[HN Gopher] Ask HN: Recommend a maths book for a teenager?
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Ask HN: Recommend a maths book for a teenager?
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?
Author : andyjohnson0
Score : 142 points
Date : 2020-07-02 11:03 UTC (1 days ago)
| tuukkah wrote:
| I appreciated getting from maths to CS with Structure and
| Interpretation of Computer Programs:
| https://mitpress.mit.edu/sites/default/files/sicp/full-text/...
| noir_lord wrote:
| Engineering Mathematics - K.A Stroud
|
| It's sometimes useful to see the context of mathematics and it's
| purpose beyond the intrinsic beauty.
| logicslave wrote:
| The classic text on analysis is Principles of Mathematical
| Analysis by Rudin. Its very difficult and leaves it to the reader
| to understand the terse proofs. It starts from the beignning,
| with no math background assumed about the reader. The terse
| proofs are written in such a way to force the reader to gain deep
| mathematical intuition. Some of the proofs are elegant and
| beautiful. I would absolutely recommend it. You can see a pdf
| here:
|
| https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_math...
| bordercases wrote:
| > 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 wrote:
| 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.
| tobinfricke wrote:
| 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).
| btrettel wrote:
| 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...
|
| 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.
| LordOmlette wrote:
| 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-.
| [deleted]
| layoutIfNeeded wrote:
| 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/...
| tobinfricke wrote:
| Maybe a textbook like _Topology_ by Armstrong, or _Galois Theory_
| by Ian Stewart.
| guidoism wrote:
| Arithmetic by Paul Lockhart
|
| https://www.hup.harvard.edu/catalog.php?isbn=9780674237513
| dTal wrote:
| 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.
| graycat wrote:
| 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
|
| Since the book is new, I've only looked through it -- it looks
| like a good selection and arrangement of topics. And Boyd is
| good, wrote a terrific book, maybe, IMHO likely, the best in the
| world, on convexity, which is in a sense is _half_ of
| _linearity_.
|
| Some course slides are available at
|
| http://vmls-book.stanford.edu/
|
| For reference for more, have a copy of
|
| Richard Bellman, _Introduction to Matrix Analysis: Second
| Edition_.
|
| Bellman was famous for dynamic programming.
|
| For computations in linear algebra, consider
|
| George E. Forsythe and Cleve B. Moler, _Computer Solution of
| Linear Algebraic Systems_
|
| although now the Linpack materials might be a better starting
| point for numerical linear algebra. Numerical linear algebra is
| now a well developed specialized field, and the Linpack materials
| might be a good start on the best of the field. Such linear
| algebra is apparently the main yardstick in evaluating the highly
| parallel supercomputers.
|
| After linear algebra go through
|
| Rudin, _Principles of Mathematical Analysis_ , Third Edition.
|
| He does the Riemann integral very carefully, Fourier series,
| vector analysis via exterior algebra, and has the inverse and
| implicit function theorems (key to differential geometry, e.g.,
| for relativity theory) as exercises.
|
| All of this material is to get to the main goals of measure
| theory, the Lebesgue integral, Fourier theory, Hilbert space and
| Banach space as in, say, the first, real (not complex) half of
|
| Rudin, _Real and Complex Analysis_
|
| But for that I would start with
|
| Royden, _Real Analysis_
|
| _sweetheart_ writing on that math.
|
| Depending on the math department, those books might be enough to
| pass the Ph.D. qualifying exam in Analysis. It was for me: From
| those books I did the best in the class on that exam.
|
| Moreover, from independent study of Halmos, Nering, Fleming,
| Forsythe, linearity in statistics, and some more, I totally blew
| away all the students in a challenging second (maybe
| intentionally flunk out), advanced course in linear algebra and,
| then, did the best in the class on the corresponding qualifying
| exam, that is, where that second course was my first formal
| course in linear algebra.
|
| Lesson: Just self study of those books can give a really good
| background in linear algebra and its role in the rest of pure and
| applied math.
|
| No joke, linear algebra, and the associated vector spaces, is one
| of the most important courses for more work in pure and applied
| math, engineering, and likely the future of computing.
| thecolorblue wrote:
| 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.
| coeneedell wrote:
| For something that's a little more fun to read and covers
| fundamental topics. (Foundations for higher mathematics) I'd
| recommend Godel, Escher, Bach by Douglas Hofsteader. It changed
| the way I approach problems to this day.
| phonebucket wrote:
| 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 wrote:
| 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.
| Consultant32452 wrote:
| 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.
| analbumcover wrote:
| 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.
| SamReidHughes wrote:
| 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.
| KenoFischer wrote:
| 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]
| new2628 wrote:
| "Proofs from the book" is very neat.
| giantg2 wrote:
| This isn't bad. I'm surprised it's expensive now.
|
| https://www.amazon.com/No-bullshit-guide-math-physics/dp/099...
| ivan_ah wrote:
| 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.
| soVeryTired wrote:
| 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.
| lanstin wrote:
| 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.
| rramadass wrote:
| 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 wrote:
| 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.
| zakk wrote:
| I suggest "What Is Mathematics?" by Richard Courant and Herbert
| Robbins.
|
| https://en.wikipedia.org/wiki/What_Is_Mathematics%3F
| blendo wrote:
| 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
| asknthrow2020 wrote:
| 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.
| rokobobo wrote:
| Seconded. For what it's worth, Harvard's Math 55 uses that as
| its textbook.
| pmiller2 wrote:
| 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...
| kolinko wrote:
| When I saw the title, this was the first book that came to my
| mind. Very nicely written, possibly the best math book I've
| read (as a Computer Science MSc)
| javajosh wrote:
| 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...
| wolfi1 wrote:
| "What is Mathematics" by Courant, a classic
| Someone wrote:
| For broadening their view:
|
| - Proofs and Refutations by Imre Lakatos
| (https://en.wikipedia.org/wiki/Proofs_and_Refutations) (makes you
| think about what a proof really is)
|
| - The World of Mathematics: not a lot of math proper, doesn't
| have much depth, but lots of examples of applied math, interwoven
| with mentions of the history of mathematics
| (https://www.amazon.com/World-Mathematics-Four-Set/dp/0486432...)
| francasso wrote:
| 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.
| MperorM wrote:
| During the first year of my undergrad someone introduced me to
| Godel, 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 :)
| [deleted]
| pvitz wrote:
| I read it when I was 16 and it was just wonderful. I can also
| recommend it.
| javajosh wrote:
| I have to disagree with you here, and strongly. I don't think
| _Godel, 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.
| [deleted]
| msla wrote:
| > I don't think Godel, 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 wrote:
| 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 Miserables_ , and I would never ever give the former
| to a teenager.
| carlosf wrote:
| Can't go wrong with Spivak's Calculus.
| galkk wrote:
| There's a good gift, a bad gift, and a book though
| javajosh wrote:
| 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...)
|
| 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.
| fxtentacle wrote:
| 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.
| prof-dr-ir wrote:
| 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?
| jostylr wrote:
| 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.
| mike00632 wrote:
| I think "Godel, Escher, Bach" is the perfect book.
| tjr wrote:
| 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...
| debbiedowner wrote:
| 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.
| gen220 wrote:
| 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.
| wqTJ3jmY8br4RWa wrote:
| 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.
| enriquto wrote:
| 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.
| iansinke wrote:
| 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...
| foolmeonce wrote:
| 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.
| JoeMayoBot wrote:
| The OpenStax series are free. I've found the explanations very
| clear and detailed:
|
| https://openstax.org/subjects/math
|
| Some are even downloadable to a Kindle (for free) on Amazon.
| enhdless wrote:
| _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.
| generationP wrote:
| _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 Korner, just because of the writing. Seriously,
| he can make the axiomatic construction of the real number system
| read like a novel; open
| https://web.archive.org/web/20190813160507/https://www.dpmms...
| on any page and you will see.
|
| _Proofs from the BOOK_ by Aigner and Ziegler is a cross-section
| of some of the nicest proofs in reasonably elementary (read:
| undergrad-comprehensible) maths. Might be a bit too advanced,
| though (the writing is terse and a lot of ground is covered).
|
| _Problems from the BOOK_ by Andreescu and Dospinescu (a play on
| the previous title, which itself is a play on an Erdos 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 wrote:
| 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.
| 0x11 wrote:
| > 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. """
| tostitos1979 wrote:
| Surely your joking Mr Feinman. I was a child prodigy eons ago and
| wished I read that when I was a teen.
| SquishyPanda23 wrote:
| 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.
| nbernard wrote:
| _The Pleasures of Counting_ by T. W. Korner. 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.
| seesawtron wrote:
| Jordan Ellenberg's "How not to be wrong". Recommended even for
| non teenagers.
| mhh__ wrote:
| 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 wrote:
| If you like pictures, another couple nice books are Nathan
| Carter's _Visual Group Theory_ and Marty Weissman's
| _Illustrated Theory of Numbers_ , both of which should be
| accessible to motivated high school students.
|
| http://web.bentley.edu/empl/c/ncarter/vgt/
|
| http://illustratedtheoryofnumbers.com
| exmadscientist wrote:
| 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....
| SMAAART wrote:
| Buy them 2 books as follows:
|
| #1: your "The Princeton Companion.." or any of the great
| suggestions that you got here
|
| AND THEN
|
| #2: "Godel, 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.
| pera wrote:
| _Mathematics: A Discrete Introduction_ by Edward R. Scheinerman:
|
| https://books.google.com/books/about/Mathematics_A_Discrete_...
|
| I bought this book when I was ~16 because I wanted to learn some
| discrete maths, but it actually touches many different
| interesting topics that you don't see in secondary school
| (including some cryptography!).
| jchallis wrote:
| 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 wrote:
| 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_.
| montalbano wrote:
| 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
|
| 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
| anirudhcoder wrote:
| https://www.amazon.com/Mathematical-Circles-Dmitry-Fomin/dp/...
|
| 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 wrote:
| This would be better for a 10-14 year old in middle school or
| early high school.
| enriquto wrote:
| 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 wrote:
| 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.
| scythe wrote:
| >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...
|
| 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 wrote:
| 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. :)
| pgtan wrote:
| "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.
| ljf wrote:
| 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 wrote:
| 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.
| Koshkin wrote:
| Calculus by M.Kline would be not a bad start. For a broad (yet
| detailed) overview, Mathematics by Aleksandrov et al. is
| exceptional.
| ColinWright wrote:
| "The Joy of X" by Steven Strogatz
|
| "Euler's Gem" by Dave Richeson
|
| "A Companion to Analysis" by Tom Korner
|
| "Elementary Number Theory: A Problem Oriented Approach" by Joe
| Roberts
| impendia wrote:
| 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.
| [deleted]
| jgwil2 wrote:
| _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...
| mci wrote:
| _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...
| maurits wrote:
| "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
| scorecard wrote:
| 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/
| crawftv wrote:
| 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.
| ColinWright wrote:
| A list:
|
| https://www.topicsinmaths.co.uk/cgi-bin/sews.py?SuggestedRea...
|
| For a single suggestion, "How to Think Like a Mathematician" by
| Kevin Houston.
|
| A second suggestion: "A Companion to Analysis" by Tom Korner.
|
| 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.
| Phithagoras wrote:
| "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
| njkleiner wrote:
| 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.
| jameshart wrote:
| Have they worked through everything Martin Gardner ever wrote?
| tobinfricke wrote:
| _The Road to Reality_ by Roger Penrose.
|
| https://www.nytimes.com/2005/02/27/books/review/the-road-to-...
| Koshkin wrote:
| 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__ wrote:
| 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.
| analbumcover wrote:
| I would be absolutely flabbergasted if a layman could get
| through _Road to Reality_. Even those with an engineering or
| computer science background would struggle considerably.
|
| That said, it serves as a great overview of physics for
| mathematicians and perhaps as very casual intro to twistor
| theory for physicists.
| nightchalk16 wrote:
| http://discrete.openmathbooks.org/dmoi3.html
| Tempest1981 wrote:
| 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
| bmking wrote:
| Maybe this one "[The Pea And The Sun](https://www.amazon.com/Pea-
| Sun-Mathematical-Paradox/dp/15688.... It reads in a nice flow and
| shows theoretical math in an understandable way even though it
| covers a very complex theorem.
| tromp wrote:
| Surreal Numbers by Knuth is great, although not related to
| analyis/calculus:
|
| https://www.amazon.com/Surreal-Numbers-Donald-Knuth/dp/02010...
| laksmanv wrote:
| Check out betterexplained.com
| speedcoder wrote:
| The Kingdom of The Infinite Number
| gramie wrote:
| You could check out Burn Math Class: https://www.amazon.com/Burn-
| Math-Class-Reinvent-Mathematics/...
|
| or Calculus Made Easy:
| https://www.math.wisc.edu/~keisler/keislercalc-09-04-19.pdf
| Tempest1981 wrote:
| This may be too basic, but "The Magic of Math" by Arthur Benjamin
|
| https://www.goodreads.com/book/show/24612214-the-magic-of-ma...
| airstrike wrote:
| A bit on the lighter side, I do recommend The Man Who Counted
| which I read as a kid and absolutely loved
|
| https://www.amazon.com/Man-Who-Counted-Collection-Mathematic...
|
| I read the original in Portuguese but would assume it's just as
| good in English, given overwhelmingly positive reviews on Amazon
|
| See also https://en.wikipedia.org/wiki/The_Man_Who_Counted
|
| It won't really teach him math per se, but if my experience is
| any indication, it will get him hooked on developing intuition
| and he'll find beauty in otherwise mundane topics such as
| arithmetic. It's an incredibly engaging story aimed at younger
| readers but fun for people of all ages - think Arabian Nights
| with a character that loves math.
|
| Come to think of it, I've got to buy it again and re-read it one
| of these days
| bobmaxup wrote:
| Jan Gullberg - Mathematics: From the Birth of Numbers
|
| https://www.amazon.com/gp/product/039304002X
|
| Amazon.com Review What does mathematics mean? Is it numbers or
| arithmetic, proofs or equations? Jan Gullberg starts his massive
| historical overview with some insight into why human beings find
| it necessary to "reckon," or count, and what math means to us.
| From there to the last chapter, on differential equations, is a
| very long, but surprisingly engrossing journey. Mathematics
| covers how symbolic logic fits into cultures around the world,
| and gives fascinating biographical tidbits on mathematicians from
| Archimedes to Wiles. It's a big book, copiously illustrated with
| goofy little line drawings and cartoon reprints. But the real
| appeal (at least for math buffs) lies in the scads of problems--
| with solutions--illustrating the concepts. It really invites
| readers to sit down with a cup of tea, pencil and paper, and
| (ahem) a calculator and start solving. Remember the first time
| you "got it" in math class? With Mathematics you can recapture
| that bliss, and maybe learn something new, too. Everyone from
| schoolkids to professors (and maybe even die-hard mathphobes) can
| find something useful, informative, or entertaining here.
| --Therese Littleton
| linguae wrote:
| 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.
| ARandomerDude wrote:
| Ordered, thank you for the recommendation!
| murkle wrote:
| Mathographics by Robert Dixon
| https://www.amazon.co.uk/Mathographics-Robert-Dixon/dp/06311...
|
| IIRC it explains how to make the pictures
| screye wrote:
| 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...
| tdsamardzhiev wrote:
| 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.
| bhntr3 wrote:
| 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...)
| npr11 wrote:
| 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.
| dmd wrote:
| https://en.wikipedia.org/wiki/How_to_Solve_It
| alimw wrote:
| 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 :) )
| sgillen wrote:
| 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 wrote:
| Chaos by James Gleick is also a good intro to some chaos maths
| and a great bio read, with the stories of Feigenbaum wrestling
| with the new ideas.
| JoshTriplett wrote:
| 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".
| KenoFischer wrote:
| Thought of another one: _Quantum Computing Since Democritus_ by
| Scott Aaronson
| uptownfunk wrote:
| First read https://artofproblemsolving.com/news/articles/avoid-
| the-calc...
|
| 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...
| uptownfunk wrote:
| Also art of problem solving vols 1-2 are now classics for that
| age.
| snicker7 wrote:
| 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?
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