Mathematics in Mesopotamia
Important note: On the problems that require computations, the steps to find the answer must be included in your submission. An answer with little or no explanantion will receive little or no credit. For instance: in problem 5, you have to write the numbers first in base 60 (with numerals in decimal), and then convert them to decimals showing all the steps.
- List at least three methods used by ancient civilizations to record numbers.
- Give a (reasonable) lower bound of the number of cuneiform tablets that have been found. List your sources. Extra credit: State how many of those are mathematical.
- Recall that Mesopontamian number system was in base 60. Are we still using base sixty to count or measure? If yes, what are we counting or measuring in base 60. Do we use any other base (other than 60 and 10) to count or measure something?
- What is the largest number that can be written in the number system used in Ancient Mesopotamia (in cuneiform)?
- How do the numbers 62 and 3 can be distinguished when written in cuneiform?
- The two photos on the left are of the Babylonian tablet YBC 7289. We can see three numbers, one labeling a side and two labeling the diagonal of the square. Convert these three number to decimal, explain what the numbers labeling the diagonal represent and discuss how accurate are they. (Recall that the decimal separator - or period- was not indicated. If we do not indicate the decimal separator in the hindu-arabic number system, we would write 125 for 1.25 or 5/4. )
- Extra Credit Prove that 1/n has a finite expansion in base 10 if and only if n=2k5r.for some non-negative integers k and r (Note: These are the numbers that appear on the Babylonian table of reciprocals)
- Extra Credit Prove that 1/n has a finite expansion in base 60 if and only if n=2k3l5r.for some non-negative integers k,l and r (Note: These are the numbers that appear on the Babylonian table of reciprocals)
- Extra credit: For each positive integer k, characterize the positive integers n such that 1/n has a finite expansion in base k.
Interesting reads : The key to cracking long-dead languages?, First Report of the Arrival of the Rosetta Stone in England, The Rosseta Stone discussed in the Khan Academy
Many of these problems are inspired in the work of the historian Eleanor Robson.
Image Credits: Top: Urcia, A., Yale Peabody Museum of Natural History, http://peabody.yale.edu
Bottom: A modified version of Bill Casselman's photo of YBC 7289, with hand tracings to emphasize the cuneiform markings.
Sample quiz 2: (These are some of the homework problems )
Important note: On the problems that require computations, the steps to find the answer must be included in your submission. An answer with little or no explanation will receive little or no credit. )
- What is the largest number that can be written in cuneiform?
- How do the numbers 62 and 3 can be distinguished when written in cuneiform?
- Find the statement of one of Zeno’s paradoxes and discuss it in your own words.