HW3 and Sample Quiz 2

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. 

  1. Explain why Mesopotamian scribes wrote numbers on clay tablets and not on paper? List at leas three  methods (excluding clay tablets) used by ancient civilizations to record numbers. 
  2. 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?
  3. What is the largest number that can be written in cuneiform? 
  4. How do the numbers 62 and 3 can be distinguished when written in cuneiform?
  5. 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. 
  6. Extra Credit (Converted to Extra credit on Oct 5, because there was a mistake in the previous statment) 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)
  7. Write the numbers 1/120 and 1081/1800 in the Babylonian number system (in cuneiform) 
  8. 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 (to be moved to course schedule) 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.

Mayan Mathematics 

  1. Given the Mayan date, say, (10,12,100) in the Calendar Round (the three wheels  13x20x365) find the date 2.2.5.11.15 days later. (We ar following the notation used in class). 
  2. Given a certain number of days, say 35788, find the corresponding calendric date, (of the form a.b.c.d.e)

Helenic Mathematics Before Euclid

  1.  What is the most important feature of Greek mathematics, beginning with Thales, which we did not find in earlier cultures? What makes this feature important?
  2. Extra credit: Find a proof of the Pythagorean theorem not discussed in class. 


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. For instance: in problem 3, you have convert the numbers to decimals showing all the steps. 

  1. What is the largest number that can be written in cuneiform? 
  2. How do the numbers 62 and 3 can be distinguished when written in cuneiform?
  3. Write the numbers 1/120 and 1081/1800 in the Babylonian number system (in cuneiform) 
  4. Given a certain number of days, say 35788, find the corresponding calendric date, (of the form a.b.c.d.e)