Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's Elementary Mathematics from an Advanced Standpoint and the books of the Gelfand Correspondence School - school-level books with a university ethos.

share|improve this question
    
I don't think that this question is really appropriate for MO. –  Andy Putman Sep 20 '13 at 19:02
17  
I realize that MO has changed since the old days, but still I think this question is appropriate. Insightful books on elementary mathematics are quite uncommon, and I'd like to see more of them. –  John Stillwell Sep 20 '13 at 19:24
    
Mathoverflow questions that guide up to the lists of great books -- I adore. I always add to favorites. This is totally appropriate! –  Olga Nov 11 '13 at 5:08

12 Answers 12

Geometry and the Imagination by Hilbert and Cohn-Vossen.

share|improve this answer
    
Yes, an excellent suggestion. –  Todd Trimble Sep 20 '13 at 19:32
    
Agree with Todd. –  Colin McLarty Sep 21 '13 at 0:15
1  
Excellent book indeed, but I would not call this "elementary mathematics". –  Alexandre Eremenko Sep 21 '13 at 4:13
    
Well, the book certainly starts off elementary, with discussions of conics, culminating in a beautiful argument that slices of cones have foci. I think it was meant to be understood by non-mathematicians. –  Todd Trimble Sep 21 '13 at 13:24

If first-order logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "Godel, Escher, Bach", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of encoding "n is a power of 10" as a predicate in Peano arithmetic is a wonderful one, quite tough even for professional mathematicians, especially if one is to avoid any form of the Godel numbering trick.)

share|improve this answer
    
I agree that it is a good and accessible book with significant mathematical content. One of the dangers is that you look for other books which attempt such holistic approaches in other sciences and you do not find them. –  The Masked Avenger Sep 20 '13 at 22:08

What Is Mathematics? An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins

Lessons in Geometry by Jacques Hadamard, and its companion books: Hadamard's Plane Geometry and Hadamard: elementary geometry. solutions and notes to supplementary problems by Mark Saul.

share|improve this answer
1  
Finally someone took the trouble to republish Hadamard's Lessons! And translate Perepelkin's solutions, even. Wow!! –  darij grinberg Sep 21 '13 at 2:19

Mathematics: A Very Short Introduction by Timothy Gowers. It is very short and indeed very insightful. It is not a textbook, but includes some school-mathematics topics. From the cover:

The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school.

share|improve this answer

I really like Concrete Mathematics by Knuth, Graham and Patashnik, and the introductions to number theory by Rose and by Hardy&Wright: you will find there many interesting school-like problems (but the whole books may not be suitable).

In geometry, I can suggest Hartshorne's Geometry: Euclid and beyond.

Books like Géométrie projective by Pierre Samuel or Artin's Geometric algebra contain a lot of algebra, but it is geometric instead of abstract, so you may judge they are on the safe side.

share|improve this answer
    
The book Concrete Mathematics (which I admire greatly) is by Knuth, Graham, and Patashnik. –  Todd Trimble Sep 21 '13 at 12:51
    
Oh thank you very much for pointing this out! This is indeed a delightful book! –  Michael Grünewald Sep 21 '13 at 13:13

I recommmend How to prove it by Daniel J. Velleman. The book introduces the basic logic and proof method to beginners and have many good examples and exercises to make students better understanding on what is a proof in the very elementary mathematics.

share|improve this answer

Euclid's elements. i find it much more useful than Klein's books, but that may mean i misunderstand the question. indeed after many years of perusing them, i find Klein's "from an advanced standpoint" books more of a polemic than a useful text. Euclid on the other hand introduces many of the main ideas of modern mathematics.

share|improve this answer

Mathematics Made Difficult by Carl E. Linderholm.

share|improve this answer
1  
It's a curious choice. I cannot deny that it discusses elementary mathematics in an insightful way. And yet, it is so wry or ironic about how those insights are formulated that it basically subverts the presumptive purpose of discussing elementary mathematics in an insightful way! (When I first saw the book as an undergraduate, I wasn't really in on the "joke".) –  Todd Trimble Sep 21 '13 at 12:56

Walter Prenowitz and Meyer Jordan, Basic Concepts of Geometry.

share|improve this answer

I always enjoyed "How to Solve It: A New Aspect of Mathematical Method" by G. Pólya.

It doesn't really cover all that much mathematics, it just helps you structure your thoughts in a mathematical sense.

But it depends a lot on your actual needs.

share|improve this answer

I personally enjoyed these books:

How To Solve It by George Polya

Geometry Revisited by H. S. M. Coxeter , Samuel L. Greitzer

share|improve this answer

How about Lawvere&Schanuel's "Conceptual Mathematics: A First Introduction to Categories"? See, e.g., http://books.google.fr/books?id=h0zOGPlFmcQC&lpg=PP1&dq=lawvere&pg=PP1#v=onepage&q=lawvere&f=false

share|improve this answer

protected by François G. Dorais Sep 21 '13 at 20:53

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.