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I'm interested in elementary books written by good mathematicians. For example:

  • Gelfand (Algebra, Trigonometry, Sequences)
  • Lang (A first course in calculus, Geometry)

I'm sure there are many other ones. Can you help me to complete this short list?

As for the level, I'm thinking of pupils (can be advanced ones) whose age is less than 18.

But books a bit more advanced could interest me. For example Roger Godement books: Analysis I & II are full of nice little results that could be of interest at an elementary level.

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    $\begingroup$ You should also mention the background of students who are going to read these books. How much math do they know? Are they familiar with abstract algebra or analysis? or just elementary algebra and elementary high school geometry? $\endgroup$ – Mohan Feb 25 '13 at 16:22
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    $\begingroup$ @Mohan In fact I was not thinking of books that pupils could read (without the help of a teacher). $\endgroup$ – ranousse Feb 25 '13 at 16:45
  • $\begingroup$ I thought of books where I could find nice little results (often not given in popular textbooks) that could be of interest in my courses (for instance as exercices for advanced pupils) $\endgroup$ – ranousse Feb 25 '13 at 16:54
  • $\begingroup$ How much math do they do know? Elementary Algebra and elementary geometry? $\endgroup$ – Mohan Feb 25 '13 at 16:56
  • $\begingroup$ The things I'm looking for in these books are : a nice presentation of a classical theory, different proofs of classical theorems, simple proofs of difficult results in a simple cases, exercices. These things are very elementary and not much harder than a classical course. $\endgroup$ – ranousse Feb 25 '13 at 17:01

20 Answers 20

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How to Solve it By George Polya.

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  • $\begingroup$ Ok I'll do that. $\endgroup$ – Mohan Feb 25 '13 at 15:48
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An introduction to theory of numbers by Hardy and Wright

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A course of Pure Mathematics by G.H. Hardy.

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    $\begingroup$ This is what got me interested in math when I was around 18. I definitely second this suggestion. $\endgroup$ – nigel Feb 25 '13 at 21:19
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By the way, I don't think age is necessarily the optimal criterium. And at 17 -18, some are already in college. So ability and intent are perhaps more relevant.

If you are not constrained to books per se, here is a link to a free download of what can be considered verbatim transcripts of lectures on real analysis by Fields Medal winner Vaughan Jones.

Here you will find a great mathematician artfully taking the student along assuming no prior knowledge, giving just the right amount of guidance each step of the way.

I personally feel they are akin (although on a smaller scale) to Feynman's lectures on physics: where a real master knows just how to present challenging material to (at the outset) beginning students.

As well, real analysis can be considered the transitional material going from a somewhat mechanical approach to a conceptual, rigorous study of math.

https://sites.google.com/site/math104sp2011/lecture-notes

Also here are books on geometry from Berkeley Math Circle:

  1. Kiselev's Geometry: Book 1, Planimetry Translated from Russian by Alexander Givental Published by: Sumizdat This is a wonderful, easy-going introduction to plane geometry, which was used for decades as a regular textbook in Russian middle schooles. It has been translated from its original Russian to English by one of UC Berkeley's very own math instructors, Professor Alexander Givental. Price: $25

Highly Recommended for BMC Intermediate and Advanced

  1. Kiselev's Geometry: Book 2, Stereometry Translated from Russian by Alexander Givental Published by: Sumizdat This is the second volume of the famous Kiselev's work. A marvelous self-contained exposition on stereometry that proved to be a favorite for generations of students and mathematicians in Russia. Thanks to our UC Berkeley Professor Alexander Givental this book is now available in English. Price: $15

and a link to their recommended publications:

http://mathcircle.berkeley.edu/index.php?options=bmc|recommendedbooks|Recommended%20Books

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Concise introduction to pure mathematics by Martin liebeck

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Introduction to Calculus and Analysis by Courant

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Hilbert, Geometry and the Imagination

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Geometry Revisited by H Coxeter

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Introduction to Geometry By Coxeter

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  • $\begingroup$ Yes, in fact I had this book by age 18. $\endgroup$ – GEdgar Feb 25 '13 at 17:16
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Geometry Euclid and Beyond by Hartshorne

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The Shape of Space by Jeffrey R. Weeks.

To get an idea for some of the topics covered in this book, check this out.

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Not sure how easy an 18 year old might grasp this, but

Paul R. Halmos' Naive Set Theory is definitely a keeper. (obviously on set theory; relatively non-axiomatic)

This is a little more elementary and I think is definitely a good read especially since most high school students live in the world of pre-rigorous mathematics; I think everyone's first exposure to rigorous math is through proofs:

David C. Velleman's How to Prove it (introductory set theory and proof-writing)

EDIT: Age is not an indicator of ability.

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V. Arnol'd's book Mathematical Methods of Classical Mechanics is superb.

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  • $\begingroup$ -1 Do you really think it is elementary? $\endgroup$ – Artem Jan 7 '16 at 4:53
  • $\begingroup$ Yes, or I would not have mentioned it. $\endgroup$ – MJD Jan 7 '16 at 13:19
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Walter Rudin Priciples of Mathematical Analysis

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    $\begingroup$ Perhaps a bit much for 18-year-olds $\endgroup$ – GEdgar Feb 25 '13 at 17:15
  • $\begingroup$ Yep agreed, missed that part :P $\endgroup$ – EhBabay Feb 25 '13 at 19:06
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André Weil's Number theory for beginners is wonderful.

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  • $\begingroup$ Well this one is not really that elementary I would say. $\endgroup$ – Hui Yu Feb 25 '13 at 16:40
  • $\begingroup$ @HuiYu, aren't you confusing it with Weil's Basic Number Theory, which despite the title, is not at all elementary? $\endgroup$ – lhf Feb 25 '13 at 18:08
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Serre's "A Course in Arithmetic".

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    $\begingroup$ Is it elementary? $\endgroup$ – Seirios Feb 25 '13 at 16:22
  • $\begingroup$ It depends what you mean by "elementary". This one definitely falls under the clause "But books a bit more advanced could interest me" by the OP. $\endgroup$ – Álvaro Lozano-Robledo Feb 25 '13 at 16:37
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    $\begingroup$ I also find this one "too much" advanced (and not just "a bit" advanced). $\endgroup$ – ranousse Feb 25 '13 at 16:44
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    $\begingroup$ This book requires knowledge of abstract algebra and complex analysis. It's meant for advanced undergraduates. But it's an awesome book(Anything by Serre is great). $\endgroup$ – Mohan Feb 25 '13 at 16:50
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Calculus by Michael Spivak.

It's very rigorous, but it starts from ground zero.

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"Mathematics - Form and Function", by Saunders MacLane could be read by a bright 18-year old.

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My suggestion is:

Note that this is not the (much harder) book Algebra by the same authors. I was fortunate in that this book was in my school library when I was doing A-levels (so I was around 17 or 18), and it continues to influence my interest in abstract algebra.

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