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 with our numbers. The work includes two different Egyptian techniques for multiplication, their technique for division, their special way of representing fractions, and their special way of solving basic algebraic equations. A good deal of what we know about ancient Egyptian math comes from problems preserved on papyri, so we look at a variety of these problems.

As in all the other major chapters, the student is challenged with “problems” and “exercises”. Problems require calculations, and they echo sample problems presented in step-by-step detail in the text. Answers to odd-numbered problems and solution techniques appear at the end of each chapter. Exercises generally require extended thought and can safely be ignored by a student who is not particularly curious. For the curious, a collection of suggestions and solutions appears in Appendix A.

Despite the aforementioned chronological facts, the chapter on ancient Chinese mathematics comes before the chapter on Babylonian mathematics because in my experience every student finds Chinese math easier than Babylonian math, not that either one is easy. Naturally, the starting point with Chinese math is the calligraphic number system, which is different in substantial ways from all the number systems a student is likely to know: our decimal system (using Arabic numerals), the Roman numerals he or she learned as a child, and the Egyptian numbers from the previous chapter. The key feature of this system is a multiplicative element, where one character tells you by how much to multiply another character.

After a discussion of how the Chinese represented numbers and did basic arithmetic, the focus of the chapter is problem solving. Chinese records include a rich selection of algebra and geometry problems dressed up in fanciful stories. Sometimes, the premise seems absurd, like a farmer needing to buy exactly 100 animals for exactly 100 coins, but the stories seem intended to guide aspiring students to competently handle systems of equations, rates, similar triangles, areas and volumes, and division and remainders. Problems range from constructing simple magic squares to factoring huge numbers that arise in equations with variables raised to the fourth power. An especially interesting category involves a branch of advanced arithmetic called “number theory” and a concept that we know today as the Chinese Remainder Theorem.

Since the Babylonian cuneiform number system is so different, a great deal of Chapter 5 is devoted to to the ins and outs of base 60 arithmetic. Number representation, place value, addition, subtraction, multiplication, division, and fractions are explained. Since the symbols for the Babylonian digits are difficult to draw, an expedient compromise has been made that helps both me and the student: the digits are shown as simple triangles rather than drawn more subtly as they appear on the thousands of hardened clay tablets that today are testimony to ancient Babylonian brilliance. I highly recommend visiting internet sites that show the actual tablets, and two of the videos I recommend (in a section listing Recommended Websites and Videos after Further Reading) show modern researchers patiently etching cuneiform wedges into soft clay.

The scope of Babylonian math is broad, so only a few really exceptional topics are covered: a dazzling calculation involving the area of a trapezoid, a technique for solving quadratic equations, formulas for square roots and cube roots, and an astonishing list of triangles that conform to the Pythagorean theorem, etched into a clay tablet many centuries before Pythagoras was born. A great enduring mystery about the Babylonians is why they recorded their mathematics in the base 60 format. Some speculations are discussed in the chapter, and in Appendix B I present my own theory on the subject.

The chapter on Greek mathematics is a scratch of the surface. Whole books have been written about Greek math, and in fact whole books have been written about individual Greek mathematicians, like Archimedes and Euclid. We have dozens of ancient Greek math textbooks fully preserved and fragments or records of the contents of many others. Today’s students hardly realize it, but to a surprising extent they are already experts in Greek mathematics, since virtually the entire math curriculum from middle school to pre-calculus is based on the work of Greek geniuses. Some brilliant Italians gave us negative numbers early in the Renaissance, some clever Frenchmen gave us probability and analytical geometry during the Enlightenment, and some Germans and Englishmen blessed us with integrals, derivatives, and logarithms, but outside of that it’s pretty much all Greek. We’ll see what Thales discovered about triangles and circles, what Pythagoras knew about triangular numbers and angles in polygons, what Hippasus found out about the square root of 2 that got him in trouble, how Hippocrates almost found a square and a circle with the same area, what Democritus calculated about cones and pyramids, the 5 shapes that Plato thought were perfect, how Euclid proved there were more prime numbers than we could ever count, how Archimedes estimated pi, and on and on and on.

Chapter 8 concentrates on the entertaining story problems Hindu mathematicians created to challenge each other and dazzle their patrons. Typical problems involve the price of fruit, the path of a flying wizard, and the length of an ever-growing snake. To solve these problems, the Hindus, like the Chinese, relied on systems of simultaneous equations, related rates, and the Pythagorean theorem. In one problem, where an exchange of gifts leads to a surprising coincidence, the solution is essentially a fundamental principle of modern matrix algebra. Influenced almost certainly by the Babylonians, the Hindus improved on the calculations of square roots and cube roots. Influenced by the Greeks, they made advances in number theory. The Hindus had a remarkable geometric method for very accurately computing the square root of 2.

Chapter 9, describing Arabian math, is our bridge to the modern world. After the fall of the Roman Empire, Europe lost both the means and the desire to preserve and expand the mathematical insights of the classical Greeks. The common term we learn in history class for this period is the “Dark Ages,” and as far as European culture is concerned it is a reasonable term. However, not far away, in the countries controlled by the Arabs, the situation was completely different. Greek mathematical manuscripts were devoured by scholars anxious to add their ideas to the Greek foundation. So, while in Europe a spectacular treatise by Aristotle was erased so that a monk could re-use the pages to write down some prayers, in Arabian libraries Greek texts were being copied, translated and annotated. (A wonderful video about Archimedes’ lost, and subsequently rediscovered, book is cited on page 230.)

The Arabs busily solved second and third-degree equations, examined geometry and trigonometry in detail, experimented with number theory, and applied the practical things they discovered to the increasingly important fields of astronomy and accounting. Astronomy was vital for regulating the calendar, for navigation, and for the Muslim’s obligation to face Mecca while praying. Accounting was useful for the growing economy and the administration of an ever-increasing territory. What the Arabs preserved and created was eventually transmitted across the Mediterranean to the Venetians, Florentines, and other politically prominent inhabitants of the Italian peninsula.

It will undoubtedly occur to many readers that something must have happened in mathematics before the Sumerians and Egyptians began their clever applications of it, and that is the subject of Chapter 2, a short and somewhat speculative chapter. The field of mathematical anthropology is wide open for research and theorizing, and it can be approached from a few distinct directions. First, discovery of new anthropological sites, or more complete interpretation of known sites, could clarify mankind’s earliest mathematical thinking. Second, it is very plausible that the remote and isolated populations around the globe still living in Stone Age societies might have the same understanding and use of numbers that our remote ancestors had. Third, other fields, including neuroscience and linguistics, might delimit dramatically what early humans were capable of.

And it is also quite likely that some readers will wonder how it is that we know what we do happen to know about the mathematical accomplishments of ancient civilizations, and this is largely a matter of archeology. Curious scientists have dug up the old cities of Egypt and Mesopotamia and have translated the words written in long dead languages. In Chapter 6, devoted to mathematical archeology, the challenges of interpreting artifacts are described, and the reader is presented several opportunities to step into the role of an archeologist, on the job, trying to make sense of a mathematical puzzle. Chapter 6 also features brief discussions of Roman numerals and Mayan numerals.

In the Introduction, I’ll address a few words to the student using this book in a class, based on my experience in teaching a class called The Development of Mathematical Thought at Westchester Community College in New York. Those words are intended to be encouraging and are in an introduction because students generally do not read prefaces, not being especially concerned about why a book was written or how a book was written.

Producing a book such as this one is a collaborative effort. John Teall, who has written his own textbooks, was a key advisor. Stan Wakefield an expert on all phases of the book business, brought my manuscript to Wiley. The Wiley editorial and production teams guided me through the process that turns a manuscript into a book. Particularly helpful and encouraging were Susanne Steitz-Filler, Sari Friedman, Divya Narayanan, Nomita Swaminathan and Shiji Sreejish.

There are many fine math professors at Westchester Community College. The first time I taught a course on the history of mathematics, Sean Simpson gave me his notes and class handouts. Jodi Cotten, who teaches a course called “The Nature of Mathematics,” made many useful comments about our shared interests. Joyce Cassidy, who teaches Geometry, presented the well-known core of that subject in a thoroughly original way. And Ray Houston, who also teaches the history of mathematics, has frequently put me on the schedule to teach my favorite course.

I had the good fortune, when I was a student, to know some real mathematical geniuses, and one of them, Jonathan Watson, has kicked mathematical ideas around with me for 50 years. Other friends, who have made things possible or made things easier or made things wonderful are Andrew Goodman, Juan Leon, Olivia Masry, Alan Koblick, and Anne Teall.

And my wife Franziska and my daughter Emily contributed to this book in several ways. Franziska knows a lot about preparing a manuscript and taking advantage of the options and subtleties of software, and Emily had the courage to take a very advanced college course in number theory and the graciousness to tell me all about it.

This book is not a comprehensive history of mathematics. There are excellent books, much longer than this one, that present a thorough survey of ancient math. There are also more specialized books that go into great depth about certain topics (the history of algebra, the development of numerals, etc.) or certain civilizations (math in ancient Greece, math in ancient China, etc.). I hope that you are interested to look at these other books one day, and some of them surely are in your college library right now.

When I start my course, I tell the students that this is obviously a math course. It’s offered within the Math Department, and the students will solve math problems. Some of the problems are practical, and some of the problems are exercises that students had to do in their schools in Egypt and Babylonia long, long ago. It’s a math class.

It’s also a history class because what we’re exploring here is where ancient mathematical ideas came from, what kinds of problems these old civilizations needed to solve, and how mastering math helped them develop their agriculture, engineering, government, economic, military, and social systems. So it’s a history class.

I tell my students it’s also a bit of an art class. Not everybody could read and write in the ancient world. In fact, it was a select minority who could. And a long period of training and education was required to train the scribes who did the counting and computing and administrative work that made ancient Egypt, ancient Babylonia, and ancient China run. I have my students practice making the numeric symbols that these old civilizations used, just as if they were in a scribe school, and this takes patience and accuracy.

It’s also a bit of an archeology class and an anthropology class. If we’re talking about the mathematics used by old civilizations, we need to know a little bit about these civilizations, and generally students get curious about how we know what we know about these civilizations—that’s where the archeology comes in. And if we go further back in history, to the nomadic, tribal, primitive kinds of societies that lived before the agricultural revolution, we get into the realm of anthropology—how and why did people even begin to think mathematically? Remarkably, there are groups of people around the world who were so isolated until modern times that we can use what we know about them to make reasonable guesses about our quite remote human ancestors.

I even tell my students this is a bit of an English class, because I ask them to write a short paper about a mathematician or about a bit of mathematics. Basically I’m looking for one idea that can be linked to one mathematician. Students can choose something from long ago (like Archimedes working out an approximation of π) or something modern (like Mandelbrot developing fractals) or something in between (like Omar Khayyam solving cubic equations). I look for clear writing that explains the idea, describes the world the mathematician lived in, and relates the idea to mathematical knowledge that came before it or grew out of it.

So, don’t expect everything from this slender book. It is a starting point. I have my own favorite websites about the history of math, but your professor may have his (or hers) and there are many good ones. I show my students videos of people making cuneiform tablets and of people building models of Platonic solids. I show the Mayan codex books that survived and I also show the Hindu derivation of the square root of 2. I read out loud to the class the view of the Crow Indians that honest people don’t need numbers larger than 1000.

So I urge you to read, think, and be curious. You’ve got an electronic calculator at your fingertips and the advantage of a clever base-10 system with 10 simple digits. But put yourself in the position of someone thousands of years ago, who needed to solve an arithmetic problem and didn’t have these wonderful and convenient things. Put yourself in that position often enough this semester, and you’ll have a profound understanding of the history of math and the development of mathematical thought.