Algebra Across Cultures

Why Did Early Cultures Develop Algebra?

For roughly 2.5 million years, people have survived and shaped the world around them using tools. From the first stone axes and arrowheads to today's airplanes and smartphones, tools form the backbone of human societies. We often think of tools as physical objects, but they are sometimes needed for more abstract work. People of the past saw certain patterns in nature, especially when they started planning cities. They didn't know what to make of those patterns, but they knew how to make the tools to understand them.

What is Algebra?

Math first arose as a method for counting and measuring objects. Counting with fingers and toes led to abstract symbols representing quantity, or numbers. Algebra is a tool to find unknown numbers through their relationships with known numbers. By substituting letters for numbers in equations, mathematicians create a formula instead of a single answer. These formulas can then be used to solve similar problems over and over again. Many cultures developed solutions to algebraic problems and shared their ideas through trade.

Algebra in its current form looks very different from the math of early people. They did not use numbers and equations as we think of them today. Instead, they explored mathematical relations in terms of shapes, or geometric algebra. This simple reasoning can be found in many cultures. It first appeared in Mesopotamia and Egypt around 3,500 years ago. Linear algebra explores straight lines, angles, and how they come together to form shapes. More complicated curves can be described with quadratic equations through square numbers and larger exponents.

The First Algebraic Math of Mesopotamia

The first known examples of algebra come from Mesopotamia, sometimes known as the cradle of civilization. Scribes wrote in cuneiform scripts on clay tablets, more durable than paper or papyrus. While they worked in many fields, most scribes received a basic education in math. In cities like Sippar and Mari, women trained as scribes as well. Many scribes were slaves, given as gifts between noble households to manage their affairs. They counted in units of 60, as opposed to our system based on multiples of 10.

The oldest piece of math ever written down can be found on a Mesopotamian tablet dating to at least 3000 BCE. It consists of two problems, showing students how to calculate the area of a field. The priceless tablet was found in a pile of mud bricks in the temple district of Uruk. Other tablets list triplet pairs of the Pythagorean theorem a thousand years before Pythagoras.[1][2]

These first scholars did not use modern algebra to solve their problems. Although they were dealing with algebraic problems, their solutions dealt with geometry rather than more flexible equations. They did not, for example, use familiar modern functions like y = mx + b to find the slope of lines. By pulling apart and rearranging rectangles and triangles, they solved complex problems through spatial reasoning and memorization. Many of the clay tablets recovered from the time are reference tables.[3]

Algebra in Ancient Egypt

The Egyptian civilization grew alongside Mesopotamia for thousands of years. Their scribes wrote on papyrus paper with reed pens dipped in ink. Because they wrote mostly on paper, the writings of the ancient Egyptians rarely survived to the present day. One rare example of early Egyptian math, the Rhind Papyrus, dates to about 1550 BCE. It contains a number of exercises in algebraic logic, similar to the training tablets found on Mesopotamian tablets.

Like the Mesopotamians, Egyptian math emphasized memorization and geometric logic. Their system was based on units of 10, similar to our own. Through their knowledge, they built massive monuments able to withstand the test of time. Papyri show that the Egyptians were capable of calculating the volume of a pyramid as it was being built, a sophisticated operation for their time. The concepts pioneered in Egypt and Mesopotamia would later travel to the schools of Greece. There, they formed the foundation of Western mathematics.[1][4]

The Geometric Algebra of Greece

The Greeks found philosophy in the study of nature, including the natural rules that govern shapes and numbers. Many philosophers contributed to this study, and their names can still be seen in modern algebra. Pythagoras, for example, created a cult around his philosophy and mathematics. He is credited with the Pythagorean theorem, or the relationship between the sides and angles of right triangles, a^2 + b^2 = c^2. His followers considered numbers to be the divine, underlying order of the universe. Women were allowed to join their order and studied as equals. They reportedly sacrificed an ox after Pythagoras solved the 47th problem of Euclid.[5]

The most important Greek contributor to algebra was a man named Diophantus. Diophantus lived from about 200 CE to 300 CE. He published Arithmetica, a collection of 13 texts examining the nature of equations with two or more unknown numbers.[6] In 1637, Pierre de Fermat wrote his famous last theorem in the margins of a volume of Arithmetica. He proposed that no three positive whole numbers could resolve the equation an + bn = cn when n is greater than 2. He never made his proof public, and the riddle stood unanswered until 1995, 358 years later.[7]

Chinese Mathematics

China's history of mathematics is less well known in the Western world. They too followed geometric algebra using a system very similar to that of the Mesopotamians. The major difference lay in their use of base 10. Chinese courts were printing books of math problems by 200 BCE. Like Egypt, the ancient Chinese could calculate the volume of an incomplete pyramid, which they used to construct elaborate underground tombs.

Confucianism, a philosophy that survived many dynasties, ranked officials through difficult civil service exams. As a result, most Chinese math was practical in nature, meant to serve the state. Western concepts of algebra arrived in 1607 with Jesuit missionaries. The Jesuits, while denouncing Chinese mathematics as inferior, copied several of their texts on linear algebra. These theories were ahead of Europe's at the time. They then taught the lessons back to Chinese students as the work of a German author. By denying Chinese people their own cultural heritage, the Jesuits hoped to win more converts.[3][8]

Indian Contributions to Algebra

The number system we use today was developed in India. It reached Europe through Arabic traders and included one of the first concepts of zero. Like other cultures before them, Indian scholars were limited by geometry. If an answer could not be found in lines and angles, it could not be found at all. Indian numerals proved key for Islamic scholars to venture into more abstract fields of math that would lead to modern equations. In this way, they set the stage for the development of modern algebra.[9]

Arabic Al-Jabr and Muhammad ibn Al-Khwarizmi

After the fall of the Roman Empire, the Arabic world grew into its new role as a hub of trade and learning. Arts and sciences flourished in a society of philosophers, artists, and poets. Their religion forbid the depiction of religious figures in art, so they turned to geometry to adorn their buildings. Arabic scholars, adopting Indian numerals, were the first to work with numbers as decimal fractions.

Many of the terms now used in algebra come from Persian mathematician Muhammad ibn Al-Khwarizmi. His works introduced both Indian numerals and Arabic decimals to Europe. These numbers at last allowed mathematicians to move past geometric algebra. They began to explore concepts through equations instead of shapes, manipulating functions in new and flexible ways. The word "algebra" is based on the term al-jabr, or completion, a method of simplifying equations. Similarly, "algorithm" is based on Al-Khwarizmi's name.[10]

Developing Modern Algebra

While men like Al-Khwarizmi studied the patterns of nature, others were busy waging war and politics. Over time, power swung from the Middle East to Renaissance Europe. To Europeans of the day, the world and all its knowledge were one immense frontier. Sailors set out seeking unknown lands and riches. Philosophers probed the meaning of life and the rights of every human being. Scientists explored the laws of chemistry and physics. Mathematicians followed their lead, exploring the rules that govern reality through algebraic equations.

Algebra as we know it today emerged in the 16th and 17th centuries. Scholars like François Viète and René Descartes began working with more complex equations containing two variables. With this tool, they could chart precise curves for the first time. Renaissance academics established a universal language of algebra, which quickly spread across the globe. Ever since, people from many different cultures have studied and contributed to algebra and the more complex math it supports. By joining this ancient tradition, modern students can master an essential tool to shape the world around them.[9]

Bibliography

1. Victor Katz and Karen Hunger Parshall, Taming the Unknown: A History of Algebra From Antiquity to the Early Twentieth Century (Princeton: Princeton University Press, 2014, 12-32.

2. Karen Rhea Nemet-Nejat, Daily Life in Ancient Mesopotamia (Peabody , MA: Hendrickson Publishers, 2008), 150-153. - new

3. Ibid., 306-307.

4. Bartel L. van der. Waerden, Geometry and Algebra in Ancient Civilizations (Berlin: Springer, 1983), 64-69.

5. W. T. Stace, A Critical History of Greek Philosophy (New York, NY: Oia Press, 2015), 48-49.

6. Thomas Little Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra (Cambridge: University Press, 1885), 110.

7. Lisa Rezende, Chronology of Science (New Delhi: Viva Books, 2010), 108.

8. Roger Hart, The Chinese Roots of Linear Algebra (Baltimore, MD: Johns Hopkins University Press, 2011), ix.

9. Katz, 8-9.

10. John L. Berggren, Episodes in the Mathematics of Medieval Islam (New York, NY: Springer, 2017), 6-9.

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