Babylonian ‘algebra’ from Crest of the Peacock

 

Discussion question: How could one state a general mathematical principle in a time before the development of algebra and algebraic notations? Is mathematics all about generalization and abstraction? Think about various areas of mathematical knowledge – number theory, geometries, calculus, graph theory – imagine how could you imagine stating general or abstract relationship without algebra?

In the time of Babylonians, there were no symbols for the variables and the mathematical operators. That is, they did not have the algebra of our time. Taking advantage of base 60 system, Babylonians had developed extensive tables of multiplications, and also tables of reciprocals (terminating sexagesimal fractions). These developments were believed to be started by the need in accounting matters on the trade. However we also found that they did have techniques to solve mathematical problems involving squares and cubes. Instead of symbols they used words and geometric shapes. They also had technique to find approximate square root and cube roots of some numbers. 

These findings from the Babylonian time show that it is possible to state some general mathematical principles. However it has limitations.

To state this for any positive values of a and A without using a symbol is possible. It would need several sentences. Suppose the word “product” is available or invented to represent the result of multiplying two numbers. Suppose also that we have the words “difference” and “sum” of two numbers; as well as “reciprocal”, “half” and “square root” of a number.

 Suppose we know the ‘difference’ between side of your square and side of my square. My side is bigger than yours. Suppose we also know the ‘product’ of side of my square and side of your square. Then, the following procedure calculates side of my square and side of your square.

Step (1): Half the ‘difference’, and calculate its ‘square’.
Step (2): Add the ‘product’ to the ‘square’ obtained in the step (1).
Step (3): Calculate the ‘square root’ of the result from step (2).
Step (4): Side of my square is the result of step (3) added to the half of the ‘difference’.
Step (5): Side of your square is the result of step (3) less the half of the ‘difference’.

 Without having a symbol for any mathematical operation means, further development will be slowed and limited. We can write the functional relationship between two variables x and y if they are related in a polynomial, reciprocal or root function of a friendly degree to that time, and also we have names to represent the coefficients. However, to describe in details about a family of quadratic functions by varying any one of the coefficients will require a good mastering of the language.

Babylonian-style base 60 multiplication table for the number forty-five

 

Examples for multiplying two numbers 

                    

Pairs of two numbers that multiply to 45

The Crest of the Peacock Discussion

 The first chapter from The Crest of the Peacock is very interesting to read. The author tries to assemble and make connection of developments in mathematics across the world in a timely fashion. The perspective of the author is non-Eurocentric, which provides alternative way to see how mathematics has been developed as a global effort. There are many details in the first chapter I encountered for the first time. I would like to share a few things I find surprising to know. 

·       Human body of knowledge including mathematics is almost negligible in any measure (magnitude, depth, speed) compared to the size of knowledge and data stored in the natural universe. Our development in science and mathematics results in not just improvement in our living, but also destruction of the planet earth and its environment which is our only world and which is not our design and construction. Science and mathematics are also used as tools in hegemony, colonization, and other conflicts such as wars, data and media control. So it was a surprise to find that enormous magnitudes of affords in creating history, arts, and activities have been used to make Europe superior for its achievement in science and math in the last 400 years. This tells something about our human nature.

• Even though I graduated from the major in Operations Research which comprises mostly optimizations utilizing programs and algorithms. I never know that the word “algorithm” was from the famous Muhammad ibn Musa al-Khwarizmi. I knew al-Khwarizmi’s books about algebraic equation solving and the number system but I never realized this special connection. He definitely influenced the western world deeply by introducing the decimal system in his mathematics text books. 
  
  
•  I was surprised by the importance of having a “meeting place” for ancient scholars to gather and discuss their research. The “meeting place” served as crucial role of hosing big ideas from big peoples. Places like Alexandria or Jund-i-Shapur acted as huge research schools where knowledge was created, synthesized, upgraded, refined, and if possible, translated. It is interesting to know that scholar like Pythagoras even traveled to India to learn their mathematics. The idea of international students studying at a foreign university can be traced back more than two thousand years ago! 
  
  
•   I was surprised by the fact that the academic exchanges between China and India was scarce before the spread of Buddhism into China. I always thought geographically, neighboring countries should have more chance to exchange information. Also, I didn’t know that Indian astronomers were hired to teach at Astronomical Board of Changan (the place I am from)!  When I was doing elementary school in China, we were taught that Chinese mathematics also was influenced greatly by the Islamic world. However, different from India, Islamic religion didn’t spread across China like Buddhism did. This is a wonder for me considering both India and the Islamic world being neighbors of China.  
  
  
•  According to the author, the principle of place value was discovered independently about four times in the history of mathematics! Mesopotamian used place-value notational system to base 60; Chinese used positional principles in rod numerical computations; Indian used place-value decimal system; and the Maya used positional number system to base 20. In particular, I found that the Maya was one of the two civilizations where the concept of zero originate (another is India which I knew). I was also surprised by the accuracy on Maya’s astronomical observations and calendar constructions (without having proper equipment, how amazing!).

Base 60

 We have seen that Babylonian mathematics had a base 60 (sexagesimal) place value system. That is, the same symbol could be used in the units column, the 60s column, the 3600s column, … with different meanings. Write a blog post about why you think the Babylonians chose base 60 rather than the base 10 system we are used to (even though they did have a special symbol for 10). Engage in a ‘speculative phase’ and a ‘research phase’. Begin by thinking, wondering and speculating for yourself, and only after that, go to doing some research online or in the library.

Speculation phase:
In the positional numeration system, the value of a digit in a numeral depends on its position. Suppose we have a number 2323111.0 written in base-4 system. Its value in base-10 system is

One advantage of larger base in positional numeral system is the ability to represent larger value in shorter numeral. So using base 60, people can write large valued numerals shorter than the writings in the smaller bases such as 10. However, there are many integers larger than 10. The choice of 60 could be related to other important activities of that time in that region. The development of number system might be tied to other social, intellectual activities such as calendar, time recording, trade, cosmology, architecture or even religious and belief system in those days. The number 60 could be an important number in those activities. Number 60 is still used in time keeping in which we can sense the difference between 60 and 10. For example, if we have 10 hours per day, 100 minutes per hours, then a lot of things we are taking as easy measures will be different. A quarter of a day will be 2.5 hours instead of 6 hours, a third of an hour will be 33.333333...minutes instead of 20 minutes. With 60, we have many whole number fractions than with 10.

If we still use the base 60 system our daily computation, we will have to remember all these sixty digits of the system if these symbols are designed with no relationship between them. On the other hand, large numbers can be expressed in relatively shorter numerals. In fact, we are still using pseudo base 60 and 360 system in time keeping and trigonometry that we feel convenient. By using 360 degrees in a cycle, the directional measures are easy to record and transmit. For example, 60 degrees counter clockwise from due north cannot be written such easily in base 10 or 100 systems. The numbers 60 and 360 have many whole number divisors then the numbers 10 and 100.

Research phase:
According to “A history of mathematics, an introduction” by Victor J. Katz, the Mesopotamian civilization in the Tigris and Euphrates river valley began sometime in the 5^th millennium BCE, initially with many small city states. 

Tigris and Euphrates river valley (Google map & Wikipedia)

These small city states were unified under some dynasties. During the dynasty of Ur around 2150 to 2000BCE, very centralized bureaucratic state was produced. It was a large system supported by scribes (record keepers) and scribal schools to train the members of the bureaucracy. After the collapse of Ur dynasty around 2000 BCE, the succeeding small city states continued to develop writing. Writing was needed to manage labour and flow of goods. They did cuneiform writing (early writing on the clay tablets using a stylus) to keep records and do calculations. Thousands of these tablets were excavated in 19^th and 20^th centuries. A large number of these tablets contain calculations, and many hundreds have been translated.

The base 60 system found was made of 59 digits without zero. These symbols were designed with the base 10 sub-system. They had a symbol for one and a symbol for 10, and other digits were built by combination of these two symbols. Sometimes the digit for one represented 60. The system is base 60 and positional. According to Victor J. Katz, the base 60 system became the standard system used throughout Mesopotamia in 3^rd millennium BCE. Most of the mathematics study of these tablet writings in his book is based on those from the old Babylonian period (the time of Hammurabi, the rulers of Babylon).

Babylonian digits representing from 1 to 59 (Wikipedia) 

For the value 16929 in base 10, the base 60 user only need to write three digits. To write the number 4 x 60^2 + 42 x 60 + 9  = 16929  in base 10 system, the scriber would put three digits on the moist clay as follows:

To write a base 60 number  4 x 60^2 + 42 x 60 + 9 , the Babylonian scribe would leave a blank space between the first and third digit. It would look like this.

However if the zero is the last digit, it is difficult to know. The reader would need to understand the context. 

We do not know why base 60 system was developed and applied in those days. One possible reason according to Victor J. Katz is that 60 has many small integer divisors. There are 12 positive integer divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Therefore 1/2, 1/3, 1/4,..., 1/30 of 60 can be written easily. These divisions of 120, 180, 360, etc. are also easily written.

In “The Sumerians, their history, culture, and character” by Samuel Noah Kramer, the Sumerians who also lived in lower Mesopotamia around or before the time of Babylonians had base 60 system numeral. They had measures of lengths in multiple of 2, 6, 10, 30 and 1800; and measure of capacities (volume) and weight in the multiple of 60, 180, and 300, and others. So it is possible that Babylonian also used these measuring methods and therefore chose the base 60 system.

According to Victor J. Katz, in Euclid (232-283 BCE) geometry the angle measure was a right angle. Other angles were referred to as parts or multiples of a right angle. Sometime before 300 BCE, Babylonians introduced division of the circumference of a circle into 360 parts or degrees. They also initiated the base 60 division of degree to minutes, and minute into seconds. Why they did that was not clear. It may be due to divisibility of 360, or due to 360 being close to 365 days in a year.

Chinese 60 year calendar cycle is based on the combinations of a cycle of 10 heavenly stems and 12 earthly branches according to travelchinaguide.com.

Today we continue to use the base 60 system in our time keeping and in trigonometry due to the convenience it offers. Whenever we need the convenience of this kind, the base 60 system will be one option. 

Discussion and reflection on "integrating history of mathematics in the classroom"

 Mathematics is an integral part of human history. Whenever I am reading mathematics out of the school context (freedom from constraints and failure-success rules) I enjoy learning theories along with important events on its development history. Within the school environment in which grades are important for personal future and time is to be optimized for that purpose, I think students need motivation. When the historical part is blended in assignment and graded, then it would be a motivational factor. Better motivation would be when students come to understand the usefulness or need of learning some historical events behind the theory they are studying. If properly integrated, history of math allows us to capture the steps involved in shaping of a theory which strengthen and enrich our understanding of the theory. Many of the historical events can be considered as parts of the theory itself. For example, there are different notations used in differential calculus such as Leibniz, Newton, and Lagrange. When we multiply two large numbers we look for computationally efficient ways. Different techniques were developed in different parts of the world at different times. Learning them is learning a history itself partly because we see the faster algorithms in the later points on the time line as well as magnitude of time and afford invested (in development) for an increase in computing speed.

Behind each body of human knowledge, there is an underlying human population on a single planet who are going to feel the influence of development on them even if most of us are not the developers and decision makers of theories or technologies. With respect to educational materials, peoples should have the right to know what is going on. (1) The magnitude of jargons used in the paper is a concern for us because it limits the readership. (2) In the argument in support for history in mathematics education, in part (3) of (d), the author correctly argued that failures, mistakes, uncertainties or misunderstandings have been building blocks of mathematicians. However in today school exam system, failure at the exam will be failure. These two particular things are where I stopped for thinking.

After reading this article, with respect to whether or not integrating history in mathematics education, my idea remains the same. I am in support of it. However, more important things are the details and larger picture of consequences. Some objects not regarded as (academic) math in the past have become part of math. Today, if art is not math, it could become part of math in the future. Suppose in history study we find that two groups of humans produced much different amounts of contributions to mathematics, then what occur in our minds? Do we quietly assign rating (aka give grading) to them and why?