Mathematics is about numbers, trigonometry, and calculus, right? If you think so, then you have little idea of what is mathematics.
On this page i present some of the math books i've read that i highly recommend to ANYONE. No, you did not remember your highschool algebra or trigonometry, i know. Even if you do, you can forget them. What you need here is a curious mind. What you will learn is in fact the most advanced modern mathematics, in sharp contrast to the formulaic drivel taught in high school or college that so troubled our young and bored our minds. After reading one or two books here, you'll come out with a fresh idea how mathematics was never what you thought it is about. Trust me.
In this New Age age there's lots of math popularizing books or highschool math education programs that have highschool students participate in puzzles-solving and block-arranging in the names of teaching mathematics. More often than not, the stately goal of teaching mathematics through painless games mutates into meaningless team-spirit-building social activity. Although math is not about formula roting, but nor is it game playing. Watch out for these dumbed down numb books and programs. The utmost importance in math education is to have teachers who are genuinely interested in math. Approaches, methodologies, and gimmicks matter less. Far less.
What you will learn from the books on this page are solid mathematics.
I'm sure you have heard of 4th dimension type of thing in science fictions. Ever wonder what exactly is the 4th dimension? Start with Edwin A Abbott's fiction Flatland (1884) amazon
This superb short fiction gently leads you into the origin and gist of concepts of dimensionality. The whole book is online at: FLATLAND: A Romance of Many Dimensions.
There's an annotated version Annotated Flatland amazon by Ian Stewart. (2001) I recommend that you read the text completely by itself before reading the annotations.
This wildly popular book begets sequels by different people. One is Flatterland (2002) amazon by Ian Stewart. Another is Sphereland: A fantasy about curved spaces and an expanding universe amazon (1983) by Dionys Burger. Recently dimension-writing veteran Rudy Rucker mathematician also joined the fray and produced Spaceland amazon (2002)
Of the above, i've read Flatland several times over the years, and have read half of Ian Stewart's Flatterland, which is pretty good.
After you have read the original Flatland, you need to read Shape of Space amazon by Jeff Weeks. The second edition just came out in 2001. This book introduces the layman the cream of modern topology.
Given a bunch of dots connected by lines, is there a way to traverse all the dots without repetition? Or, is there a way to trace all the lines without repetition? You may have run into these type of puzzles. These kind of questions are the subject of Network Theory (commonly known as Graph Theory), whose application in industry such as airline route planning or wire-laying scheme involves billions and billions of money at stake, among others. Welcome Introduction to Graph Theory amazon (1994) by Richard J Trudeau. This is an excellent, lovely book. All you need in advance is a curious mind. You don't need to know fractions or anything in order to read this book. It is this book, that introduced yours truly this extremely game-oriented branch of math. Yes, graph theory is a giant branch of math with no ending in sight.
Once upon a time, there's a hateful thing called Calculus.
What Is Calculus About amazon (1978) by W W Sawyer
Oh Calculus — the headache of your youth! This is an excellent explanatory book i read in the early 1990s when i was learning calculus. Calculus is the stepping stone to a branch of real mathematics called Analysis. In essence, analysis is the study of the concept of continuity and infinitesimals.
Ever wondered about things symmetric? Surprisingly, there is an entire branch of mathematics that studies symmetry. Yes, you got me right. The symmetry you see in bee combs and kaleidoscope patterns, in the zigzag patterns of brick walls, the decorative patterns in Persian carpets or Celtic knots, the repetition in spiral stairs, the left/right mirror image of faces and animals, the swastika symbol or airplane and boat propellers or the fancy automobile stainless wheels, and also the three dimensional pattern on the soccer ball or the Geodesic dome .
One might wonder What is there to study about symmetry? It turns out lots. Mathematicians did not one day started to study symmetry. Rather, when trying to solve puzzling equations such as x^5 == y, mathematicians have gradually developed a body of theories, and over the centuries these got refined and abstracted, then they realized their theories is simply about symmetry. In the 1700s, their theories are aptly named Theory of Equations. In 1800s, it's abstracted form became Group Theory.
A very good intro to group theory for the non-professional is: Groups and Their Graphs amazon (1975) by I Grossman and Magnus W. It is this book that taught yours truly group theory. Your interest in group theory depends on how acquainted with math you are. Although group theory can be taught easily to children, but nevertheless it is very abstract. I recommend that you lean group theory only when you feel a need, which will come naturally when you read enough math. Otherwise, you may feel the subject quite pointless and dry. Another, even easier to read book than the above is Mathematical Groups amazon by Tony Barnard and others.
Once you learned group theory, you might find the following book interesting: Conceptual Mathematics: A First Introduction to Categories amazon by F William Lawvere and Stephen Hoel Schanuel. This book is about Category Theory, which is one of the new branch of math invented in the 1900s. Essentially, it is the mathematical study of analogy. Intritruly'sguing, you say? Yes indeed. It is amazing that even “analogy” can be codified by mathematicians to become a useful and deep theory.
Knot Book (2001), by Colin C Adams. amazon
Knot? Yes, one of the most modern branch of mathematics is the study of knots. Trefoil knots, figure-eight knots, you name it. These studies have far-reaching consequences from chemistry to cosmology. Take out a string and play, for the field is young, and you just might discover something unprecedented.