Elementary books by good mathematicians

Question

Elementary books by good mathematicians

4.7k Views Asked by Bumbble Comm At 03 Apr 2026 - 5:50

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.

Original Q&A

There are 19 best solutions below

**Bumbble Comm** · Answer 1 · 2013-02-25 14:59:28

Bumbble Comm On 25 Feb 2013 - 2:59

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

**Bumbble Comm** · Answer 2 · 2013-02-25 15:21:13

Bumbble Comm On 25 Feb 2013 - 3:21

How to Solve it By George Polya.

**Bumbble Comm** · Answer 3 · 2013-02-25 15:29:13

~~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.

**Bumbble Comm** · Answer 4 · 2013-02-25 15:31:30

Bumbble Comm On 25 Feb 2013 - 3:31

An introduction to theory of numbers by Hardy and Wright

**user12802** · Answer 5 · 2013-02-25 15:50:06

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:

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

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

**Bumbble Comm** · Answer 6 · 2013-02-25 15:50:43

Bumbble Comm On 25 Feb 2013 - 3:50

Solving Mathematical Problems: A Personal Perspective by Terence Tao

**Bumbble Comm** · Answer 7 · 2013-02-25 15:59:24

Bumbble Comm On 25 Feb 2013 - 3:59

Geometry Revisited by H Coxeter

**Bumbble Comm** · Answer 8 · 2013-02-25 16:02:08

Bumbble Comm On 25 Feb 2013 - 4:02

Introduction to Geometry By Coxeter

**Bumbble Comm** · Answer 9 · 2013-02-25 16:06:22

Bumbble Comm On 25 Feb 2013 - 4:06

Geometry Euclid and Beyond by Hartshorne

**student** · Answer 10 · 2013-02-25 16:08:16

student On 25 Feb 2013 - 4:08

The Shape of Space by Jeffrey R. Weeks.

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

**Bumbble Comm** · Answer 11 · 2013-02-25 16:12:25

Bumbble Comm On 25 Feb 2013 - 4:12

Walter Rudin Priciples of Mathematical Analysis

**Bumbble Comm** · Answer 12 · 2013-02-25 16:12:25

Bumbble Comm On 25 Feb 2013 - 4:12

André Weil's Number theory for beginners is wonderful.

**Bumbble Comm** · Answer 13 · 2013-02-25 16:15:12

Bumbble Comm On 25 Feb 2013 - 4:15

Concise introduction to pure mathematics by Martin liebeck

**Bumbble Comm** · Answer 14 · 2013-02-25 16:15:22

Bumbble Comm On 25 Feb 2013 - 4:15

Serre's "A Course in Arithmetic".

**Bumbble Comm** · Answer 15 · 2013-02-25 16:28:18

Bumbble Comm On 25 Feb 2013 - 4:28

Introduction to Calculus and Analysis by Courant

**Bumbble Comm** · Answer 16 · 2013-02-25 16:37:36

Bumbble Comm On 25 Feb 2013 - 4:37

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

**Bumbble Comm** · Answer 17 · 2013-02-25 20:59:18

Bumbble Comm On 25 Feb 2013 - 8:59

Hilbert, Geometry and the Imagination

**Bumbble Comm** · Answer 18 · 2013-03-06 14:25:56

Bumbble Comm On 06 Mar 2013 - 2:25

Calculus by Michael Spivak.

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

**Bumbble Comm** · Answer 19 · 2013-03-06 15:28:22

Bumbble Comm On 06 Mar 2013 - 3:28

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

Elementary books by good mathematicians

There are 19 best solutions below

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