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I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would prefer texts written by well known mathematicians. My first recommendation to him was Courant's book "What is mathematics?" for an introduction while I look for more systematic study texts. I am specially interested in an introductory study of calculus and elementary geometry (for introductions to number theory I think A. Baker or Hardy's books will fit). English, French or Spanish are ok.

Edit. Please note that I'm not interested in problem-solving books, nor lists of problems. Thank you anyway for those answers (Arnold's list seems interesting, even though for other purposes).

Vinteuil
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    I am aware that this question may not be appropriate for mathoverflow, so I will understand if you remove it, but before doing it let me explain that I placed it here because my primary interest is the opinion of professional mathematicians. In particular I am not interested in migrating to mathstackexchange or to pedagogical forums. So I would appreciate you left it open at least a couple of days. Also, I have read http://mathoverflow.net/questions/4023/text-for-an-introductory-real-analysis-course but I think most answers there are thinking in a reader with more background. – Vinteuil Apr 27 '15 at 08:13
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    You could try texts from the "new mathematical library" (it was new in the 1960s!). For example, Coxeter and Greitzer wrote a book "geometry revisited", and there are many more by many others. The books were aimed at high school students, and required lots of effort but little knowledge. – user1729 Apr 27 '15 at 08:22
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    Any books by Martin Gardner ... (I read those myself at about that age) – kjetil b halvorsen Apr 27 '15 at 12:10
  • The following question seems pretty similar http://mathoverflow.net/questions/186244/math-books-for-advanced-high-school-students // I now note you gave an answer there. I think it could help if you explained more specifically what you are looking for and what woudl be the difference. Else, we will get just again about the same list (which is about the same as quite a few thers), as initial answers show. –  Apr 27 '15 at 12:27
  • @user1729, I had not considered the "new mathematical library" since I thought the texts there were not enough "textbooks", but I have already take a look to Coxeter and Greitzer and it is precisely the kind of books I am looking for. I still have to look at other texts in that series. – Vinteuil Apr 27 '15 at 16:40
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    @quid, sorry if not clear enough the differences. Both questions refer to students of similar age, but what I look for is essentially the opposite to that other question. I asked for systematic study texts, while in that other question I understand that this is precisely what is not desired. In fact many answers there are orientated to problem solving (as some in this one, though this is not what I wanted). I have edited the question with the hope it would be more clear what I am asking. – Vinteuil Apr 27 '15 at 16:52

3 Answers3

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I would recommend Mathematical Omnibus: Thirty Lectures on Classic Mathematics, by D. B. Fuks and Serge Tabachnikov: http://books.google.ru/books?id=bomkJMq2H9sC&source=gbs_similarbooks and also books from the series Kvant Selecta: http://books.google.ru/books/about/Kvant_Selecta.html?id=aJLcCFyLwhEC&redir_esc=y

P.S. For systematic study of elementary geometry a good book is Kiselev's Geometry (in two parts): http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202 combined with Prasolov's "Problems in Plane and Solid Geometry": http://students.imsa.edu/~tliu/Math/planegeo.pdf

For an introductory study of basics calculus, I recommend Spivak's "The Hitchhiker's Guide to Calculus": http://www.amazon.com/dp/0883858126/?tag=stackoverfl08-20 and then "Analysis by Its History", by Ernst Hairer and Gerhard Wanner: http://www.springer.com/gp/book/9780387945514

Zurab Silagadze
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I support very much the idea to recommend the Fuks and Tabachnikov book as in the answer above, and in addition to this two book by Vladimir Arnold:

  1. Arnold's problems book for school students, which is freely available via http://imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf . Note that it is also available in many languages on the same web site.

  2. Arnold's Catastrophy Theory, which gives a great introduction to dynamical systems and related topics.

Martin Peters
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  1. Courant and Robbins, What is mathematics?

  2. Hilbert, Cohn-Vossen, Geometry and imagination.

Alexandre Eremenko
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