Concrete Mathematics by Knuth and Patashnik (already mentioned for u/pmiller2) if the kid likes numbers. That's perhaps the guiding thread of the book -- it's about the beautiful (yet usually very elementary and natural) things you can do with numbers.
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).
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.
If the kid likes plane geometry and is interested in further math, I’d highly recommend Yaglom’s books Geometric Transformations. They are a series of (hard) problem-focused books which teach the ideas of transformation geometry in service of solving various construction problems.
In general transformation geometry is drastically underemphasized in American (and possibly other countries’) secondary and early undergraduate math education.
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.
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.
I'm going to go a completely different direction from other recommendations and say Concrete Mathematics by Knuth and Patashnik. They will definitely be able to use skills from analysis and calculus here, but there are so many additional tools in this book that it's very much a worthwhile digression. The marginal notes are great, as well!
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".
This is more about level of preparation / past experience than age per se. The OP describes a “bright, self-motivated child in their late teens who is into maths” and is a few years ahead of their peers. The mentioned book might seem a bit easy or elementary for this particular kid. The two Berkeley Math Circle books might be better. https://mathcircle.berkeley.edu/books
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.
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
My father (and me) would always recommend Zeldovich's "Higher mathematics for beginners" for learning analysis at the upper high-school level. This particular book does not seem to be available in translation, instead there is a reworked version with Yaglom (who was a brilliant science educator himself):
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.
During the first year of my undergrad someone introduced me to Gödel, Escher, Bach. I thought it was mind blowing at the time and still find it to be an incredible introduction to formal systems, thinking mathematically and understanding the concept of proofs.
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 have to disagree with you here, and strongly. I don't think Gödel, Escher, Bach is a good book. Hofstaeder is clearly very smart, curious, and open-minded, and I love all those things, but the book itself is just so pretentious and sort of pointless. It's precisely the wrong kind of book you want to give a bright teenager, because it will only encourage them to get a head-start inserting their head up their own arsehole, metaphorically speaking.
> I don't think Gödel, Escher, Bach is a good book. Hofstaeder is clearly very smart, curious, and open-minded, and I love all those things, but the book itself is just so pretentious and sort of pointless.
I'm curious: Do you feel this way because it isn't a math textbook?
Not at all. My own recommendation, God created the Integers, isn't a math textbook. I doubt Hofstadter himself would claim GEB had a point - it was more of an intellectual fugue put to paper. If GEB was a novel it would be more along the lines of Finnegan's Wake than Les Misérables, and I would never ever give the former to a teenager.
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.
If I could give my high school self only one math book, it would have to be Seven Sketches in Compositionality by Fong and Spivak. Did every exercise over winter break in college and realized along the way that I had been hustling through math courses and olympiad problems without appreciating any beauty in the structure of mathematics. It completely changed my life and, at least in my eyes, dissolved the assumption that “applied” math must be less rigorous or “pure” math must be less practical. Not only did it immediately recast my basic intuition about what math “is” (and what numbers “are” or what processes “do”) but with a bit more effort toward studying category theory, I came to see my previous encounters with more advanced topics like forcing in set theory or the Legendre-Fenchel transform used in physics/economics in a completely new light. What is truly wild to me is that Seven Sketches has no real prerequisites, and I could have just as easily read it when I was 14. This book should be the basis of a mandatory course for a math-loving high schooler. Instead of rushing to learn linear algebra and real analysis in high school, I wish I had gained the wonderful perspective of Fong and Spivak—I would have fallen truly in love with math much sooner, found a deeper perspective in my courses much faster, and enjoyed all of it so much more along the way.
Hope someone sees this and shares the book with a high schooler—it’s also available for free online!
I second this. Do t be scared off by it being for math Olympiads. A lot the first volume deals with concepts across much of the field. Lots of practice and ideas for logs, exponents, word problems.
And it comes with a solution guide which helped me a lot.
The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities is a graded problem book that will teach them the principles and practice of mathematical proofs like no other book. Here is its MAA review: . A pirate PDF is a Google search away. Take a look and see if you like it.
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.
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....
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".
"...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.
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.
I'm surprised nobody has yet mentioned An Infinitely Large Napkin by Evan Chen . It's a fantastic, very dense primer and overview of a large variety of university-level topics in mathematics. It was originally targeted at high school students with an interest in higher mathematics, and while the later chapters have strayed somewhat from that goal, one of the best things about Napkin is that it does its best to justify why we introduce certain ideas and abstractions. Generally, it tries to give a high-level overview without sacrificing technical rigor. I highly recommend it.
Plus, it's a free PDF on the internet! Doesn't get better than that.
For a deep, but deeply entertaining introduction to extraordinarily high-level concepts that remain useful tools of thought forever - Godel, Escher, Bach. That belongs on everyone's bookshelf.
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.
Abstract Algebra by Pinter and Introduction to Topology by Mendelson are two fantastic books, published by Dover, that are too elementary to be used as university textbooks on those subjects but as a result are great for a more casual reader. They are well motivated and rarely omit details. They would serve as a great introduction to undergraduate math.
Maths tutor here. At that age I was very inspired by Fermat's Last Theorem by Simon Singh. It's not a technical book but gave me my first idea of what mathematicians actually do and how the process works. This book motivated me to major in maths.
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.
Yeah I read that as a 19 year old and spent hours? Days? Making graphics of fractals on my HP-28s (at Oxford studying maths but with no friends). Awesomely inspiring and technical enough to get going even without the internet.
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.
I'd consider something by John Stillwell. For example, Numbers and Geometry, which investigates the connections between number theory and plane geometry -- two subjects which your child has probably seen, but not seen related.
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.
I suggest Infinite Powers by Steven Strogatz. It doesn't matter if they already took a calculus course, I guarantee it's a much better way to make them appreciate the the subject than any textbook. And if they don't know calculus yet, that just makes it even better!
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-.
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.
Probably not anyone's first choice, completely unknown in the US and not truly a maths book, so much as a physics book.
Problems in general physics by IE Irodov  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.
For something that's a little more fun to read and covers fundamental topics. (Foundations for higher mathematics) I'd recommend Gödel, Escher, Bach by Douglas Hofsteader. It changed the way I approach problems to this day.
If you want them to look beyond Analysis, would an intro to discrete math maybe be what you're looking at?
Discrete Math With Ducks (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
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.
Sanjoy Mahajan's The Art of Insight in Science and Engineering (available online, https://mitpress.mit.edu/books/art-insight-science-and-engin...). Takes a very pragmatic look to doing mathematics, while not pulling many punches on how advanced the mathematics gets. Would have appreciated this a lot, math books are usually split into dry theory where you have to already know math to be able to properly read them and books that are simplified to death for people who are forced to study math and don't want to.
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.
>What would you have appreciated having been given at that age?
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:
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.
I would give God Created The Integers as a reference to read up to where you are in math, not as something to get through. So, you could get a feel for what Euclid wrote, or what Descartes wrote, either after or during learning those lessons. As you move through your education, you can keep moving through the book, Cauchy, Galois, Riemann, etc. Anyway, that's the context in which I would give it. BTW Cantor's original diagonal proof is in GCTI. :)
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!).
Since you mention analysis, I recommend Yet Another Introduction to Analysis and Metric Spaces by Victor Bryant. They should be at the right level and a lot of fun.
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.
Measurement by Paul Lockhart. Written by the author of the well known Mathematician's Lament essay , which deplores the state of school maths education, in response to questions like "Well, what are you going to do about it?".
> 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
> 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.
> 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?
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.
I would recommend highly "What is Mathematics" by Richard Courant and Herbert Robbins. This is very accessible book for high schoolers who are keen and interested in mathematics, and will expose the reader to a broad array of topics and pique the interest and awaken the imagination and instill the beauty of mathematics. This in turn can drive the reader to find out more and fall in love with the subject.
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.
I remember Pi in the Sky by John Barrows very fondly. It has more of a focus on geometry and logic.
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.
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
I'm neither bright nor a teenager but I have been enjoying working through Spivak's Calculus. I picked it up because it was recommended as a good intro to pure mathematical thinking for someone who knows calculus. I've found it challenging but it has delivered on that promise.
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.
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.
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).
"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
The university track will already put him on rails for a while. I believe your instinct on the encyclopedia should be followed because he should gain breadth early on to be sure that he has enough insight not to prematurely specialize.
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.
If they like calculus and can stand proofs, I’d recommend a Course of Pure Mathematics by Hardy. It totally blew my mind when I was that age, to see how everything was “connected” by proofs, starting with real numbers. Despite being proof heavy, I found the writing style singularly legible and comprehensible.
Metamathematics by Kleene. Fairly accessible math, mostly new and developed from the start it takes one into compatibility theory and formalization of maths in a way that makes Godel easy to understand and just full of cool ideas that are very relevant to today’s world of computers and the limits to certainty.
"Uncle Petros and Goldbach's Conjecture" by Apostolos Doxiadis.
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.
If you want to get away from analysis, I've found that cryptography can be quite an engaging subject. If you have the right book, it can have the rigors of more mathematical subjects, while being accessible without extensive background and having visible real-world applications. I unfortunately don't have much experience with books in this area, but I do like https://files.boazbarak.org/crypto/lnotes_book.pdf (plus it's free ;) ).
[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]
A little off-topic (and perhaps more useful for younger students) but you could do worse than introduce them to the achievements of Gauss (though they are probably somewhat familiar), who as a teenager had discovered and rediscovered several important theorems - his foundational Disquisitiones Arithmeticae was written at 21.
The books by Tent are mathematically at a 4th or 5th grade level. They’re sort of like Jean Lee Latham’s bio of Nathaniel Bowditch (you learn a fictionalized life story but you can’t really grok the person’s contribution). Of course, most mathematicians lead far more boring lives than Bowditch did in his youth, so the kind of kid who is reading Harry Potter by 4th or 5th grade is going to find them very dry.
Problem-Solving Strategies by Arthur Engel. It's more than a textbook and not easily absorbed. The book + the internet is a powerful combination for not just learning cool math skills but building mental models/ problem framing lenses that will benefit them later in life
Polya's How to Solve It changed the way I thought about learning mathematics. His treatment of random walks in one dimension (eventually all walks return to the same point) vs three dimensions (where they can escape) really affected my mental model of the world.
Instead of How To Solve It, which is organized dictionary–style with short sections on particular named problem solving topics, and is somewhat hard to interpret for novices without guidance, let me recommend Polya’s other two books (each 2 volumes), Mathematical Discovery and Mathematics and Plausible Reasoning.
"The Symmetries of Things"  by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss.
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.
The princeton companion is nice to have around, but you do not really read it end to end.
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.
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.
My 13 year old and I have through parts of the first three chapters of "An Illustrated Theory of Numbers". I would reckon that if your student is motivated and at upper high school level, they would have the sophistication to go at it alone. It is just a beautiful book also, with lots of exercises and the associate website http://illustratedtheoryofnumbers.com/ also has Python notebooks if they are into programming.
There is no such thing as 'maths'. It is called 'math' which is short for 'mathematics'. I have an advanced degree in mathematics. No one ever called mathematics 'maths' while I was in college. Absolutely no one. They would have been laughed out of the room.
I enjoyed "our mathematical universe" by max tegmark. It's not the books intention to teach mathematics, but rather explain how the author sees a link between mathematics and the universe.
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!
Discrete Mathematics with Applications by Susanna S. Epp. is one of my favorite textbooks. Discrete math is considered a sophomore-level college subject, but it's really not that challenging, and the textbook is extremely thorough and understandable.
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.
There are many books on that front, particularly the ones related to recreational math or intro higher math (see https://mathvault.ca/books for instance). Spivak's Calculus as an intro would be an interesting start, though Stewart's Calculus is dense but more accessible.
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 :) )
Introduction to Logic: And to the Methodology of Deductive Sciences, by Alfred Tarski. One of my favorite math books, which convinced to pursue an undergrad in math. Being an intro logic book it's completeley self-contained and may not not even feel like a math book, but yet a great intro into foundational stuff.
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.
Gödel's Proof and GEB blew my mind in high school, gave me the motivation to actually attend my math classes, because it finally showed me there was more interesting stuff at the end of the math tunnel. Of course I only understood a fraction of it then (still do), but it was eye-opening.
The little LISPer is the book I wish I encountered junior/senior year. For someone coming from more of a traditional math/logic education than anything else, it would have been nice to have that introduction to thinking about computation before classes in C.
For analysis you absolutely MUST read Principles of Mathematical Analysis by Walter Rudin. Covers everything and is literally a gold standard text in modern analysis. "Baby Rudin" is essentially the analysis bible that all subsequent texts worked off of.
I heard a bunch of students complain about how tough the text for our Real Analysis class was, so I was surprised to find it felt pretty readable to me. Turns out it was a new prof that semester who decided to go with "Mathematical Analysis, Second Edition by Tom Apostol" instead of Baby Rudin.
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.
When I was younger, I received a book about video game physics as a gift. The combination of applied mathematics and, well, games really hooked me for that year. In the end, I built my own physics simulation and collision detection engine after school.
I enjoyed "The Mathematical Tourist" by Ivars Peterson although it might be more "descriptive" than you are looking for. I found it quite inspiring, probably early in high school (forget when exactly I got it - maybe even earlier).
This may not be exactly what you are looking for but you should checkout the cartoon introduction to economics by Yoram Bauman. Its a good book to start an interest in economics, it is not deep at all but could lead to other sources.
If you are planning on majoring in math (or related), why not get a head start and get some textbooks corresponding to actual courses you would like to take at the college/university you are planning/hoping to attend?
I think I would have really enjoyed Mathematics and its History by Stillwell. It does a good job connecting analysis, algebraic geometry and number theory following the historical evolution of modern topics.
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:
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.
Of the books mentioned in this thread so far I think I'd have been most excited about the Princeton Companion to Mathematics as a birthday present.
- Your goal of the gift is something more than a plain textbook. The Princeton Companion is something your child will return to throughout their math career. It will be an anchor book that will remind them of your support for them when they were still a budding mathematician.
- Relatedly, the book is far too broad to be consumed as a textbook. Hence it will be more like a friend (or companion :) ) on their journey. Even a really amazing textbook (like Baby Rudin) in contrast is just a snapshot of where they are now.
I second this! Gardner would be something more recreational and fun than a lot else out there. The Colossal Book of Mathematics is a good way to get a lot in one place. The Night is Large is also great but less math-focused.
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.
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
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
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.
Introduction to Matrix Analysis:
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.
#1: your "The Princeton Companion.." or any of the great suggestions that you got here
#2: "Gödel, Escher, Bach: an Eternal Golden Braid" by Douglas Hofstadter. Best if you can get an old, old beat up paper copy at Amazon. Tell him that if he's lucky it will take him a lifetime to actually "get it". Tell him to keep the book in sight, bedroom, studio.. why not, bathroom. And to just read it not sequentially but at random. That is the best present to a mind thirsty for knowledge.
He might not appreciate it right not, he will appreciate it 30 years from today, if he's lucky.