PREFACE

Many people simply dread mathematics. The idea that the history of mathematics might be interesting or that math problems could be solved as a recreation astonishes them.

I like math. When I see a mathematical idea for the first time, I wonder: Where did that come from? Who came up with that, and why? And my effort to answer these questions often leads to a pleasant and intellectually exciting investigation of books and websites. Some of my students shrug anyway, and say: Who cares? I tell them why they should care. Every idea that expands their minds makes them better people, more capable of doing the tasks they have chosen to do, and more interesting to talk to during lunch.

The history of any subject is a worthwhile study, and while art, science, music, and medicine have long been taught at colleges, in recent decades there has been a proliferation of courses on the history of art, the history of science, the history of music, and the history of medicine. The history of mathematics is enjoying a similar surge in popularity, although perhaps not for the best of reasons. Just as you can have no artistic talent yourself but still enjoy and excel at a course in the history of art, or be inept at scientific work but still excel at a course in the history of science, you might find comfort in the history of math despite finding no comfort at all in math itself. Many of my students have stumbled into my classroom just because they were scared to take calculus or statistics.

I see the history of math as closely allied to the field of recreational mathematics. In recreational math, we solve problems not because they are in a textbook, but because they are part of a story that contains a mathematical puzzle. Recreational math is about solving brain-teaser-type problems in a clever way. It is like solving crossword puzzles, and some of the newer puzzles that are in vogue, like kenken and sudoku. It's problem solving without the drudgery, and finding a solution gives the solver a feeling of satisfaction and success. In a college course in the history of mathematics, there are a lot of recreational math problems to solve, using clever techniques which, when properly applied, often lead to a feeling of satisfaction and success. The ancient Egyptians, Chinese, Babylonians, Greeks, Hindus, and Arabs had a wide range of mathematical interests, and there are plenty of concepts and calculations to discover and admire, and even to understand and emulate. When we try to do math their way, we are doing it primarily as a game, not for practical reasons.

Many of my students have said that they wanted a small book that summarized the historical background behind the mathematical accomplishments of ancient civilizations and also explained how to do representative math problems, and that is why I wrote this book. It is intended to do exactly those two things, briefly to give a bit of the history and to show solutions to typical problems, so that the students would have a guide to fall back on when they themselves tried to do Babylonian multiplication or Egyptian fractions or any similar operation based on ancient methods.

In many ways, this is a "how to" book. How to solve a Chinese problem in a Chinese way. How to solve an Arabian problem in an Arabian way. It is also a "why" book. Why did the Greeks make such astonishing progress in mathematics? Why did the Hindus want to calculate the square root of 2 so precisely? It is also a "proof" book. I will show a proof that verifies that an Egyptian algebra technique works. I will show a proof that verifies that a Babylonian geometry technique works. But mostly, I hope that you find this an interesting "story" book, which ties together the hows and whys and the proofs in a way that makes the history of mathematics a pleasant subject to think about and want to know more about.

I have a great deal of enthusiasm for the history of mathematics, partly because I like to tinker with numbers. The study of ancient math makes it clear that other people were enjoying tinkering with numbers thousands of years ago. Yes, mathematics developed because there were practical problems to solve in agriculture, government, and war, but once the early mathematicians had worked out the math they needed for their business, they apparently played around with numbers a lot. So many of the problems preserved in ancient writing are not practical at all-they are frivolous, gimmicky, and recreational, meant to entertain and let the cleverest mathematicians show off a little bit.

There have been a few times in my lifetime of tinkering with numbers that I have "discovered" something wonderful and surprising, and I thought I had an original result that the mathematical community would be thrilled to know about. But when I researched the history of each topic, I found that others had found these wonderful and surprising results long ago. My insight into the sums of squares and cubes was known by the Arabs 1000 years ago. My insight into a different way to define a circle was known by the Greeks 2000 years ago. My insight into a way to find right triangles was outdone by the Babylonians 3000 years ago because they had a way to find even more right triangles than I had found.

It would be greatly satisfying to me to know that this book had in some way inspired you to tinker with numbers. More likely though, this book is a means for you to pass some time or pass a course, and I can be somewhat satisfied if the book entertained or taught you something and caused you to be curious about ancient people or appreciative of their skills or perhaps in awe of their skills-they did this mathematical stuff without calculators and computers.

Chapters 3-5 and 7-9 in this book cover events in the history of mathematics in six important civilizations: Egyptian, ancient Chinese, Babylonian, Greek, Hindu, and Arabian. They are presented in a rough chronological order, related to when these civilizations made significant mathematical progress (which we know about!), but the chronology is very rough, and the reader should feel absolute freedom to acquire the information in any different sequence. In fact, historians will tell you that the earliest cultural activities that we recognize as essential to a "civilization" occurred in Mesopotamia, over 5000 years ago, under the direction of people we know as Sumerians. Within a few hundred years, Sumerian control had given way to Babylonian control, and via outstanding archeological good fortune, we know a lot about Babylonian mathematics.

I present Egyptian mathematics first. Egyptian civilization is nearly as old as Mesopotamian civilization, and Egyptian math is much easier than Babylonian math, a key consideration when your audience is new to the overall subject.

Shortly after the Egyptians had mastered their environment, another civilization sprang up in the region of the Indus River; but in this case, there is no archeological good fortune to speak of, and while the remains of cities assure us that these people were well acquainted with mathematics-you hardly get architecture and engineering without math-there is little to be said about this civilization. Instead, the descendants of these people, the Hindus, are treated in a later chapter.

After covering Egyptian math, I turn to Chinese math. China was one of four civilizations that emerged in the next time interval, still over 4000 years ago, but unlike the other three, its mathematics is well-documented. The Elamites in Iran and the Hittites in Turkey left us nothing particular in math (their work may have been absorbed by the Babylonians), and the Minoans in Crete are an enigma, although eventually their ideas may have influenced Greece. Even to say China's math is well-documented is something of an exaggeration: Chinese civilization flourished for thousands of years, and the mathematical history we know comes mostly from the middle and later periods, not the earliest. Of course, the same points about longevity and documentation apply to Egypt and Babylonia also.

Next I present Chapter 5 on Babylonian math. It simply cannot be put off any longer, despite its thoroughly different approach to representing numbers.

The math of classical Greece comes next. The Greeks undoubtedly benefited from the work of the Egyptians and the Babylonians who came before them. They may have also interacted advantageously with the Hindus, who were their contemporaries, although geographically distant. The final group to be discussed is the Arabs, who built their mathematics on a foundation laid down in both Greece and India.

I decided to limit the content of this book to events that occurred before AD 1000. The volume of math that comes after this date dwarfs the volume of math that comes before it, and my intent is to present the right amount for a one semester introductory course.

For many students, the most intimidating aspect of the study of ancient math is the unfamiliar number systems, but this obstacle is easy to overcome and curiously not an essential element to many parts of the story. We can do multiplication the Egyptian way either with their hieroglyphic numbers or with our everyday numerals, so understanding their method is more important than knowing their penmanship. The Internet is a fantastic resource for seeing all the old number systems, and with sufficient patient practice, any student can replicate the symbols well enough.

Chapter 3 includes a few illustrations with Egyptian hieroglyphics, but most of the work is done...