
Ask HN: Recommend a maths book for a teenager? - andyjohnson0
I&#x27;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 &#x2F; early undergrad level.<p>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&#x27;m thinking something along the lines of <i>The Princeton Companion to Mathematics</i>, Spivak&#x27;s <i>Calculus</i>, or Moor &amp; Mertens <i>The Nature of Computation</i>.<p>What would you have appreciated having been given at that age?
======
generationP
_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).

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...](https://web.archive.org/web/20190813160507/https://www.dpmms.cam.ac.uk/~twk/Number.pdf)
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.

~~~
jacobolus
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.

------
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.

------
pmiller2
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!

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

[https://www.amazon.com/Concrete-Mathematics-Foundation-
Compu...](https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-
Science/dp/0201558025)

~~~
javajosh
Great pick. Note: you can get it for $20 less AND support a local used book
store if you buy it from alibris. [https://www.alibris.com/Concrete-
Mathematics-A-Foundation-fo...](https://www.alibris.com/Concrete-Mathematics-
A-Foundation-for-Computer-Science-Ronald-L-Graham/book/1273349)

------
anirudhcoder
[https://www.amazon.com/Mathematical-Circles-Dmitry-
Fomin/dp/...](https://www.amazon.com/Mathematical-Circles-Dmitry-
Fomin/dp/0821804308)

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".

~~~
jacobolus
This would be better for a 10–14 year old in middle school or early high
school.

~~~
enriquto
Hmmm. There's no way my 10 year old daughter would read beyond the first page
of that thing. For 15-16 it's great.

~~~
jacobolus
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](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.

------
bobmaxup
Jan Gullberg - Mathematics: From the Birth of Numbers

[https://www.amazon.com/gp/product/039304002X](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.

------
0-ary
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!

~~~
shaunxcode
link to read online:
[https://arxiv.org/pdf/1803.05316.pdf](https://arxiv.org/pdf/1803.05316.pdf)

------
dynamic_sausage
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):

[https://archive.org/details/MIRZeldovichYAndYaglomIHigherMat...](https://archive.org/details/MIRZeldovichYAndYaglomIHigherMathForBeginners1987)

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

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.

------
MperorM
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
:)

~~~
javajosh
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.

~~~
msla
> 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?

~~~
javajosh
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.

------
zakk
I suggest "What Is Mathematics?" by Richard Courant and Herbert Robbins.

[https://en.wikipedia.org/wiki/What_Is_Mathematics%3F](https://en.wikipedia.org/wiki/What_Is_Mathematics%3F)

~~~
blendo
My high school math professor recommended this to me 40 years ago.

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](http://www.ams.org/notices/200111/rev-blank.pdf)

------
scorecard
Art of Problem Solving is popular with the Math Olympiad types. I see that
others on this thread have recommended it already.

[https://artofproblemsolving.com/](https://artofproblemsolving.com/)

~~~
crawftv
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.

------
mci
_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:
[0]. A pirate PDF is a Google search away. Take a look and see if you like it.

[0] [https://www.maa.org/press/maa-reviews/the-cauchy-schwarz-
mas...](https://www.maa.org/press/maa-reviews/the-cauchy-schwarz-master-class-
an-introduction-to-the-art-of-mathematical-inequalities)

------
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.

~~~
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://web.bentley.edu/empl/c/ncarter/vgt/)

[http://illustratedtheoryofnumbers.com](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](https://www.amazon.com/gp/product/1107499437)

------
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".

------
javajosh
God Created the Integers: The Mathematical Breakthroughs That Changed History.
Stephen Hawking. I bought mine for cheap on alibris
([https://www.alibris.com/God-Created-the-Integers-The-
Mathema...](https://www.alibris.com/God-Created-the-Integers-The-Mathema...))

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.

------
uptownfunk
First read [https://artofproblemsolving.com/news/articles/avoid-the-
calc...](https://artofproblemsolving.com/news/articles/avoid-the-calculus-
trap)

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/dp/0321797094/ref=cm_sw_r_cp_api_i_K4R.Eb1SBS1F6)

~~~
uptownfunk
Also art of problem solving vols 1-2 are now classics for that age.

------
btrettel
I worked through a lot of this partial differential equations book during
downtime while working in a gas station after my freshman year of college:

[https://www.amazon.com/Differential-Equations-Scientists-
Eng...](https://www.amazon.com/Differential-Equations-Scientists-Engineers-
Mathematics/dp/048667620X)

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.

------
montalbano
Spivak is an excellent choice but may be too advanced depending on his level.
I would also strongly recommend any of the books in the Art of Problem Solving
series:

[https://artofproblemsolving.com/store/list/aops-
curriculum](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

------
3PS
I'm surprised nobody has yet mentioned An Infinitely Large Napkin by Evan Chen
[1]. 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.

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

~~~
sitkack
Thanks for alerting me to Evan Chen, what a wonderful person and skilled
mathematical educator!

------
dTal
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.

------
analbumcover
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.

------
wdutch
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.

------
layoutIfNeeded
I remember being blown away by this book as a teen: James Gleick - Chaos:
Making a New Science [https://www.amazon.com/Chaos-Making-Science-James-
Gleick/dp/...](https://www.amazon.com/Chaos-Making-Science-James-
Gleick/dp/0143113453)

~~~
lanstin
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.

------
impendia
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.

------
VitalyAnkh
I recommend John Stillwell’s The Four Pillars of Geometry
[https://www.springer.com/gp/book/9780387255309](https://www.springer.com/gp/book/9780387255309).
This book is suitable for all who want to cultivate a interest in mathematics.

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.

------
LordOmlette
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-.

------
prof-dr-ir
In response to the question about the best book to learn [subject] from, the
best answer I ever received was: "the third book".

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?

------
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.

------
screye
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 [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://smile.amazon.com/Problems-General-Physics-I-
Irodov/dp/8183552153?sa-no-redirect=1)

------
tromp
Surreal Numbers by Knuth is great, although not related to analyis/calculus:

[https://www.amazon.com/Surreal-Numbers-Donald-
Knuth/dp/02010...](https://www.amazon.com/Surreal-Numbers-Donald-
Knuth/dp/0201038129)

------
debbiedowner
Princeton companion regular and applied version 100% is the one book I wished
I got in HS. Shows how big the world is which is very useful at that age.

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.

------
seesawtron
Jordan Ellenberg's "How not to be wrong". Recommended even for non teenagers.

------
coeneedell
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.

------
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.

------
ian-g
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[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...](https://www.maa.org/press/maa-reviews/discrete-mathematics-
with-ducks-0)

------
rsaarelm
Sanjoy Mahajan's _The Art of Insight in Science and Engineering_ (available
online, [https://mitpress.mit.edu/books/art-insight-science-and-
engin...](https://mitpress.mit.edu/books/art-insight-science-and-
engineering)). 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.

------
scythe
>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_ :

[https://www.amazon.com/Fractal-Geometry-Mathematical-
Foundat...](https://www.amazon.com/Fractal-Geometry-Mathematical-Foundations-
Applications/dp/111994239X)

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.

~~~
javajosh
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. :)

------
pera
_Mathematics: A Discrete Introduction_ by Edward R. Scheinerman:

[https://books.google.com/books/about/Mathematics_A_Discrete_...](https://books.google.com/books/about/Mathematics_A_Discrete_Introduction.html?id=DZBHGD2sEYwC)

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!).

------
gramie
You could check out Burn Math Class: [https://www.amazon.com/Burn-Math-Class-
Reinvent-Mathematics/...](https://www.amazon.com/Burn-Math-Class-Reinvent-
Mathematics/dp/0465053734#reader_B017QL8UHU)

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

------
bmitc
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.

------
mkl
_Measurement_ by Paul Lockhart. Written by the author of the well known
Mathematician's Lament essay [1], which deplores the state of school maths
education, in response to questions like "Well, what are you going to do about
it?".

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

------
jldugger
> 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](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](https://amzn.to/3eTz5RL)) is sort of a layman's
book on the subject of prediction and statistics.

------
0x11
> 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. """

------
iansinke
Around that age, I read "The Heart of Mathematics", by Edward Burger and
Michael Starbird. It's a really fun book which introduces a wide variety of
math concepts while being amusing to read.

[https://www.amazon.com/Heart-Mathematics-invitation-
effectiv...](https://www.amazon.com/Heart-Mathematics-invitation-effective-
thinking/dp/1931914419)

------
phonebucket
William Dunham has two books which are great: 1- Euler (The Master of us all)
2- Journey Through Genius.

John Stillwell’s Mathematics and Its History.

Needham’s Visual Complex Analysis.

------
super_mario
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.

------
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...](https://www.amazon.com/Man-Who-Counted-Collection-
Mathematical/dp/0393309347)

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](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

------
jostylr
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.

------
bhntr3
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.

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.physicsforums.com/forums/science-and-math-
textbooks.21/))

------
giantg2
This isn't bad. I'm surprised it's expensive now.

[https://www.amazon.com/No-bullshit-guide-math-
physics/dp/099...](https://www.amazon.com/No-bullshit-guide-math-
physics/dp/0992001005/ref=mp_s_a_1_1?dchild=1&keywords=no+bullshit+math&qid=1593720395&sr=8-1&pldnSite=1)

~~~
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).

------
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.

------
tuukkah
I appreciated getting from maths to CS with Structure and Interpretation of
Computer Programs: [https://mitpress.mit.edu/sites/default/files/sicp/full-
text/...](https://mitpress.mit.edu/sites/default/files/sicp/full-
text/book/book.html)

------
secabeen
A group of us maintain an annotated bibliography of math textbooks used at the
University of Chicago. The entry level ones (like Spivak's Calculus) would be
good to check out: [https://github.com/ystael/chicago-ug-math-
bib](https://github.com/ystael/chicago-ug-math-bib)

~~~
jfarmer
Spivak was my 1st-year college textbook and it convinced me to major in math.
I highly recommend.

------
MathematicalArt
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.

Because there are many books with this title, I will link to Amazon:
[https://www.amazon.com/Encyclopaedia-Mathematics-Michiel-
Haz...](https://www.amazon.com/Encyclopaedia-Mathematics-Michiel-
Hazewinkel/dp/155608000X/ref=sr_1_12?dchild=1&keywords=Encyclopaedia+of+mathematics&qid=1593874486&s=books&sr=1-12)

------
ljf
[https://en.m.wikipedia.org/wiki/Flatland](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.

------
gen220
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.

------
lanstin
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.

------
pgtan
"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.

------
jgwil2
_How to Prove It_ by Velleman [0]. Should help with the increasing emphasis on
proofs.

[0] [http://users.metu.edu.tr/serge/courses/111-2011/textbook-
mat...](http://users.metu.edu.tr/serge/courses/111-2011/textbook-math111.pdf)

------
KenoFischer
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](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]

------
mellosouls
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.

[https://www.storyofmathematics.com/19th_gauss.html](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...](https://www.goodreads.com/book/show/837010.The_Prince_of_Mathematics)

~~~
aoki
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.

------
yaksha13
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

------
jchallis
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.

~~~
jacobolus
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_.

------
Someone
For broadening their view:

\- Proofs and Refutations by Imre Lakatos
([https://en.wikipedia.org/wiki/Proofs_and_Refutations](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...](https://www.amazon.com/World-Mathematics-Four-
Set/dp/0486432688))

------
noriuday
1) "The Science of Programming", by David Gries, is an excellent book dealing
with mathematical proof based approach to programming.

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.

------
enriquto
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.

------
pgreenwood
"The Symmetries of Things" [1] 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.

[1] [https://www.amazon.com/Symmetries-Things-John-H-
Conway/dp/15...](https://www.amazon.com/Symmetries-Things-John-H-
Conway/dp/1568812205)

------
ColinWright
A list:

[https://www.topicsinmaths.co.uk/cgi-
bin/sews.py?SuggestedRea...](https://www.topicsinmaths.co.uk/cgi-
bin/sews.py?SuggestedReading)

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.

------
njkleiner
This might be a bit of a different take than the other comments here, but I
highly enjoyed reading Things to Make and Do in the Fourth Dimension by Matt
Parker when I first became interested in maths.

------
ljw1001
I would second the recommendation for _Who is Fourier? A Mathematical
Adventure._ It's an unusual and engaging introduction to waves, Fourier
coefficients, and transforms. The slope is gentle but not dumbed down.
[https://www.amazon.com/Who-Fourier-Mathematical-
Adventure-2n...](https://www.amazon.com/Who-Fourier-Mathematical-
Adventure-2nd/dp/0964350432#:~:text=In%20Who%20is%20Fourier%3F,exponentiation%2C%20differentiation%2C%20and%20integration).

------
charlescearl
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/](http://illustratedtheoryofnumbers.com/)
also has Python notebooks if they are into programming.

------
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.

------
ColinWright
"The Joy of X" by Steven Strogatz

"Euler's Gem" by Dave Richeson

"A Companion to Analysis" by Tom Körner

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

------
dmd
[https://en.wikipedia.org/wiki/How_to_Solve_It](https://en.wikipedia.org/wiki/How_to_Solve_It)

------
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.

------
ask1200
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!

------
tjr
I'm going to guess that for the OP, their reader is already past this level,
but sharing anyway for the benefit of others, as I think it's a great book for
roughly around that age:

[https://www.amazon.com/Prof-McSquareds-Calculus-Primer-
Inter...](https://www.amazon.com/Prof-McSquareds-Calculus-Primer-
Intergalactic/dp/0486789705/)

------
jtolmar
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.

------
bmking
Maybe this one "[The Pea And The Sun]([https://www.amazon.com/Pea-Sun-
Mathematical-Paradox/dp/15688...](https://www.amazon.com/Pea-Sun-Mathematical-
Paradox/dp/1568813279\)"). It reads in a nice flow and shows theoretical math
in an understandable way even though it covers a very complex theorem.

------
R3G1R
There are many books on that front, particularly the ones related to
recreational math or intro higher math (see
[https://mathvault.ca/books](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.

------
EdwardWarren
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.

~~~
dpk666
"Maths" is the standard Anglophone term outside of North America.

------
noir_lord
Engineering Mathematics - K.A Stroud

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

~~~
Koshkin
Seconded. Starts with bare essentials and goes a long way. It’s amazing how
much you can learn from this book. (Also, his Advanced Engineering
Mathematics.)

------
wqTJ3jmY8br4RWa
Mathematics: Its Content, Methods and Meaning (3 Volumes in One) Paperback –
by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrent’ev

The best book.

------
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 :) )

------
csense
At that age, I enjoyed Number Theory by George E. Andrews

[https://www.amazon.com/Number-Theory-Dover-Books-
Mathematics...](https://www.amazon.com/Number-Theory-Dover-Books-
Mathematics/dp/0486682528/ref=sr_1_2?dchild=1&keywords=andrews+number+theory&qid=1593829462&sr=8-2)

------
humanendeavor
Your child is probably beyond this but it's one of my favorite pre-calculus
books

Mathematics, a human endeavor by Harold R. Jacobs

[https://openlibrary.org/books/OL5699810M/Mathematics_a_human...](https://openlibrary.org/books/OL5699810M/Mathematics_a_human_endeavor)

------
scott_russell
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.

------
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.

------
tobinfricke
_The Road to Reality_ by Roger Penrose.

[https://www.nytimes.com/2005/02/27/books/review/the-road-
to-...](https://www.nytimes.com/2005/02/27/books/review/the-road-to-reality-a-
really-long-history-of-time.html)

~~~
Koshkin
This book is totally inappropriate for learning mathematics, especially for a
teenager. A _well-prepared_ layman, maybe, but not a teenager who just wants a
good introduction to the subject.

~~~
mhh__
It's an inspired book, but it's not a textbook.

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.

------
nikisweeting
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.

------
maurits
"Calculus made Easy" comes to mind. Probably not the best suggestion here, but
it is available on Gutenberg. [1]

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

------
Tempest1981
For broadening his view, and sparking some fun and joy of maths, try "Humble
Pi" by Matt Parker:

[https://www.goodreads.com/book/show/39074550-humble-
pi](https://www.goodreads.com/book/show/39074550-humble-pi)

------
murkle
Mathographics by Robert Dixon [https://www.amazon.co.uk/Mathographics-Robert-
Dixon/dp/06311...](https://www.amazon.co.uk/Mathographics-Robert-
Dixon/dp/0631148272/)

IIRC it explains how to make the pictures

------
foolmeonce
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.

------
Tempest1981
This may be too basic, but "The Magic of Math" by Arthur Benjamin

[https://www.goodreads.com/book/show/24612214-the-magic-of-
ma...](https://www.goodreads.com/book/show/24612214-the-magic-of-math)

------
asknthrow2020
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.

~~~
mennis16
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.

------
fxtentacle
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.

------
JoeMayoBot
The OpenStax series are free. I've found the explanations very clear and
detailed:

[https://openstax.org/subjects/math](https://openstax.org/subjects/math)

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

------
tobinfricke
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).

------
tobinfricke
Maybe a textbook like _Topology_ by Armstrong, or _Galois Theory_ by Ian
Stewart.

~~~
lanstin
Anything by Iam Stewart is a good read.

------
lalos
Not strictly about math but still recommended and specially for younger folks:
Logicomix.
[https://en.wikipedia.org/wiki/Logicomix](https://en.wikipedia.org/wiki/Logicomix)

------
thecolorblue
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.

------
snicker7
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?

------
ppg677
I wish I read this before taking college-level Calculus

[http://www.gutenberg.org/files/33283/33283-pdf.pdf](http://www.gutenberg.org/files/33283/33283-pdf.pdf)

------
francasso
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.

------
FabHK
+1 for _Proofs from the BOOK_ , _Visual Group Theory_ , and _Flatland_ ,
already recommended by others.

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

------
guidoism
Arithmetic by Paul Lockhart

[https://www.hup.harvard.edu/catalog.php?isbn=9780674237513](https://www.hup.harvard.edu/catalog.php?isbn=9780674237513)

------
lazyant
This is a beautiful book, especially if he or she is into magic: _Magical
Mathematics: The Mathematical Ideas That Animate Great Magic Tricks_ by Persi
Diaconis, Ron Graham

------
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...](https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf)

~~~
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.

------
devchris10
Not a book recommendation but maybe applied math or probability/statistics
towards investing or AI/ML. A fast feedback loop can do wonders for learning.

oraclerank.com kaggle.com

------
SquishyPanda23
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.

Here's why:

\- 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.

------
thanatos519
For the younger, less-motivated child: "Mathematics: A Human Endeavor"

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

------
jameshart
Have they worked through everything Martin Gardner ever wrote?

~~~
pjungwir
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_.

~~~
pjungwir
I thought of a few others that are "fun":

\- _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. :-)

------
tdsamardzhiev
I got Spivak as a first Calculus book and it felt a bit over my head, but if
one already has some knowledge and appreciation of analysis, it'd be a great
gift.

------
zhte415
Lots of book recommendations here already, so a complementary idea:

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

------
galkk
There's a good gift, a bad gift, and a book though

------
nightchalk16
[http://discrete.openmathbooks.org/dmoi3.html](http://discrete.openmathbooks.org/dmoi3.html)

------
Consultant32452
I passed the AP calc exam with calculus for dummies. It was great, though I'm
not sure that kind of title is received well as a gift.

------
genghizkhan
This might be a silly recommendation, but "Higher Algebra" by Hall and Knight
was brilliant for me when I was around 15 or so.

------
Koshkin
Calculus by M.Kline would be not a bad start. For a broad (yet detailed)
overview, Mathematics by Aleksandrov et al. is exceptional.

------
new2628
"Proofs from the book" is very neat.

------
abnry
Godel Escher Bach is a good place to start. Or any book about Godel. That's a
great place to blow a kids mind.

------
KenoFischer
Thought of another one: _Quantum Computing Since Democritus_ by Scott Aaronson

------
laksmanv
Check out betterexplained.com

------
itsshreyarora
Informal math book called An Infinitely large Napkin is amazing for fun math

------
mike00632
I think "Gödel, Escher, Bach" is the perfect book.

------
the_burning_one
Why not give them all, in a handy pdf or e-pub format? ;)

------
wolfi1
"What is Mathematics" by Courant, a classic

------
carlosf
Can't go wrong with Spivak's Calculus.

------
bade
What is Mathematics? by Courant and Robbins

------
ThefinalResult
Walter Rudin real and complex analysis

------
speedcoder
The Kingdom of The Infinite Number

------
apengwin
Art of problem solving volume 2!!

------
tostitos1979
Surely your joking Mr Feinman. I was a child prodigy eons ago and wished I
read that when I was a teen.

------
graycat
Linear algebra, and more than one such book.

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](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/](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.

------
RhysU
Linear Algebra Done Right

------
bilbobagends
Polya’s How to Solve It.

------
SMAAART
Buy them 2 books as follows:

#1: your "The Princeton Companion.." or any of the great suggestions that you
got here

AND THEN

#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.

------
soVeryTired
I stumbled on Q.E.D by Feynman at a young age - it had a deep influence on me.
I also read parts of "the mathematical experience" by Davis and Hersch, and
"Godel, Escher, Bach" by Douglas Hofstadter.

It's not really maths, but _Spacetime Physics: Introduction to Special
Relativity_ would have been great for me at that age.

The Princeton Companion is a cool book, but it'd be better suited to a
graduate in mathematics.

