Throughout my life, I have always had a passion for magic. Whether I was watching other magicians or performing magic myself, I was fascinated with the methods used to accomplish amazing and impressive feats, and I loved learning its secrets. With just a handful of simple principles, I could even invent tricks of my own.
I had the same experience with mathematics. From a very early age, I saw that numbers had a magic all their own. Here’s a trick you might enjoy. Think of a number between 20 and 100. Got it? Now add your digits together. Now subtract the total from your original number. Finally, add the digits of the new number together. Are you thinking of the number 9? (If not, you might want to check your previous calculation.) Pretty cool, huh? Mathematics is filled with magic like this, but most of us are never exposed to it in school. In this book, you will see how numbers, shapes, and pure logic can yield delightful surprises. And with just a little bit of algebra or geometry, you can often discover the secrets behind the magic, and perhaps even discover some beautiful mathematics of your own.
This book covers the essential mathematical subjects like numbers, algebra, geometry, trigonometry, and calculus, but it also covers topics that are not so well represented, like Pascal’s triangle, infinity, and magical properties of numbers like 9, π, e, i, Fibonacci numbers, and the golden ratio. And although none of the big mathematical subjects can be completely covered in just a few dozen pages, I hope you come away with an understanding of the major concepts, a better idea of why they work, and an appreciation of the elegance and relevance of each subject. Even if you have seen some of these topics before, I hope you will see them and enjoy them with new perspectives. And as we learn more mathematics, the magic becomes more sophisticated and fascinating. For example, here is one of my favorite equations:
eiπ + 1 = 0
Some refer to this as “God’s equation,” because it uses the most important numbers in mathematics in one magical equation. Specifically, it uses 0 and 1, which are the foundations of arithmetic; π = 3.14159 . . . , which is the most important number in geometry; e = 2.71828 . . . , which is the most important number in calculus; and the imaginary number i, with a square of −1. We’ll say more about π in Chapter 8, and the numbers i and e are described in greater detail in Chapter 10. In Chapter 11, we’ll see the mathematics that help us understand this magical equation.
My target audience for this book is anyone who will someday need to take a math course, is currently taking a math course, or is finished taking math courses. In other words, I want this book to be enjoyed by everyone, from math-phobics to math-lovers. In order to do this, I need to establish some rules.
Rule 1: You can skip the gray boxes (except this one)!
Each chapter is filled with “Asides,” where I like to go off on a tangent to talk about something interesting. It might be an extra example or a proof, or something that will appeal to the more advanced readers. You might want to skip these the first time you read this book (and maybe the second and third times too). And I do hope that you reread this book. Mathematics is a subject that is worth revisiting.
Rule 2: Don’t be afraid to skip paragraphs, sections, or even chapters. In addition to skipping the gray boxes, feel free to go forward anytime you get stuck. Sometimes you need perspective on a topic before it fully sinks in. You will be surprised how much easier a topic can be when you come back to it later. It would be a shame to stop partway through the book and miss all the fun stuff that comes later.
Rule 3: Don’t skip the last chapter. The last chapter, on the mathematics of infinity, has lots of mind-blowing ideas that they probably won’t teach you in school, and many of these results do not rely on the earlier chapters. On the other hand, the last chapter does refer to ideas that appear in all of the previous chapters, so that might give you the extra incentive to go back and reread previous parts of the book.
Rule π: Expect the unexpected. While mathematics is a seriously important subject, it doesn’t have to be taught in a serious and dry fashion. As a professor of mathematics at Harvey Mudd College, I can’t resist the occasional pun, joke, poem, song, or magic trick to make a class more enjoyable, and they appear throughout these pages. And since this is a book, you don’t have to hear me sing—lucky for you!
Follow these rules, and discover the magic of mathematics!