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Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend?

It would be nice if you could answer in the following way:

Title: The Poincaré Conjecture: In Search of the Shape of the Universe
Author: Donal O'Shea
Short description: The history of the Poincaré Conjecture.
(Perhaps something like "difficulty level": + (no prior knowledge of math, as the book mentioned above), ++ (some prior knowledge of math is helpful), +++ (Roger Penrose: Road to Reality (?))

I hope this is appropriate for MO, since I think is of interest to mathematicians (at least for those who want to buy a popular math book for some aunt :-) ).

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Title: Godel, Escher, Bach: an Eternal Golden Braid

Author: Douglas Hofstadter

Short Description: It's mildly debatable whether this is in fact a book about mathematics, but any mathematician who has read this book will understand why I recommend it and any who has not should. Probably best for those with either a philosophical or musical bent.

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    $\begingroup$ I think Hofstadter would himself say that it is not about mathematics. And I'd say that he was wrong. $\endgroup$
    – Dan Piponi
    Commented Dec 11, 2009 at 22:56
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    $\begingroup$ I must be one of the few people who disrecommend the book. He draws some philosophical conclusions using arguments which have subtle philosophical gaps which are hard for a layman (to philosophy) reader to spot, and gives no hint there could be controversy. It is a legitimate argument that the gaps I mentioned are not actually gaps at all, but also a legitimate argument that they are. (Mathematical analogy for the kind of subtle error: confusing an abstract group with one of its faithful matrix representations.) $\endgroup$
    – Alexander Woo
    Commented Dec 12, 2009 at 2:24
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    $\begingroup$ Admittedly I'm no expert on philosophy. I just think GEB has a lot of interesting ideas in it, and Hofstadter presents them very artfully. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 12, 2009 at 3:57
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    $\begingroup$ Indeed a very nice book. However, although I know many people who read it, I don't know a single one who doesn't have a degree in math (non of them had when they read it). Is your aunt up to it ? $\endgroup$
    – David Lehavi
    Commented Dec 13, 2009 at 6:49
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    $\begingroup$ I agree with @Woo that there are suspect philosophical arguments in GEB. Hofstadter would probably say that these arguments are more important than the mathematics. And yet I would still recommend GEB for the mathematics it contains. $\endgroup$
    – Dan Piponi
    Commented Feb 24, 2010 at 22:22
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Title: Flatland

Author: A. Square / Edwin A. Abbot

Short Description: Imagine how life would be in less than three dimensions.

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    $\begingroup$ Jean-Pierre Luminet's "The Wraparound Universe" is a relatively new publication that's kind of in the same direction. Only it's souped-up by general relativity, geometrization of 3-manifolds and some serious physical experiments. I just got it in the mail today but it looks promising. $\endgroup$
    – Ryan Budney
    Commented Dec 18, 2009 at 6:24
  • $\begingroup$ Bundle it with "Flatland:The Movie", flatlandthemovie.com. Your aunt will not be disappointed! $\endgroup$
    – Victor Protsak
    Commented Jun 6, 2010 at 23:28
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    $\begingroup$ I first read Flatland when I was 11 or 12, and to me it was an invitation to imagine encounters with beings who live in more than three dimensions! It was a real eye-opener. $\endgroup$
    – Todd Trimble
    Commented Jun 7, 2010 at 1:53
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    $\begingroup$ It must be mentioned that at least some knowledge of Victorian society is needed to fully appreciate the satire, though. $\endgroup$
    – Ketil Tveiten
    Commented Dec 9, 2010 at 8:51
  • $\begingroup$ Much more satisfying for a modern reader is The Planiverse, by Dewdney (<a href=en.wikipedia.org/wiki/The_Planiverse>see Wikipédia</a>) $\endgroup$
    – Feldmann Denis
    Commented Jun 14, 2012 at 15:58
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Title: Logicomix: An epic search for truth

Authors: Apostolos Doxiadis, Christos Papadimitriou

Artists: Alecos Papadatos, Annie Di Donna

Short description: A comic book biography of Bertrand Russell, focusing on his work on the foundation of mathematics. (About 345 pages. I just started reading it, so I haven't formed a firm opinion on it yet, but I like what I see so far.)

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    $\begingroup$ This is a great book. They take some liberties with historical accuracy (as they explain in an appendix) but turn it into a compelling story. The characterization of the main players is excellent. $\endgroup$
    – Konrad Swanepoel
    Commented Dec 12, 2009 at 10:32
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Title: Mathematics: A very short introduction

Author: Timothy Gowers

Short description: As the title says, very short. Gives the non-mathematical reader a good idea what mathematics is all about in just about 100 small pages.

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    $\begingroup$ I've read it, and it is near perfect (there are just one or two paragraphs that I would have said otherwise). $\endgroup$
    – Jose Brox
    Commented Dec 12, 2009 at 19:13
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Title: The Man Who Loved Only Numbers

Author: Paul Hoffman

Short Description: A biography of Paul Erdős. No previous knowledge of math.

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    $\begingroup$ Although, those that knew him claimed he loved many things... $\endgroup$
    – Micah Milinovich
    Commented Dec 11, 2010 at 17:23
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    $\begingroup$ This is the most delightful biography I have read, of anyone in any field. $\endgroup$
    – shreevatsa
    Commented Aug 6, 2012 at 14:08
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"classic"

Title: Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem (US)
Fermat's Last Theorem: The story of a riddle that confounded the world's greatest minds for 358 years (UK)

Author: Simon Singh

Short Description: The history of Fermat's Theorem, from the famous note "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." to the solution of the Taniyama-Shimura-Conjecture by Andrew Wiles.

By the way, the video Fermat's Last Theorem (1996) is based on the book.

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    $\begingroup$ I'd also recommend whatever else Simon Singh has written, it's all good. The most mathematical of his other works is The Code Book (aka. The Cracking Code Book, apparently the same book), which I suppose is mathematical enough. $\endgroup$
    – Ketil Tveiten
    Commented Dec 9, 2010 at 9:00
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Title: A Mathematician's Apology

Author: G. H. Hardy

Short Description: In the style of Plato and Xenophon, G. H. Hardy offers a justification for pure mathematics.

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    $\begingroup$ Really? I personally don't think this book has aged well. In particular his remarks about number theory being useless for warfare are completely off in light of WWII. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 11, 2009 at 22:23
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    $\begingroup$ Let me instead counter-suggest Ian Stewart's "Letters to a Young Mathematician," which is in some sense a response to Mathematician's Apology (which is why I'm not putting it down as a separate answer). $\endgroup$
    – Qiaochu Yuan
    Commented Dec 11, 2009 at 22:47
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    $\begingroup$ I also didn't like the snobbery that Hardy displays prominently throughout the book. $\endgroup$
    – Akhil Mathew
    Commented Dec 11, 2009 at 23:03
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    $\begingroup$ I think you're being somewhat unfair about the snobbery. Hardy is without doubt a fool in a number of respects, (as well as being a phenomenal mathematician) but this book was written quite obviously in despair, and that I think gives it a unique, touching quality most popular books about math fail to attain. $\endgroup$
    – Kevin McGerty
    Commented Dec 12, 2009 at 5:15
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    $\begingroup$ @KM: Agreed. The combination of lucid mathematical insight and unguarded description of personal darkness make it an absolutely unique and unforgettable book. I try to return to it once every few years as with Catcher In the Rye, Hamlet and other great literary works. (It is remarkable to me that some people simply repeat Hardy's gloomy pronouncements about old mathematicians as though they came out of a sober how-to guide.) $\endgroup$
    – Pete L. Clark
    Commented Dec 12, 2009 at 16:41
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Title: The Book of Numbers.

Author: John Conway and Richard Guy.

Short Description: A beautifully illustrated book with all sorts of facts about all sorts of numbers. The difficulty is variable throughout the book: parts are written at a + level, while other parts are closer to +++.

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Title: Fearless Symmetry

Authors: Avner Ash and Robert Gross

Short Description: Very much a +++ text, these guys actually got a pop-math account of Galois representation theory published! By far the most technically demanding pop-math book I've ever read, (and one that took me about three years to actually finish) it nonetheless makes for a compelling "non-technical" introduction to a beautiful subject.

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    $\begingroup$ The table of contents of this book is enough to make me want to read it! $\endgroup$
    – Qiaochu Yuan
    Commented Dec 13, 2009 at 20:28
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    $\begingroup$ This is a pop book? I just looked at the table of contents, and it looks like I could learn a lot from this...and now I'm off to add it to my wishlist on amazon... $\endgroup$
    – Charles Siegel
    Commented Dec 15, 2009 at 13:56
  • $\begingroup$ I'm with Qiaochu on this. That is a captivating table of contents. $\endgroup$
    – Simon Rose
    Commented Dec 10, 2010 at 18:28
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Title: Journey Through Genius

Author: William Dunham

Short Description: A fencepost history of mathematics. For each highpoint it describes some fun history and then the actual math. Examples of topics covered are Heron's formula for triangular area, Euler's evaluation of $\zeta(2)$ and Cantor's set theory.

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    $\begingroup$ Dunham is a great writer. I also liked his book "Euler: The Master of us All" a lot. $\endgroup$
    – Kevin H. Lin
    Commented Dec 12, 2009 at 15:46
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    $\begingroup$ This book made me want to study mathematics! $\endgroup$
    – Max Lonysa Muller
    Commented Jun 9, 2010 at 17:31
  • $\begingroup$ I do not know of any better choice than this one. It is spellbinding. $\endgroup$
    – Steven Landsburg
    Commented Dec 9, 2010 at 4:34
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    $\begingroup$ I don't like the title - that genius cult in mathematics is a horrible thing in my opinion. $\endgroup$
    – Peter Arndt
    Commented Dec 9, 2010 at 7:38
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    $\begingroup$ Good point about the title, I still love the book though. $\endgroup$
    – Noah Snyder
    Commented Dec 9, 2010 at 15:54
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Title: Innumeracy: Mathematical Illiteracy and its Consequences

Author: John Allen Paulos, mathematician and well-known skeptic (in the good, modern sense of the word).

Short description: Paulos explains for the general public (and he does it fairly well) why it should understand a little more mathematics and know how to do Fermi calculations and educated probabilistic guesses. The dangers coming from pseudosciences are also highlighted, something very needed nowadays in my opinion.

Math level: +. Some chapters are not that easy to read without a mathematical background, more because of the great number of calculations and logic steps involved in the explanations than because of the math level of those per se.

Price: 9.89$ at Amazon.

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    $\begingroup$ Seconded. This is a very necessary book. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 12, 2009 at 1:23
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    $\begingroup$ +1 I enjoyed skimming this book, which I found lying around a nonmathematician's house, so it's also appropriate for this question. $\endgroup$
    – Akhil Mathew
    Commented Dec 13, 2009 at 0:19
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Title: The Lady or the Tiger?

Author: Raymond Smullyan

Short Description: If your aunt is young enough not to scorn the fairy-tales, this book is a charming collection of logic puzzles, presented under a guise of an entertaining story. The book presuppose no knowledge of mathematics (so difficulty is +), but it ends with a version of Godel's incompleteness theorem. Your aunt will be delighted!

Remark: Other books by Smullyan are also great, and will relieve you from wondering of what to give to your aunt for the next few Christmases.

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    $\begingroup$ Agreed, Smullyan is great. I also liked Chess Mysteries of Sherlock Holmes, although this isn't really a book about chess, but retrograde analysis. $\endgroup$
    – Tony Huynh
    Commented Feb 24, 2010 at 21:34
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Title: Proofs and Refutations - The Logic of Mathematical Discovery

Author: Imre Lakatos

Short Description: A Plato-esque dialogue between a teacher and his students, aiming to discuss how mathematical discovery happens, how mathematical arguments take form and how mathematical knowledge arises. The discussion is based around the Euler-characteristic, starting off with a concrete conjecture about polyhedra, expanding into the result in full-blown generality through investigation, trial-and-error, the occasional side-track and creative injections as it happens in real life. The book is also sprinkled with mathematical in-jokes, and interesting historical facts about the result. This is a real gem that I keep on my bedside table. Probably at the higher end of the ++ spectrum.

I'm not sure if I'd get the book for my aunt. It takes mathematical reasoning skills to get though the argument and an understanding of why the questions above are worth answering in the first place. I might get it for a philosophically inclined friend instead.

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  • $\begingroup$ That is indeed a very good book, but it is (a version of) a doctoral dissertation in philosophy. I don't think anybody can read it who do not have some experience with mathematical proof! $\endgroup$
    – kjetil b halvorsen
    Commented Oct 9, 2012 at 17:31
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Title: Letters to a Young Mathematician

Author: Ian Stewart

Description: A beautiful book in which Stewart tries to convey in the form of letters from a mathematician to his grand daughter what sort of things does the profession of mathematics involves. It is very interesting since the letters advance from the time in which the grand daughter is in high school up until she is a professional mathematician doing research. I would recommend it without a doubt.

From my own experience it has been really nice to come to this book at different times during the past years. I started college as an engineering student but on my third year I started taking courses from the mathematics program and I bought and read this book when I was just beginning. I had no real idea of what pure mathematics was all about (since I was used to the kind of calculus courses in which the emphasis is on computation rather than proving things) and this book gave me at least some perspective and a few hints of what may be ahead of me.

Just for the record, I ended up switching my major to mathematics.

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Title: Gamma : Exploring Euler's constant

Author : Julian Havil

Short description: Provides a fascinating history of a constant that doesn't get nearly as much attention as $e$ or $\pi$. Definitely has more math than most books intended for a general audience but I feel the book is accessible to those who persevere with it. And the reward is lots and lots of beautiful mathematics.

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    $\begingroup$ What a fantastic book! It's easily one of the main driving forces that got me into analytic number theory. $\endgroup$
    – Alex R.
    Commented Feb 25, 2010 at 3:53
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Title: Prime Obsession

Author: John Derbyshire

Short Description: A book about the Riemann Hypothesis. Its been a while since I read this book, but I remember it being well-written and fun to read. It is very accessible, explaining everything. I think half of the chapters have no math (just facts about Riemann's personal life and such), while the rest is avery gentle introduction to math ideas (like infinite series).

http://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259/ref=sr_1_2?ie=UTF8&s=books&qid=1260570418&sr=8-2

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    $\begingroup$ I would rather read the account of a pro: Marcus du Sautoy, The Music of the Primes. $\endgroup$
    – José Hdz. Stgo.
    Commented Dec 11, 2009 at 23:30
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    $\begingroup$ The Music of the Primes and Prime Obsession are about essentially the same topic, but they're very different books. The Music of the Primes is a "+", but Prime Obsession is probably "++" (for difficulty level). I thought both were reasonably entertaining. $\endgroup$
    – Darsh Ranjan
    Commented Dec 15, 2009 at 23:31
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Title: Uncle Petros and Goldbach's Conjecture

Author: Apostolos Doxiadis

Short Description: It is a very fun novel based on the life of a fictional character who spends most of his live trying to prove Goldbach's conjecture. It talks about some other famous problems in math. This novel is fun for mathematicians and non-mathematicians. No knowledge of math is required.

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  • $\begingroup$ Indeed it is one of the best math-fiction story ever! $\endgroup$
    – Yannic
    Commented Jun 14, 2012 at 18:19
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Title: The Curious Incident of the Dog in the Night-Time.

Author: Mark Haddon.

Description: A mathematically gifted kid tries to solve the mystery of a dog murdered in the night-time.

Remark: Although this isn't a math book per se, it actually does contain a lot of interesting and non-trivial mathematics. Also, it's very well illustrated. As a final selling point, you have to respect any book whose chapters are indexed by the primes.

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    $\begingroup$ In my opinion, this looks more like a murder novel than an introductory or public mathematical book for an aunt to learn some math, any way, it is quite an enjoyment to read the book. $\endgroup$
    – awllower
    Commented Feb 17, 2011 at 8:55
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Title: The Symmetries of Things

Authors: John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss

Description: The authors begin by introducing the general concept of geometric symmetry / regular tiling, and then pose the problem of classifying all possible symmetries of the plane. They provide an elegant topological classification, and along the way introduce the notions of orbifold and classify the compact surfaces with boundary. It's then easy enough to also classify the discrete symmetries of the sphere. In part II, the authors introduce the notion abstract group, and classify the prime-order "color symmetries" of the plane, which are tiling patterns with different-colored tiles. Part III consists of a discussion of higher-dimensional tilings, including the four-dimensional Archimedean solids.

Part I is suitable for an interested amateur with no specific prior knowledge. Part II is at an advanced undergraduate level, although it does contain new results. Part III, which is about half the book, is research level. The book is printed in full-color.

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Title: Symmetry and the Monster

Author: Mark Ronan

Short Description: The history of the classification of finite simple groups. Reads like a good novel.

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    $\begingroup$ A beautiful book. Though I'm biased, as an undergrad at Rutgers, I met a few of the major players in this story. One of my first recommendations to people when they ask me "You're an algebraist? What's that?" $\endgroup$
    – Charles Siegel
    Commented Dec 15, 2009 at 13:58
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Title: The Mathematical Experience

Authors: Davis and Hirsh

Short Description: A really accessible and funny introduction to the philosophy of mathematics. I think the description of the "ideal mathematician" is particularly hilarious.

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Title: The Code Book

Author: Simon Singh

Description: This is a popularized history of cryptography and cryptanalysis. It runs the range from shift-ciphers to modern public key cryptosystems and quantum key distribution. No prior knowledge of maths is necessary, but can be helpful.

It has been quite some time since I read this book, but I found it incredibly engaging at the time. I know it isn't strictly focused on mathematics, but I am surprised that no one else has mentioned it.

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  • $\begingroup$ In my opinion the greatest fragment of the book is explanation of reconstruction of linear letters of B ( or A, I do not remember) of Cretan people. $\endgroup$
    – kakaz
    Commented Mar 11, 2010 at 15:25
  • $\begingroup$ This book also exists under the name The Cracking Code Book, apparently. $\endgroup$
    – Ketil Tveiten
    Commented Dec 9, 2010 at 9:03
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Title: Indiscrete Thoughts

Author: Gian-Carlo Rota

Description: General discussion on Mathematics, Princeton of the fiftees, lives of Artin, Feller and of Rota's intimate friend Stanislaw Ulam; fantastic bedtime reading. Also includes philosophical flavour like '10 things I had been taught' type.

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  • $\begingroup$ This is one of my favorite books, but the philosophy section is a little too heavy to be called "popular" writing. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 27, 2009 at 8:04
  • $\begingroup$ Actually, I think some of the philosophy section would be of more interest to a general cultivated audience than most of the rest of the book, which might be interesting to mathematicians and academics but perhaps not to many others. $\endgroup$
    – Todd Trimble
    Commented Oct 9, 2012 at 19:36
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Title: What is mathematics?

Author: Herbert Robbins and Richard Courant

This book is a very nice introduction to mathematics, it covers basic number theory, analysis, algebra, geometry and topology.

I'am very surprised that i couldn't find it on this list already.

(from a duplicate answer - feel free to edit) This would be for someone who has some mathematical ability, and really wants to understand what math is. Courant goes through essentially all of mathematics, starting at a very elementary level, but getting to some very deep and important stuff. He often does real proofs, and doesn't dumb it down, but does explain things conceptually very well, including sometimes giving just ideas or justifications for really difficult things, like the prime number theorem. I use this when I teach our senior proof seminar, just to force the math majors to own a copy.

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  • $\begingroup$ I copy-pasted a description from a duplicate answer, since the previous description was a bit brief. Anyone who wants to clean it up should do so. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 10, 2012 at 3:08
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Title: The honors class: Hilbert's problems and their Solvers

Author: Ben Yandell

Short description: The title says it all. It gets dense at times, but it has an enormous range of topics and the biographies are very interesting. Probably a ++.

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  • $\begingroup$ This is really a great book! $\endgroup$
    – Spinorbundle
    Commented Dec 12, 2009 at 10:32
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Title: Euler's Gem: The Polyhedron Formula and the Birth of Topology

Author: David S. Richeson

Short Description: V-E+F=2. Requires little (no?) preparation, but some willingness to think. I gave it to a few people who wanted to know more about what math really was (as opposed to adding longer and longer columns of numbers, as my mother believes), and those who made the investment of their time came away with a much better understanding.

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  • $\begingroup$ I finally got around to reading this! It is really, really fantastic. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 26, 2010 at 5:52
  • $\begingroup$ I agree fully with Qiaochu Yuan's comment above. It's a wonderful book, starting with polyhedra and ending with a touch of algebraic topology (and the Poincaré Conjecture in the epilogue). $\endgroup$
    – J W
    Commented Mar 28, 2012 at 13:57
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Title: What is the name of this book?

Author: Raymond Smullyan

Short description: Lots and lots of logical puzzles (the kind with people always telling the truth, people always lying, people who sometimes tell the truth and sometimes lie, etc.) It is really entertaining, and it serves well as an introduction to the logic of propositions.

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Title: King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry

Author: Siobhan Roberts

Short Description: Biography of Coxeter.

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Title: Everything and More: A Compact History of Infinity

Author: David Foster Wallace

Short description: History of the idea of infinity, mostly leading up to Cantor's set theory ideas. It's been awhile since I've read this but I would say the mathematics is pretty decent for popular writing and the writing is really enjoyable if you like Wallace's style (and maddening if you can't stand him, I'm sure). This is the kind of math book you could get for someone who enjoys serious literature.

No prior knowledge is necessary, but the math-phobic could struggle.

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    $\begingroup$ Unfortunately Wallace didn't fully understand much of what he wrote about, and made some serious errors. $\endgroup$
    – Michael Weiss
    Commented May 27, 2017 at 18:32
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Title: Men of Mathematics

Author: E. T. Bell

Short Description: Short biographies (~30?) of important mathematicians. Excellent relations of the death of Archimedes, Galois' temper, Newton's eccentricities, &c. Also try his (Edit: actually L. Hogben's) book Mathematics for the Millions.

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    $\begingroup$ A fun read, but not to be taken seriously as history. $\endgroup$
    – Gerry Myerson
    Commented Dec 9, 2010 at 8:04
  • $\begingroup$ You are absolutely right, Gerry. Not that good history must always be literally true . . . $\endgroup$
    – user1241
    Commented Dec 10, 2010 at 7:44
  • $\begingroup$ Early editions of this book are not exactly politically correct by today's standards. $\endgroup$
    – Micah Milinovich
    Commented Dec 11, 2010 at 17:30
  • $\begingroup$ Belated comment: good history should have regard for truth/accuracy $\endgroup$
    – Yemon Choi
    Commented Jun 15, 2012 at 18:36
  • $\begingroup$ A couple of my friends chose mathematics as there career becuause of this book :) $\endgroup$
    – Amir Asghari
    Commented Dec 18, 2013 at 13:58
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