MAT336: History of Mathematics

Study Guide: Quiz 4

Mesopotamian Mathematics

Note: This is a study guide. The quiz will consist of three or four questions covering the material below. If you understand the ideas and facts in the non-computational questions and can work through the computational problems, you will be well prepared.

Part 1: Quiz-Style Questions

  1. Write 43 and 2403 first in base 60 and then in cuneiform. What Mesopotamian numeral or numerals are used in each sexagesimal place?
  2. (Related to the previous question.) Write 3605 and 65 in base 60, then in cuneiform.
    • (a) What are the numerals forming these numbers?
    • (b) Assume the numeral for 0 had not yet appeared in Mesopotamia. One of these could easily be misread as the other on a damaged tablet, or if written by a careless scribe. Explain why.
  3. Express the numbers A. B. and C. first in base 60 and then in Hindu-Arabic numerals:
    Babylonian numerals and three numbers

    Babylonian numerals and three numbers

  4. A scribe writes a number as 20. Give three different values this could represent, and a situation in which each would be the right reading.
  5. The √2 tablet gives the computation of the diagonal of a square of side 30. If a scribe needed the diagonal of a square of side 7, what would he do? (You only need to indicate, not perform the computations) What scholarly explanation discussed in class accounts for who wrote the tablet and why (1;24,51,10)60 is written on the tablet?
  6. Convert 1/16 and 21/80 into base 60. Show your steps.
  7. Convert 7/5 and 33/50 to sexagesimal (base-60) notation. Show your steps.
  8. Convert 1/9 into base 60. Does the process stop? Explain how you know.
  9. (Optional) Why does 1/20 have a terminating sexagesimal expansion, while 1/7 does not?
  10. Why would tables (e.g., multiplication tables, tables of reciprocals, squares, square roots, etc.) be especially valuable in a system built around repeated numerical procedures?
  11. What feature of the Mesopotamian number system provided a natural framework for developing systematic algorithms and carrying out highly accurate computations?
  12. We inherited 360° circles and 60-minute hours from Mesopotamia. Does this continued use prove that these were the best choices? Or was it a mix of tradition and efficiency, with the balance hard to determine?
  13. In Old Babylonian geometry, each geometrical figure has a defining component (typically a particular length of a curve or segment), from which its other parameters are computed. A coefficient is a fixed numerical constant that relates one of these parameters (for example, a length or an area) to the defining component: multiplying the defining component (or its square) by the coefficient yields that parameter.

    The defining component of a circle was the circumference, and the coefficient for the diameter was 1/3.

    • (a) Write a formula expressing the diameter d in terms of the circumference c using this coefficient. Provide a numerical example.
    • (b) Give a modern interpretation of this coefficient. What does it tell us about Babylonian knowledge of the relationship between circumference and diameter?
  14. The following is a problem from a Babylonian tablet, 1900–1600 BCE. "I found a stone but did not weigh it. After I subtracted 1/7, added 1/11 [of the difference], and then subtracted 1/13 [of the previous total], it weighed 1 mina. What was the stone's weight?"
    • (a) Set up an equation with unknown x whose solution answers the problem.
    • (b) Do you think this problem had a practical application or was mathematics for its own sake?
  15. Two squares problem (Babylonia, c. 1800–1600 BCE): "I have added the areas of my two squares; [result] (25,25)60. [The side of] the second square is 2/3 the side of the first plus 5." Set up the equation(s) needed to solve this problem in the Hindu-Arabic number system.
  16. Most surviving mathematical tablets are tables or problems. Give one example of each and explain what this tells us about how mathematics was practiced.
  17. Why have so many Mesopotamian written records survived? How does the material used for writing affect what historians know about a civilization?
  18. The Behistun Inscription is called the "cuneiform Rosetta stone." What feature made it useful for decipherment?
  19. Give a brief discussion of the account of the development of writing discussed in class, including the proposed relationship between accounting tokens and writing.
  20. King Šulgi's praise poem presents mathematical skills as royal achievements linked to "knowledge and comprehension." What does this reveal about how Mesopotamian culture valued mathematics?
  21. Explain one concrete way in which the environment could have pushed societies toward abstraction (numbers, calendars, or writing).
  22. (Not for lecture 2.) Find the reciprocals in base 60 of 18 and 32. (Recall: in this context, the product of a number with its reciprocal can be any power of 60.)

    Remark: Tables of reciprocals were important in Mesopotamian mathematics because they allowed scribes to perform division by using the relation A ÷ B = A × (1/B).

  23. (Not for lecture 2.) Square-root algorithm (Babylonian method): Starting from guess a = 3/2, compute one iteration to approximate √2. Explain the update step in words. Sketch the corresponding figure(s).

Part 2: Reflection Questions

  1. How does a written law code (like Hammurabi's) change the nature of justice compared to an oral tradition?
  2. Compare Egyptian and Mesopotamian approaches to fractions. How did their different number systems lead to different computational strategies?
  3. A Mesopotamian scribe needs to divide 47 by 7 using only multiplication and reciprocal tables. Explain the problem the scribe faces and what strategies might work.
  4. Mesopotamian mathematics that we know from clay tablets appears primarily in administrative and practical contexts (canal digging, field measurement, wages). What does this tell you about who was doing mathematics and why?
  5. How did the clay tablet medium shape what kind of mathematics Mesopotamians could do? Would their methods have been different on papyrus or parchment?

Quiz Problem Rubric

Points Criteria
3 Correct answer with reasoning/work shown
2 Partially correct with some reasoning shown
1 Correct answer without reasoning/work OR significant attempt with some understanding
0 Incorrect or blank

Notes