Convert the following
Convert the numbers from
the first column of Plimpton 322 in the indicated rows into fractional
form in lowest terms.
(a) ...sexagesimal numbers
from Plimpton 322 into decimal form.
(i) [38, 11]
(ii) [59, 1]
(iii) [27, 59]
(v) [3, 31, 49]
(b) ...decimal numbers
into sexagesimal form.
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!)
(i) row 11
(ii) row 13
(iii) row 15
(iv) row 3
(v) row 10
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
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) 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?
Return to the calendar
(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.
Copyright (c) 2000.
Daniel E. Otero