# What are some great books for exploring mathematics? (not kids' books and not textbooks)

People often think of math as facts and procedure - dry stuff. But it is so much more, even at basic levels. What books about mathematics have you been inspired by? There are some real treasures out there - what are they?

If your interest is in children's books, please check out this question.

What are some great books for exploring mathematics?

Two good books that I liked when I read them years ago are

• Simon Singh: The Code Book. This is a great introductory book to cryptography. The book is not very mathematical heavy, but cryptography is very related to number theory, so I think the book works well and can function well as an inspirational book.

• Simon Singh: Fermat's Enigma. This is a book about Fermat's Last Theorem/ Simon Singh does a great job (IMO) at surveying the history behind FLT. He manages well to bring the mathematics down to an elementary level.

• I would suggest also a book on mathematical logic. There a lot of good books out there. I think that studying mathematical logic can inspire because it looks behind mathematics to the world of logic that mathematics formally is built upon. There are some great formal treatments out there. One concrete suggestion is Gödel's Proof by Ernest Nagel and James Newman. This book discusses the famous (infamous?) theorems of Gödel and works very well as a soft and gentle introduction to logic. There is also a list of references for mathematical logic here: https://math.stackexchange.com/questions/140681/where-to-begin-with-foundations-of-mathematics.

• It seems that many mathematicians got inspiration to study mathematics from studying problem books. There are, of course, a bunch of these out there. One is The USSR Olympiad Problem Book. I believe that a good problem book can foster an the inquisitive mind that is needed in mathematics. One experiences the joy of problem solving.

• To add another book on logic, I very much enjoyed Bertrand Russell's Introduction to Mathematical Philosophy when I was in high school – Steven Gubkin Mar 16 '14 at 20:17
• Pretty much anything is by Simon Singh is a pure gem. – Gamma Function Mar 19 '14 at 0:03

William Dunham's Journey Through Genius is, ultimately, about a bunch of facts, but it's written very well and can be inspiring to a budding math student.

How to Lie with Statistics is just a classic and deserves to be mentioned, even thought it's not really "math-heavy".

The Kaplans' The Art of the Infinite is genuinely playful and seeks to present math as the opposite of dry.

Also, I think an infrequently-tapped way to draw interest to math is through stories about its people. To that end:

• +1 for "Journey Through Genius". – Gamma Function Mar 19 '14 at 0:04
• and another +1 for "The Man Who Loved Only Numbers" – Benjamin Dickman Jun 25 '14 at 19:32

For a recent suggestion, check How Not to Be Wrong by Jordan Ellenberg.

Lying in the "simple and profound" quadrant, the book also gives deserved attention to Condorcet, in addition to providing a very readable book for a wide audience. Rather than my saying more, let me direct you to some recent reviews/responses:

LA Times

Salon

Scientific American

WSJ

For another author who demonstrates an ability in both the mathematical and literary world, I must recommend Alan Lightman, and, specifically, Einstein's Dreams.

(Incidentally, for better or worse, free copies of the latter are easily found online.)

Martin Gardner's The Colossal Book of Mathematics. It contains many of his best columns from Scientific American on recreational mathematics. My favorite chapter is the April Fool's day chapter, which includes a 'counter-example' to the Four-Coloring Theorem.

• To add to this, there are many collections of Martin Gardner's columns and based on my (limited) experience, they are all good for exploring mathematics. – Andrew Stacey Mar 17 '14 at 9:50

I strongly reccomend The Cauchy-Schwarz Master Class by J. Michael Steele. It could be read by advanced high school students who did well in calculus and have a strong interest in mathematics although it is probably better suited for first year undergraduate math majors. It reads like a novel that contains plenty of challenging exercises.

Another book that is usually brought up in such conversations is Concepts of Modern Mathematics by Ian Stewart. It is my understanding that it is geared towards a popular audience and attempts to give a brief overview of modern mathematics, but I have yet to read it so I can't give it any more of a review.

I highly recommend Paul Lockhart's Measurement

In terms of a book which inspires, I will offer Love and Math: The Heart of Hidden Reality by Edward Frenkel.

It is an autobiography. It explains how a knowing mentor used physics to lure him into deeper math. I don't want to say too much and spoil it, but, he goes on to explain symmetries, groups, Lie groups, Loop groups, braid groups, Lie algebras, Kac-Moody algebras in non-technical sort of heurstics in as much as is possible.

For me, the thing that is contagious about the book, is the raw passion for pure math with a hint of theoretical physics. Ultimately, the point of the book is to give you some sense of what Langlands conjecture is, how it has already been seen and/or verified and how it is still a vastly open question. The book is an invitation to join in the joy of discovery of math. Not as a drudgery, rather as an art.

Beyond this, I also like Roger Penrose's Road to Reality. Although, I will confess, I have not finished it yet. It does not read as fast as Frenkel's text. What math does that text contain? Better question: what math does it not contain?

Added 12-28-2014 I just finished my first read of Fearless Symmetry by Avner Ash and Robert Gross. A review by Dino Lorentzini is found on the AMS website. I found it to be a bit tougher to read than Frenkel's book, but, I think that is largely due to the fact this book really endeavors to give you some psuedo-technical insight into the problem of investigating the absolute Galois group to solve Diophantine equations. The major theme is to understand the structure of reciprocity theorems. Explicit examples are given. Fermat's Last Theorem is also discussed in some detail and placed in part of a larger story about a whole family of reciprocity theorems some just conjectured at this time. The book is full of wonderful stories and nice quotes. I recommend it for both style and as a leisurely way to see pretty far into modern number theory (of which I am no expert!)

I greatly enjoyed Another Fine Math You've Got Me Into by Ian Stewart. Entertaining and at a pace that any level of mathematician or non-mathematician would be comfortable with, but nevertheless discusses some very interesting and beautiful topics.

The books by Willian Dunham ("Journey through genius" (Penguin, 1991), "Euler, master of us all" (MAA, 1999), "The calculus gallery" (Princeton University Press, 2008) are the ones I've read) are outstanding. They show mathemathics in terms of the original work (more or less, using modern notation), and motivate the subject matter well. They do require some background and work by the reader.

Delicious is Nahin's "An imaginary tale: The story of $\sqrt{-1}$" (Princeton University Press, 1998).

Almost forgot Aigner and Ziegler's "Proofs from THE BOOK" (Springer, 4th edition 2012), a collection of beautiful proofs, accessible to the interested high-school/beginning college student.

Surreal Numbers by Donald Knuth is a story about two people discovering the surreal numbers and proving theorems about them.

At the books page on my blog, Math Mama Writes, you can find a list that includes about a dozen great books at adult level. (Scroll down past the kids' books to see them.)

If I had to pick my favorite, I think it would be Math Girls, by Hiroshi Yuki (along with Math Girls 2). The unnamed protagonist is a boy in high school who loves math. He helps Tetra with her math, and is challenged by the problems Miruka poses. In Math Girls 2, a few more girls join the gang. The math is challenging in these books, and the storyline makes it all the more fun.

A few other favorites:

• Chances Are: Adventures in Probability, by Michael and Ellen Kaplan. History, philosophy, science, and statistics all come together in this delightful exploration of probability.
• Euclid in the Rainforest, by Joseph Mazur. Logic, infinity and probability are the topics. Adventures in Venezuela, Greece, and New York furnish the background. Mazur has wide-ranging interests, and skillfully brings the math to life.
• Surreal Numbers, by Donald Knuth (requires well-developed math skills), requires lots of work, doing the math, and what fun work it can be! Alice and Bill are enjoying their extended vacation on an isolated tropical beach, but are getting a bit bored, when they discover a rock with two 'rules' on it. Conway has invented number through these two rules, and Alice and Bill (and the reader) are sucked in, trying to figure out how it all works. This is higher math.
• The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanige, would be unbelievable if it were fiction or even slightly fictionalized. Ramanujan was too focused on his own mathematical work to do well in school - he was kicked out college when he failed exams in his other subjects. It took him years of working as a clerk to support himself before he managed to catch the attention of a famous mathematician in England, G.H. Hardy, whose interest in him suddenly changed his life. A year later he would sail to England to begin with Hardy the work of making his mathematical results comprehensible to others.

There is a awesome list on similar topics on Maths Stack Exchange, even more awesome list on Math Overflow.

I heard a lot of good things about Smullyan's bibliography, and a lot of bad things about Gödel, Escher, Bach.

• I find that GEB makes a good companion to the text in the Theory of Computation class I teach. – Jim Hefferon Mar 19 '14 at 9:49

Love and Math , which is semi-autobiographical book by Edward Frenkel. It is not exactly a math book, but it does try to explain the Langlands program and its connections with QFT in lay's man terms. It also tries to convey the beauty of and excitement of doing mathematics

• I endorse this message :) – James S. Cook Mar 20 '14 at 4:36

an old one, but still a good one

Courant & Robbins, What is Mathematics?

It explains what mathematics is all about, for someone who may not know, but is willing to spend the time to learn.

In getting the link, I was delighted to discover that the latest edition is "Revised by Ian Stewart"

Eli Maor's book e: Story of a number was what first got me interested in mathematics when I read it back in high school. He presents the history of e, and of topics such as logarithms and calculus in a narrative that is engaging and math that is both simple enough that I could follow as a high school student but complex enough that I still come back to it to unpack its depth over a decade later. ( http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347 )

He also has a great book on infinity, To Infinity and Beyond, which covers infinity in lots of different contexts and situations, spanning across the mathematical disciplines. I used this as a textbook for a high school seminar I taught, but I think it's really more on the beginning college level - the high school students found it pretty challenging ( http://www.amazon.com/Infinity-Beyond-Cultural-History-Infinite/dp/0691025118 ). It looks like he has some more recent books on the Pythagorean Theorem, Trigonometry, and Geometry, too.

Also if the book on infinity piques your interest, there's another great book I found that comes at it more from the story of Cantor and set theory: Amir Aczel's The Mystery of an Aleph. This one has some good ties to infinity and religion, as well as a great historical component. It also really gives a good layman's introduction to the paradoxes of set theory and the continuum hypothesis. http://www.amazon.com/Mystery-Aleph-Mathematics-Kabbalah-Infinity/dp/0743422996

Finally, also on the topic of math history, Theoni Pappas has a great book, Mathematical Scandals which is an easy read and highly engaging: http://www.amazon.com/Mathematical-Scandals-Theoni-Pappas/dp/188455010X - this is one that I'd probably give to someone who said "they weren't a math person" as the other three I recommended are chock full of formulas and diagrams and lots of "mathy" content :-)

For more mathematical philosophy try:

"Godel, Escher Bach" by Douglas Hofstadter

This is a great book for exploring mindset when approaching natural mathematical problems. Many problems are entirely theoretical in nature, as in, they results that flow from axiomatic constructions. This book teaches that natural mathematics is as much about creativity of perception and the explicit recognition of the usefulness of abstractions as skill with mechanics and identities.

I've recently read T.W. Körner's "The Pleasures of Counting" and I consider it a true gem, unlike anything I've read before.

Largely, it is a book about the historical uses of applied mathematics which manages to be accessible to a clever high-school student yet rewarding to a math/science major with some courses under her belt.

Among topics covered are fighting cholera epidemics in Europe using statistics, and the birth of operations research in WW2 (the organization of convoys, how not to train pilots, why you should repair your planes less often). He explores predator-prey models through epidemiology as well as Darwinism. Dimensional analysis is used to derive power laws from mathematical biology and physics (humans and fleas can jump roughly the same height, how on earth can that be can that be??). He discusses Richardson and the genesis of weather modeling. Euclid and Galileo are in there too, as is as the obligatory section on the Enigma machine. It's an amazing book.

The math content of the stories is relished rather then elided. There are occasional many exercises fueled by the history (Requiring anything common sense to trig, probability, calculus, ODEs, group theory). Occasionally, there's a choose-your-own-adventure moment right before a crucial point in the historical narrative, when you're given a chance to reflect on a problem before learning how it was actually solved and lives saved.

It's filled with example after example of real life crisis where the use of simple mathematical tools coupled with ingenuity lead to insights that made a huge, undeniable difference. Examples showing that actually measuring things can prove common-wisdom to be wrong. The book manages to do this without resorting to synthetic, trivialized problems such as would be found in your average calculus book these days.

I found it to be a truly inspirational book; it got me curious and excited about math, and made me want to learn more. It was also a thoroughly good read. I'm convinced it has the potential to be a life-changing read to a certain kind of person at the right age.

One thing to point out is that the exercises are positively riddled with errors, so you better arm yourself with a copy of the (lengthy) errata and maintain vigilance. Other then that, I cannot recommend it enough.

How to Lie with Statistics, by Darrell Huff. Pictures by Irving Geis. http://www.amazon.com/exec/obidos/ISBN=0393310728/

The World of Mathematics. I have Volume Four, which was edited by James R. Newman in 1956. Most of the articles are 10 - 30 pages long -- long enough to take "what if" questions seriously. There are no dry mathematical derivations. Instead, there are articles like:

• Alan Turing's explanation of the Turing Test. (Can a man pretend to be a woman? Can a computer do better than a human at figuring out if a man is pretending to be a woman? When you think about it, it is really quite bizarre, especially for the 1950s.)
• "Mathematics in Warfare" -- How basic numerology can help win wars.
• "How to Hunt a Submarine" -- By the folks who figured out how to win the Battle of the Atlantic in World War II.
• Excerpts from "Flatland", Jonathan Swift, Lewis Carroll, and others.
• FYI the former was mentioned earlier in this thread. – Benjamin Dickman Feb 23 '15 at 7:09

William Poundstone's book The Recursive Universe. It's an engaging read for any intelligent reader. It's about Conway's Game of Life amongst other things.

• Welcome to Mathematics Educators! Note that the question asks: "please explain what makes this book appealing and what its math content is." – Jon Ericson Mar 25 '14 at 18:14

When I was younger I was fascinated by Jorge Luis Borges' ' Library of Babel'. It is a story about combinatorics, countability and the hugeness of finite numbers.

There is nothing even remotely close to T. Gowers Mathematics, A Very Short Introduction. (And less than \$10 so that it can be really recommended to students.)

From the preface: "It is possible to become comfortable with [infinity, the square root of –1, the twenty sixth dimension, and curved space ] without immersing oneself in technicalities."

And it's true!

And while Gowers never even mentions real numbers, he spends a whole chapter on inﬁnite decimals.

Both immense PLUS: As engineers are wont to put it, “The real real numbers are the decimal numbers”.

I love the connection between mathematics and physical models that serves as the theme in Bryant and Sangwin's delightful 2008 book How Round Is Your Circle:

Some of the best areas for mathematics, and I believe to get an interest in mathematics too, is the field of probability and statistics. Statistics is learnt fairly intuitively early on when people learn to count and do frequency tables.Here is a list of books that are good to learn probability and statistics.

A great book: Euler's Gem: The Polyhedron Formula and the Birth of Topology by David S. Richeson. It does an outstanding job at explaining serious mathematics to a general audience, it introduces Euler's Polyhedral formula as well as many concepts in topology.

The World of Mathematics (1950s). Good popular essays to describe various fields of math (e.g. operations research).

Martin Gardner column collections (several).

The Brian Hayes columns from The American Scientist (read in the mag, not aware if collected).

You might be interested in the expansive answers that were generated on math.stackexchange by the questions Book ref. request: “…starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics…” and Book series like AMS' Student Mathematical Library?.