Classic Mathematics Books for Lifelong Learners

"Challenging Mathematical Problems with Elementary Solutions" by Yaglom and Yaglom, Volume 1 and 2.

Volume 1 contains 100 problems from probability and combinatorics. Volume 2 contains 74 problems from a variety of areas including points and lines, lattices of points in the plane, topology, convex polygons, distribution of objects, nondecimal counting, theory of primes. Complete solutions are included for each problems, as well as hints.

Available from Dover so relatively inexpensive but good quality. Here are the Dover links, but of course they are available from Amazon and the other usual places. I'm linking to Dover because that will have the most complete description.

http://store.doverpublications.com/0486655369.html

http://store.doverpublications.com/0486655377.html

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"Three Pearls of Number Theory" by Khinchin. One of Khinchin's former students was seriously wounded in WWII, and to pass the time during his long recovery in the hospital he wrote to his old professor and asked if he had anything mathematical to study to pass the time.

Khinchin wrote back with three problems in elementary number theory that had recently been solved by people who were not a "great number theorist". Khinchin gave his former student the proofs along with guidance, examples, clarifications, and notes to help understand them.

Dover link: http://store.doverpublications.com/0486400263.html

Review at MAA: https://www.maa.org/press/maa-reviews/three-pearls-of-number...

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"The Enjoyment of Math" by Rademacher and Toeplitz. The MAA review has a good summary:

> This is a serious math book that has minimal prerequisites: geometry and college algebra, but no trig or calculus. It contains 28 largely independent chapters that solve a variety of famous and difficult math problems, mostly in the areas of plane geometry and number theory. The problems include: Fermat’s last theorem for exponent 4, unique factorization in number fields, a number of geometrical maximization problems including several versions of the isoperimetric problem, some transfinite numbers, the 5-color map coloring theorem, and the arithmetic mean - geometric mean inequality. There’s no analysis per se in the book, but several topics depend on the analytic ideas of continuity and variation.

> This book was first published in German in 1930 and in English in 1957 as The Enjoyment of Mathematics, and is still in print today in both languages. This implies that there is still an audience for it, but it is hard to imagine exactly what this audience is. The book was developed out of a series of public lectures and was intended as a “popular math” book. While it is very clear and well-written, the reasoning in all the chapters is very intricate (especially in the geometric problems), and the book is much more difficult than anything that appears in popular math books being written today. It’s also too difficult for a math appreciation text. The modern (2000) Preface to the German edition suggests that the book is suited for bright high-school students who are hungry for learning, and maybe this is its real audience today

https://www.maa.org/press/maa-reviews/the-enjoyment-of-math

https://www.amazon.com/gp/product/B07DMWX5FC/

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Anneli Lax New Mathematical Library is a whole series of books described thusly at the AMS site:

> Featuring fresh approaches and broad coverage of topics especially suitable for high school and the first two years of college, the volumes in this series are an excellent source of enrichment material for teachers and students. Good mathematical reading with lively exposition.

https://bookstore.ams.org/nml

I read "Ingenuity in Mathematics" by Honsberger in high school and it was good. Kind of like "The Enjoyment of Mathematics" but a lot easier.

A lot of books in this series can be good stepping stones to more advances books. For example, Olds "Continued Fractions" could be a reasonable read before then reading Khinchin's "Continued Fractions". The latter is available from Dover and is about 1/3 the price of the Olds book, so personally I'd start with Khinchin, and if it turns out a simpler intro is needed then I'd get Olds.

This is a good point to toss in a note about Dover. They like to take older books, often out of print, get the rights to them, and publish a relatively inexpensive but high quality paperback edition. The difficulty level ranges from classic elementary intro texts to advanced material for practicing mathematicians. (And not just math...they do this for physics, chemistry, and various other fields of science and engineering).

If you are interested on some math topic and want a book on it, it is usually a good idea to have a look at the Dover catalog to see if they have something about that at the level you are looking for.

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"A Book of Abstract Algebra" by Pinter, available as a Dover edition.

http://store.doverpublications.com/0486474178.html

The usual undergraduate abstract algebra stuff: groups, rings, fields, the impossibility of the classic Greek duplicating the cube and trisecting the angle problem, Galois theory and solvability by radicals.

What sets this book apart is that although it is rigorous and proves nearly everything, it takes things in smaller steps than a lot of other books, and has a lot of well chosen exercises that further cement the material, often by applying it to some interesting practical area. The exercises are grouped into sections, each of which focuses on a particular concept from the chapter, or develops and proves interesting things. One or two exercises from each of these sections usually has a solution given.

Only about $12 at Amazon. If you haven't done much proof-based math before this could be a good first proof-based book.

And the mentioned books, at least some of them, serve different purposes: they aim to inspire, to motivate, and to offer historical context and intuition. The last is especially important, as they show people how abstract concepts emerged from historically concrete endeavors.

When it comes to learning math, you can't take an elite's view. Not everyone is born Bourbaki dudes or Galois or people like them. Ordinary people like me don't just fall in love with maths. I was certainly not interested in number theory as I thought it was too fundamental for me to spend serious time on. And I'm still not. I was certainly puzzled on why my professors introduced the concept of functional in linear algebra or quotient groups in algebra or lattice equations in program analysis or category theory in model checking or probability space in probability. After all, all I wanted was to learn how to model the world to be a better programmer. And I was not able to grasp the abstractions without serious effort. I needed historical context and motivations to plow through those topics and to enjoy math.

Yes, the elites will love and excel at math for no particular reason. Yet it is the middle majority like me who will greatly benefit from the mentioned books and biographies and what not.