Plimpton 322

Exercises

1. Convert the following ...
• (a) ...sexagesimal numbers from Plimpton 322 into decimal form.
• (i) [38, 11]
• (ii) [59, 1]
• (iii) [27, 59]
• (iv) [48,49]
• (v) [3, 31, 49]
• (b) ...decimal numbers into sexagesimal form.
• (i) 180
• (ii) 10800
• (iii) 375
• (iv) 750
• (v) 13179661
2. Convert the numbers from the first column of Plimpton 322 in the indicated rows into fractional form in lowest terms.
• (i) row 11
• (ii) row 13
• (iii) row 15
• (iv) row 3
• (v) row 10
3. The digits in colored print in our transliteration of Plimpton 322 are mistakes made by the Scribe.  (Although it is pure speculation, the Scribe may have been an apprentice mathematician, and Plimpton 322 a B+ homework assignment!)
• (a) Consider the entry in column 2, row 9. What sexagesimal number should the Scribe have marked here? How did he make this error?
• (b) The entry in column 3, row 15.. What sexagesimal number should the Scribe have marked here and what was the nature of the error?
4. Why was 60 chosen as the base of a numeration system? One reason for this can be gleaned from the following computations. Form a table of the reciprocals of the numbers from 1 to 10: in one column express these as fractions; in a second, write them as exact decimals (if the decimal representation does not terminate, identify the repeating block of digits that determine the full representation); finally, in a third column, give the exact sexagesimal form (which may also require repeating blocks of digits). Which of the decimals representations have terminating forms, and which of the sexagesimal ones do? In general, many more fractions have terminating representations in the sexagesimal system than in the decimal system (for an extra challenge, try to explain why), making the latter more clumsy for denoting these values. Consequently, it can be argued that sexagesimal notation is better than decimal for doing arithmetic.
5. Another Babylonian tablet of mathematical significance is YBC 7289 (from the Yale Babylonian Collection). See a reproduction of the tablet here. It depicts a square with both diagonals drawn in. One side of the square is labeled with the number [; 30] while a diagonal is labeled with the two numbers [1; 24, 51, 10] and [; 42, 25, 35].
• (a) Verify that the product of the first two of these numbers is the third; that is, that [; 30] ¥ [1; 24, 51, 10] = [; 42, 25, 35].
• (b) What number is it that when multiplied by the length of the side of a square gives the length of a diagonal? Convert this number into sexagesimal form.

(c) What is [1; 24, 51, 10] ¥ [; 42, 25, 35]? Use this answer to determine the values that these numbers are meant to approximate.