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.