Chapter 7. Ancient Number Theory
The impossibility of getting square roots to come out even, in connection with appli-
cations of the Pythagorean theorem, may have caused mathematicians to speculate
on the difference between numbers that have (rational) square roots and those that
do not. We shall take this problem as the starting point for our discussion of num-
ber theory, and we shall see two responses to this problem: first, in the present
chapter, to find out when indeterminate quadratic equations have rational solu-
tions; second, in Chapter 8, to create new numbers to play the role of square roots
when no rational square root exists.
- Plimpton 322
Rational numbers satisfying a quadratic equation are at the heart of a cuneiform
tablet from the period 1900-1600 BCE, number 322 of the Plimpton collection at
Columbia University. The numbers on this tablet have intrigued many mathemat-
ically oriented people, leading to a wide variety of speculation as to the original
purpose of the tablet. We are not offering any new conjectures as to that purpose
here, only a discussion of some earlier ones.
As you can see from the photograph on p. 160, there are a few chips missing,
so that some of the cuneiform numbers in the tablet will need to be restored by
plausible conjecture. Notice also that the column at the right-hand edge contains
the cuneiform numbers in the sequence 1, 2, 3, 4,7, 8, 9, 10,11,12, 13,...,
Obviously, this column merely numbers the rows. The column second from
the right consists of identical symbols that we shall ignore entirely. Pretending that
this column is not present, if we transcribe only what we can see into our version
of sexagesimal notation, denoting the chipped-off places with brackets ([...]), we
get the four-column table shown below.
Before analyzing the mathematics of this table, we make one preliminary ob-
servation: Row 13 is anomalous, in that the third entry is smaller than the second
entry. For the time being we shall ignore this row and see if we can figure out
how to correct it. Since the long numbers in the first column must be the result
of computation—it is unlikely that measurements could be carried out with such
precision—we make the reasonable conjecture that the shorter numbers in the sec-
ond and third columns are data. As mentioned in Chapter 6, the Mesopotamian
mathematicians routinely associated with any pair of numbers (a, b) two other num-
bers: their average (a + 6)/2 and their semidifference (b — a)/2. Let us compute
these numbers for all the rows except rows 13 and 15, to see how they would
have appeared to a mathematician of the time. We get the following 13 pairs of
numbers, which we write in decimal notation: (144,25), (7444,4077), (5625,1024),
(15625,2916), (81,16), (400,81), (2916,625), (1024,225), (655,114), (6561,1600),
(60,15), (2304,625), (2500,729).
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