Math books are a great source of information and inspiration. On this page we review math books that can stimulate your imagination, change your prejudices, and make difficult things clear.
Here are descriptions of some fine books on Math.
Some books put you to sleep.
Some are over my head.
Here are some math books
which both teach and entertain.
Dear Visitor,

     For many years I've collected and read books on math. Some have been very technical and complicated. Some have been dull. Some have been outside my line of interest.

     But a few of them I've kept and re-read and studied and enjoyed and cherished. Many of these are the source of material that appears in our CD-ROM Solid Gold Gnarly Math

     On this page I'll list the books I think are best. Of course, what I think is best may not be what you think is best. Let me encourage you to look around on your own.

     But this list will give you a starting point.

     Many of the books are in print, and can be purchased at Amazon.com. Some are no longer in print, but these can almost always be found in used bookstores. Down below we provide links to Amazon, for both in-print and out-of-print books.

     I've organized them by their seeming usefulness. The most important for children are in the first and last groups.

Sincerely,

     P.S. I'd like to emphasize one point in connection with my list. Almost every book has some sections, often many pages, that the reader may not understand. Don't be afraid to skip over such parts. The parts you don't skip will reward you enormously.

     P.P.S. Clicking on any of the links down below will open a new browser window for you. Just close the new window when you've finished looking.

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MATH BOOKS WHICH ARE EASIEST TO READ AND ENJOY
  1. The World of Mathematics James R. Newman, Editor,
    Simon and Schuster, NY, 1956. In four volumes (Out of print.)
         If there's only one book you can buy, this should be it. It's in four volumes, and every one is fascinating. And there's something in it for everyone, young and old. It includes historical stuff (Greek Mathematicians), Arithmetic (Calculating Prodigies), the Physical World (Soap Bubbles, and On Being the Right Size), Unreasonableness in Math, Mathematical Machines (Can a Machine Think?), Mathematical Literature (What if the Law of Averages were repealed?), and a section called Amusements, Puzzles, and Fancies. It contains excerpts from several of the books I recommend below. It's a famous four volumes, and Mr. Newman did a marvelous job of putting it together.
         Even though it is out of print, you'll have no trouble finding a copy in a used bookstore somewhere. Or in amazon.com's used book section.
  2. The Mathematical Magpie, and
    Fantasia Mathematica, by Clifton Fadiman
    Simon and Schuster, NY, 1962 and 1967.
         These two wonderful books contain short stories, poems, essays, music, cartoons, and miscellany. Many different authors wrote these articles, so in a way the books resemble Newman's The World of Mathematics. But Newman was more interested in math, and Fadiman more in ideas, and curios. You'll never have to skip anything in Fadiman's books: everything is beautifully understandable! Highly recommended.
  3. Men of Mathematics by E.T. Bell,
    Simon and Schuster, NY, 1937, 593 pages.
         This is the first book on math that I ever read. Someone gave it to me when I was a boy, and I suspect it is the source of my love of mathematics. It contains some technical material, but that part can be skipped over so you can read the life stories of mathematical geniuses. Some knew kings and queens, others lived quietly. Some were practical, and invented and constructed things of use; others lived with their heads in the clouds. Some were famous in their time, some remained obscure until after their deaths. Some died young and tragically, others lived perhaps too long. All were fascinating, one way or another.
  4. How to Solve It by George Polya
    Princeton University Press, Princeton, NJ, 224 pages.
         Professor Polya was the teacher I faced in my first math class in college. He was a famous mathematician and a wonderful teacher (the two traits are not often found in one person).
         Polya, like me, knew that many kids disliked math. On page 88 of this book he writes, "When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should, first of all, stir up his curiosity, give him some desire to solve the problem." This remark is the foundation of everything I've been trying to do these past years. And Professor Polya himself has been my inspiration -- even to the point that The Professor on our CD-ROM is the way I think of him! Prof. Polya wrote 'How to Solve It', a wonderful book aimed at helping teachers with math.
         The aim of the book is not only to stir up our curiosity, but also to suggest a method for solving mathematical problems. (In fact, the method is useful in solving any problem.) Polya describes, in detail and with examples, a series of steps we should carry out to unriddle our mystery. I won't list the steps here (they appear, in simplified fashion, in our web page on Teaching Methods, and of course in our math CD-ROM Gnarly Math), but I surely encourage you to buy the book and try them out, yourselves. Some of the examples he uses to illustrate his points are outside our realm of interest, but most are easy to follow and the others can be ignored. Much of the book is in the form of questions (as put by a teacher) and answers (as expected from the student).
  5. Flatland by E.A. Abbott
    Dover Publications, NY, 1952, 103 pages, and
    The Planiverse by A.K. Dewdney
    Poseidon Press, NY, 1984, 267 pages.
         Each of these books describes a world in two dimensions. (Ours, of course, has three: left-right, ahead-back, and up-down). The first book, Flatland, was written by a schoolmaster and was originally published in 1882. The author of the second book is a computer scientist.
         The authors are, in a sense, from different worlds, and their flat worlds are different. Furthermore, Abbott was intent on making clear what a two-dimensional world would be like. Dewdney was much more ambitious: he was interested in the physical laws that would govern such a place.
         As a consequence, Dewdney's book is much more interesting because it covers so much: art, seeds, telescopes, wheels, astronomy, evolution, magnetism, eating....it's a rich place he envisions. If you buy only one book, buy Dewdney's. If you want to start out with a simple world, buy them both. (Incidentally, one of the lessons on Geometry in Solid Gold Gnarly Math shows you a world much like Abbott's.)
  6. The Cryptoclub: Using Mathematics to Make and Break Secret Codes by Janet Beissinger and Vera Pless.
    A. K. Peters, Ltd. Wellesley, Mass. 2006 197 pages.
         A past issue of our Gnarly Gnews dealt with all kinds of secret codes. And now here's a fine book aimed particularly at kids in grades 5 to 8, though I believe it will appeal to even younger children.
         The heros and heroines are young folks who find various reasons to hide what they're writing from teachers and from other kids. They use Caesar ciphers, Vignere Ciphers, and even RSA Ciphers -- which are the newest. This is an intriguing book, and I recommend it highly. It's full of examples, and of problems to be solved.
  7. The Number Devil: A Mathematical Adventure by Hans M. Enzensberger
    Metropolitan, 1998, 262 pages.
         This book is a sort of "fairy tale" in which a 12-year-old boy who hates math has twelve dreams of a little Number Devil. In twelve chapters the Devil shows him the neat mysteries and connections between prime numbers, integers, rational and irrational numbers, permutations, Pascal's triangle, and much more. The illustrations are beautifully done and very helpful.
         The Number Devil should appeal to most young folks who find math forbidding or hateful, and might also be useful to adults who have either feared or been disdainful of mathematics.
         (I should perhaps add that our CD-ROM Solid Gold Gnarly Math contains virtually all the material in this book, and much more.)
  8. A History of Pi by Petr Beckmann
    St. Martins's Press, NY, 1971, 202 pages.
         Pi is, of course, the number found when you divide a circle's circumference by its diameter. It's roughly equal to 3.1416. Beckmann starts about 2000 B.C, when the Babylonians thought pi was 3 1/8 (3.125) and the Egyptians favored 3 13/81 (3.160). He ends in 1967, when a CDC computer found pi to 500,000 decimal places. Along the way he notices that the Bible implies that pi is exactly 3. And that in 1897 the Indiana House of Representatives passed, 67 to 0, a bill that ruled that pi equals 3 1/5 (3.20). The bill then went to the Indiana Senate and was referred to the Committee on Temperance, which recommended that it be passed. (Luckily, a math professor at Purdue University happened to be visiting the State Capitol, and hastened to suggest that the Senate not act so impulsively. The bill was then tabled, and died.)
         I hope you see that this is a very entertaining book. It's a good illustration of why I think math is fun. I should add that the book also contains many fine stories of how mathematics, and a world interest in pi, have evolved over the centuries.
  9. A Mathematician Reads the Newspaper by John Allen Paulos
    Basic Books, 1995, 212 pages.
         We all get news of the world one way or another: from newspapers, radio, TV, magazines, the Internet, and word of mouth. Paulos talks about newspapers, but what he writes applies to all sources of news because (he reminds us) math is not just computations. It's a way of thinking and questioning that is available to us all.
         So he applies mathematical thinking and questioning to 50 different news items. The subjects include racial quotas, free trade, SAT scores, cellular phones, health statistics, and recipes. Youngsters might be particularly interested his discussion of UFO's, baseball, lists of the ten best or most, man-in-the-street interviews, and advertising. One chapter, entitled "More Dismal Math Scores for U.S. Students", points out the costs of our not being able to think mathematically.
  10. A History of Mathematics by D.E. Smith
    Dover Publications, NY, 1958, 2 volumes, 1300 pages.
         Bell's book Men of Mathematics is really an abbreviated history of math. But it's relatively short and therefore incomplete. Smith's book is complete, covering the development of math from prehistoric times up 'till the mid 1800's.
         Volume 1 is a survey, starting with primitive counting and geometric designs, then what is known of math in China, India, Babylon, and Egypt before 1000 BC. It then moves forward in time, showing how math evolved in various parts of the world, and describing the works and lives of the principal mathematicians. There is little detailed mathematics in this volume, but it is well illustrated and provides a fascinating story of the development of mathematical ideas.
         The second volume surveys the history of math by subject, and includes numbers, geometry, algebra, trig, and calculus. Here you'll find some mathematical detail, though none of it is very complicated.
         These volumes show that mathematicians have always been a fascinating group of individuals. As you browse through them you'll learn that Pythagoras visited Egypt, Babylon, and perhaps India; that Hypatia was murdered by fanatics (she was the first woman mathematician we know of, and a picture of her appears in Gnarly Math. She was also interviewed in an issue of the Gnarly Gnews), that Fermat was a French bureaucrat for whom mathematics was a hobby; and that Pascal invented a computing machine but gave up math at the age of 25.
  11. Innumeracy by John Allen Paulos
    Hill and Wang, NY, 1988,135 pages.
         Paulos' subtitle is "Mathematical Illiteracy and its Consequences". He defines illiteracy as "An inability to deal comfortably with the fundamental notions of number and chance," and gives dozens of examples of errors we make when our own "fundamental notions" are out to lunch. He also suggests what we can do to improve our understanding of math.
         Here are some ideas and examples he discusses: everyone should know roughly how far it is from New York to Los Angeles; each of us should be able to compute approximately how fast his or her hair grows, in miles per hour; coincidences are common, and it's not surprising that President Kennedy had a secretary named Lincoln while President Lincoln had a secretary named Kennedy; astrology is nonsense, and is but one example of the kind of fake science we hear about almost every day; the results of a poll are often unrepresentative because of the bias of the poll-taker, and it's easy to misinterpret the results of even a fair poll.
         This book is fun to read, and will give kids, and their parents and teachers some good examples of the importance of being mathematically literate.
  12. Algebra Survival Kit by Josh Rappaport
    Singing Turtle Press, Santa Fe, NM, 1998, about 150 pages.
         Rappaport is a teacher, tutor, and winner of the Outstanding Young Educator award. He describes his book as "a conversational guide for the thoroughly befuddled", and you'll find it leads you by the hand clarifying all that seems difficult and mysterious about Algebra. The author says, "I've led hundreds of nail-biting students across this [algebraic] Wilderness, and not one has failed to make the journey."
  13. Zero. The Biography of a Dangerous Idea by Charles Seife
    Viking, New York, NY, 2000, 246 pages
         There was a time when the numeral zero didn't exits. The famous Roman Numerals, for example, had numerals for 1 and five and ten and fifty and 100 and so on, but none for zero.
         This neat book tells the history of zero. According to Seife, it was invented in India (though there are those who insist that it began in Baghdad). But he gives himself a great deal of freedom in discussing his subject, and we find stories about calculus, and the artist's vanishing point, and imaginary numbers, and the temperature absolute zero, among other things.
         Most of us have never given the digit zero much thought. Here's a whole book devoted to the topic!
  14. String, Straightedge, and Shadow by Julia A. Diggins
    Whole Spirit Press, Denver, CO, 2003, 157 pages
         Here's a lighthearted history of geometry that begins in the far past before man could even write, and ends in 300 B.C. It is imaginative, nicely illustrated, and proves, in a simple way, all the important elementary geometric theorems.

MATH BOOKS FOR ADULTS, PARENTS, AND AHEAD-OF-THEMSELVES KIDS

  1. Crimes and Mathdemeanors By Leith Hathout
    A. K. Peters, Ltd, Wellesley Mass. 2007, 197 pages.
         This is a wonderful book (written by a young fellow who was born in 1991!) whose hero, the son of a District Attorney, solves various mysteries including murders. Each crime, which baffles the police and other authorities, he solves with the help of math. Each chapter contains a mystery and its solution, and all of them are fascinating.
         I'd like to be able to recommend this book to younger readers, but the solutions are all quite complicated, and will challenge even the brightest of high school kids.
  2. The Mathematical Experience by P.J. Davis and Reuben Hersch
    Houghton Mifflin Co., Boston, 1981, 440 pages. (Out of print.)
         Here you have a neat discussion about math by two university professors. They start by asking, 'What is Mathematics?' (Their first answer is 'the science of quantity and space', but they end with something else). And then they go on to talk of: The Ideal Mathematician (his work in understood by only a few specialists, he finds it hard to talk to non-mathematicians, he dreams of solving a famous problem, his writing is almost indecipherable, etc.); Prime numbers (you can't find the biggest one, as is proven in Gnarly Math); Math Teachers (it's difficult to teach it well); Intuition (a slippery concept); and Envisioning Four-Dimensional Space (with the help of a computer, one of the authors could do it!)
         This is not an easy book, but it's worth studying if you have the time and patience.
  3. Chaos--Making a New Science by James Gleick
    Penguin Books, NY, 1987, 352 pages.
         For most of the past two centuries, we have thought that, given the right equations, we could accurately predict the future. Certainly we've been able to predict where a planet will be so as to have our rocket pass by it years in the future. We can predict how an integrated circuit will work, or how a bridge will withstand traffic, or how an airplane will fly, when we're doing the design. And we've always figured that the reason we can't predict who will survive a heart attack, or what tomorrow's weather will be, is simply that we don't yet have good enough equations to describe climate and health.
         But over the past decade or two we've discovered that some things seem to have a kind of instability that may make such predictions forever impossible. The first inkling of this deplorable situation came during a study of the weather, and the resulting "chaos" (from a scientist's point of view) is often described as the "butterfly effect": a butterfly flapping its wings in a Chinese garden today can change storm clouds next month in Philadelphia.
         Gleick's book shows just how this science of chaos came about, and discusses what it means for the future. For the most part the mathematics is very simple. The story is fascinating!
         Our CD-ROM Solid Gold Gnarly Math has a lesson on Chaos that, among other things, lets you experiment with a very chaotic waterwheel!
  4. Mathematics and the Imagination by E. Kasner and James Newman.
    Simon and Schuster, NY, 1952, 380 pages. (Out of Print)
         The authors have chapters on numbers, paradoxes, geometry, puzzles, pi, probability, topology, and calculus. Their original object was to "popularize" mathematics (at a time when people were writing books and newspaper articles popularizing relativity and astronomy and othersciences), and it seems to me they were quite successful. The book will be hard to find, but if you run across it, buy it!

MATH BOOKS ESPECIALLY FOR TEACHERS AND HOMESCHOOLERS

  1. How to Solve It by George Polya
    (See above, under EASIEST TO READ...)
  2. Teaching Mathematics--a Sourcebook by M.A. Sobel and E.A. Kaletsky
         This book is aimed at teachers, but it will be useful to anyone who wants to get kids interested in math. The authors aim to give children motives to learn math. It's a fancy word, "motive", but in practice they just want to make the subject fun and exciting. And they do an admirable job of it.
         They suggest using games, puzzles, guesses, magic, stories from mathematical history, displays, experiments, and a number of visual "aids" including blocks, number cards, folded paper, checkerboards, and graphs.
         Sobel and Kaletsky have put together a very useful book. I recommend it highly.
  3. When are we Ever Gonna Have to use This? by Hal Saunders
    Dale Seymour Publications, Palo Alto, 1988, 133 pages.
         Saunders, who is himself a high-school teacher of mathematics, has given us a book full of the kinds of calculations real people must carry out every day, or at least from time to time.
         He has organized the material beautifully, with chapters on General Arithmetic (including fractions and averages), Geometry (including Area, Volume, and Pythagoras' Theorem), and Algebra. Furthermore his book is very nicely indexed, so if you want to see problems faced by an Accountant, a Carpet Cleaner, a Policeman, a Veterinarian, or a Welder, you can easily look them up. Each problem is graded as Easy, Medium, or Challenging, and he provides answers to them all.

MATH BOOKS WHICH ARE LOTS OF FUN FOR EVERYONE

     For many years Martin Gardner wrote a monthly column titled "Mathematical Games" in the magazine Scientific American. A typical column might describe a game, or a bit of magic, or a paradox, or a puzzle. Often he had a new idea that led to long-term investigations by both professional and amateur mathematicians.
     Many, if not all of those articles, have been collected in a number of books. They are often accompanied by comments and ideas supplied by folks who had something new to add to what Gardner had written.
     I own four of his books, but I don't hesitate to recommend any them all. Children will be as fascinated as adults. Below a list of those I know about. Those "Out of Print" can often be found at Amazon, or in used bookstores.

  1. Aha! Gotcha : Paradoxes to Puzzle and Delight
  2. Entertaining Mathematical Puzzles
  3. Hexaflexagons and Other Mathematical Diversions : The First Scientific American Book of Puzzles and Games
  4. Knotted Doughnuts and Other Mathematical Entertainments
  5. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications
  6. Magic Numbers of Dr Matrix
  7. Mathematical Carnival
  8. Mathematical Circus : More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments
  9. Second Scientific American Book of Mathematical Puzzles and Diversions
  10. Mathematics, Magic and Mystery.
  11. New Mathematical Diversions
  12. Time Travel and Other Mathematical Bewilderments
  13. The Unexpected Hanging : And Other Mathematical Diversions
  14. Wheels, Life, and Other Mathematical Amusements
     Incidentally, Gardner has written books on other subjects, and they, too, are uniformly interesting. For example, there is one called The Annotated Alice, which contains Lewis Carroll's two Alice books, with marvelous comments on the sources of Carroll's humor.
     We've tried to make it easy for you to buy any of these books by doing some mouse-clicking in the sections below.
     In the first section we've listed the books from the lists above that are available. We have associated with Amazon.com to make it easy for you to purchase these books. Each book name is a link that will take you to that book's page at Amazon. There you will find a brief description, along with price and other information. You are of course not obligated to buy anything when you connect to Amazon.
     Below that list, there's a link that lets you search Amazon for any book you like. You can enter "Math", or "Algebra" or "Math history" or "Fermat" or any topic that appeals to you. When you click "Search", you'll be connected to Amazon's search results for whatever it was you were looking for.
     Remember -- each time you click on a book name, the book information is shown in a new browser window. Click the X in the upper right corner of the window to close it before you choose another book.

1. Here are the Amazon.com links. Just click the book name to be connected.

By Martin Gardner:

2. Here's where you can search Amazon for any book -- for example, for all the math books that Amazon has for sale.
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