions we ask ourselves as teachers is "what do we want our students to remember about this course ten years from now?"

Our answer is sobering. From a ten year perspective most thoughts about the syllabus -- "what should be covered" -- seem irrelevant. What matters more is our wish to change the way we approach the world. """

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

Deep math is cool and all, but right now I'm working through a used copy of the Freedman, Pisani and Purves Statistics textbook https://amzn.to/2YVvU6o It's chock full of actual examples from real research and statistics, complete with citations. I just worked through some problem sets today, analyzing some twin studies establishing the link between smoking and cancer. Other topics I can recall: robbery trials, discrimination lawsuits, and coronary bypass surgery.

That said, it's an actual textbook, and expects the learning to come from engaging in problem sets. And it's far less technical than the Stats for Engineers course I barely passed. If you're looking for something less textbooky, Super Crunchers (https://amzn.to/3eTz5RL) is sort of a layman's book on the subject of prediction and statistics.

Another analysis suggestion is Creative Mathematics by H.S. Wall. It is a book that walks a high school level student through creating the proofs themselves. The topic covered is a stripped down version of analysis, calculus, and later even differential geometry. It's really brilliant. Going slow and having fun when you're a teen would be much more productive than going fast and burning out.

The Spivak book is a good suggestion and might be a little difficult depending on their actual background. The books by Gelfand mentioned by someone else (there's actually a series of them that cover algebra, functions, coordinates, trigonometry, etc.) would help provide the needed background.

The book Conceptual Mathematics is claimed to be aimed at high school students. Maybe give it a whirl and see what happens. If they know calculus, then Advanced Calculus: A Differential Forms Approach by Harold Edwards is a gem. The first three chapters should be readable, as they give heuristic discussions of the topic.

https://www.amazon.com/Heart-Mathematics-invitation-effectiv...

John Stillwell’s Mathematics and Its History.

Needham’s Visual Complex Analysis.

I would second this by "Concrete Mathematics" by Graham, Knuth and Patashnik. This is actual university course book with very formal proofs and theory, but the subject matter is still largely accessible to serious high school students and demonstrates beautiful reasoning examples throughout. It is also very practical book, after covering techniques in this book, one can often times calculate exact sums of infinite series quicker than estimating their bounds. If your high school student decides to study math at university level, the techniques and skills taught in this book will prove invaluable in broad areas of study.

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

A Programmer's Introduction to Mathematics by Jeremy Kun is wide ranging and appropriate if there is also interest in programming.

Nature and Growth of Modern Mathematics by Edna Kramer is a wonderful book if history is a passion as well.

Elements of Mathematics by John Stillwell is a broad overview of subjects. It has a crisp mathematical feel to it.

Vector Calculus, Linear Algebra, and Differential Forms by John & Barbara Hubbard is a beautiful introduction to the multi-dimensional aspects, but it is a book that should happen after knowing one dimensional calculus. .

If your child hasn't been exposed to Guesstimation, then a book on that is highly recommended. The book with that title by Weinstein and Adams is a nice guide to investigating that realm.

If the child does arithmetic from right to left, as is sadly too common, the book Speed Mathematics Simplified by Edward Stoddard is a great remedy for that.

Everyday Calculus by Oscar Fernandez could also be worth a look.

There are many good recommendations here but I do think it will be good for them to gain some exposure to pure mathematics. It's different than what's typically taught in high school so they can start to get an idea whether they actually want to be a mathematician or instead focus on applied math in an engineering discipline.

Also you're probably going to get a computing bias here. I found the threads on physicsforums.com helpful so you might ask there as well if you want a different bias. (https://www.physicsforums.com/forums/science-and-math-textbo...)

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

> 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.

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).

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

Also, "Street Fighting Mathematics" from the MIT press

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.

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.

It depends on your budget, but I would recommend the 10-volume set of “Encyclopaedia of Mathematics” (spelled just like that), which is a translation of the Soviet mathematics version. I have found that this is the resource I turn to when I want to quickly explore some new area of mathematics.

Because there are many books with this title, I will link to Amazon: https://www.amazon.com/Encyclopaedia-Mathematics-Michiel-Haz...

[EDIT: Previously I recommended Calculus on Manifolds here also, but on further reflection and reading some of the other responses I think I both misremembered the difficulty level of the book and overestimated what early-undergrad level means]

Not a math book, but a really well written, full with math history novel about the value of mathematics in a human's life. It gives you the reason, why you should know (higher) maths, even if you will won't become a mathematician.

[0] http://users.metu.edu.tr/serge/courses/111-2011/textbook-mat...

https://www.storyofmathematics.com/19th_gauss.html

This book is aimed at a young audience, though I haven't read it and cannot say whether it is age-appropriate for late-teens.

https://www.goodreads.com/book/show/837010.The_Prince_of_Mat...

- Proofs and Refutations by Imre Lakatos (https://en.wikipedia.org/wiki/Proofs_and_Refutations) (makes you think about what a proof really is)

- The World of Mathematics: not a lot of math proper, doesn’t have much depth, but lots of examples of applied math, interwoven with mentions of the history of mathematics (https://www.amazon.com/World-Mathematics-Four-Set/dp/0486432...)

2) "The Book of Numbers", by John H. Conway and Richard Guy is a beautiful book which discusses about figurative numbers amoung several other beautiful topics.

3) "Stories About Maxima and Minima", V. M. Tikhomirov has some beautiful anecdotes and interesting applications of calculus.

4) "Contemporary Abstract Algebra", by Joseph Gallian is an algebra textbook that goes beyond just teaching material. It has quotations, biographies, puzzles and interesting applications of algebra.

Spivak's calculus you bring to the beach and read it between swim and swim.

EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach readings for this summer.

A fantastic and beautifully illustrated expository work describing symmetry groups such as the 17 wallpaper groups in the plane (think Escher), and other tiling groups in for example the hyperbolic plane. Love the use of orbifold notation as opposed to crystallographic notation.

[1] https://www.amazon.com/Symmetries-Things-John-H-Conway/dp/15...

https://www.topicsinmaths.co.uk/cgi-bin/sews.py?SuggestedRea...

For a single suggestion, "How to Think Like a Mathematician" by Kevin Houston.

A second suggestion: "A Companion to Analysis" by Tom Körner.

But it depends a lot on whether you want books about math, or books of math. It sounds like you want the latter ... at some point I'll get around to putting annotations on the choices in the list that would help distinguish.

"Euler's Gem" by Dave Richeson

"A Companion to Analysis" by Tom Körner

"Elementary Number Theory: A Problem Oriented Approach" by Joe Roberts

It will be some new mathematical concepts for him, but I reckon he will be able to Google what does are.. I also find it extra motivating to learn a new mathematical tool when I know what type of problem it can solve!

https://www.amazon.com/Prof-McSquareds-Calculus-Primer-Inter...

Discrete math is also orthogonal to typical math curricula so it's unlikely to be redundant to anything they've already learned or will learn.

It's sometimes useful to see the context of mathematics and it's purpose beyond the intrinsic beauty.

The best book.

https://www.amazon.com/Number-Theory-Dover-Books-Mathematics...

Mathematics, a human endeavor by Harold R. Jacobs

https://openlibrary.org/books/OL5699810M/Mathematics_a_human...

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.

https://www.nytimes.com/2005/02/27/books/review/the-road-to-...

It's purpose to me at least was as a guide to the mathematics that was too difficult for me to understand straight away but could be considered the end goal to a given study i.e. As Symplectic Geometry is Analytical Mechanics.

That said, it serves as a great overview of physics for mathematicians and perhaps as very casual intro to twistor theory for physicists.

[1]: https://www.gutenberg.org/files/33283/33283-pdf.pdf

https://www.goodreads.com/book/show/39074550-humble-pi

IIRC it explains how to make the pictures

Point being, there may be a better analysis text for this student to start with right now- depends highly on their background/situation, but personally I am glad I didn't have to read Rudin for my first "real" math course.

https://www.goodreads.com/book/show/24612214-the-magic-of-ma...

https://openstax.org/subjects/math

Some are even downloadable to a Kindle (for free) on Amazon.

http://www.gutenberg.org/files/33283/33283-pdf.pdf

-1 for Polya's How to Solve it - I don't remember a damned thing from it.

https://www.hup.harvard.edu/catalog.php?isbn=9780674237513

oraclerank.com kaggle.com

... so probably not good for this kid, but always worth mentioning in the context of awesome math books.

I don't see any recommendations for Smullyan yet. The Lady or the Tiger is the classic I think, but I really loved Forever Undecided.

- The Annotated Turing by Charles Petzold

- Quantum Computing since Democritus by Scott Aaronson

- Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (a textbook, but an easy one to read on your own)

- Gödel's Proof by Nagel and Newman (best if he can read it with a partner and talk through the steps)

- Prime Obsession by John Derbyshire

Some of those have already been mentioned, but consider this another vote for them. :-)

Is it possible where you are to have your teenager attend maths lectures at a university as an auditing student?

IMHO long and still the best linear algebra book is

Halmos, Finite Dimensional Vector Spaces (FDVS).

It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics.

But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one.

Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.

For a first book on linear algebra, I'd recommend something easy. The one I used was

Murdoch, Linear Algebra for Undergraduates.

It's still okay if can find it.

For a first book, likely the one by Strang at MIT is good. Just use it as a first book and don't take it too seriously since are going to cover all of it and more again later.

I can recommend the beginning sections on vector spaces, convexity, and the inverse and implicit function theorems in

Fleming, Functions of Several Variables

Fleming was long at the Brown University Division of Applied Math. The later chapters are on measure theory, the Lebesgue integral, and the exterior algebra of differential forms, and there are better treatments.

Also there is now

Stephen Boyd and Lieven Vandenberghe, Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares

at

http://vmls-book.stanford.edu/vmls.pdf

Since the book is new, I've only looked through it -- it looks like a good selection and arrangement of topics. And Boyd is good, wrote a terrific book, maybe, IMHO likely, the best in the world, on convexity, which is in a sense is half of linearity.

Some course slides are available at

http://vmls-book.stanford.edu/

For reference for more, have a copy of

Richard Bellman, Introduction to Matrix Analysis: Second Edition.

Bellman was famous for dynamic programming.

For computations in linear algebra, consider

George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems

although now the Linpack materials might be a better starting point for numerical linear algebra. Numerical linear algebra is now a well developed specialized field, and the Linpack materials might be a good start on the best of the field. Such linear algebra is apparently the main yardstick in evaluating the highly parallel supercomputers.

After linear algebra go through

Rudin, Principles of Mathematical Analysis, Third Edition.

He does the Riemann integral very carefully, Fourier series, vector analysis via exterior algebra, and has the inverse and implicit function theorems (key to differential geometry, e.g., for relativity theory) as exercises.

All of this material is to get to the main goals of measure theory, the Lebesgue integral, Fourier theory, Hilbert space and Banach space as in, say, the first, real (not complex) half of

Rudin, Real and Complex Analysis

But for that I would start with

Royden, Real Analysis

sweetheart writing on that math.

Depending on the math department, those books might be enough to pass the Ph.D. qualifying exam in Analysis. It was for me: From those books I did the best in the class on that exam.

Moreover, from independent study of Halmos, Nering, Fleming, Forsythe, linearity in statistics, and some more, I totally blew away all the students in a challenging second (maybe intentionally flunk out), advanced course in linear algebra and, then, did the best in the class on the corresponding qualifying exam, that is, where that second course was my first formal course in linear algebra.

Lesson: Just self study of those books can give a really good background in linear algebra and its role in the rest of pure and applied math.

No joke, linear algebra, and the associated vector spaces, is one of the most important courses for more work in pure and applied math, engineering, and likely the future of computing.

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