The History of Mathematics: A Brief Course

182 7. ANCIENT NUMBER THEORY

By brute-force enumeration of cases, the author concludes that there will be
377 pairs, and "in this way you can do it for the case of infinite numbers of months."
The sequence generated here (1, 1, 2, 3, 5, 8,...), in which each term after
the second is the sum of its two predecessors, has been known as the Fibonacci
sequence ever since the Liber abaci was first printed in the nineteenth century. The
Fibonacci sequence has been an inexhaustible source of identities. Many curious
representations of its terms have been obtained, and there is a mathematical journal,
the Fibonacci Quarterly, named in its honor and devoted to its lore. The Fibonacci
sequence has been a rich source of interesting pure mathematics, but it has also had
some illuminating practical applications, one of which is discussed in Problems 2.4-
2.6.

Questions and problems

7.1. Compute the sexagesimal representation of the number

for the following pairs of integers (p,q): (12,5), (64,27), (75,32), (125,54), and
(9,5). Then correct column 1 of Plimpton 322 accordingly.

7.2. On the surface the Euclidean algorithm looks easy to use, and indeed it is
easy to use when applied to integers. The difficulty arises when it is applied to
continuous objects (lengths, areas, volumes, weights). In order to execute a loop
of this algorithm, you must be able to decide which element of the pair (a, b) is
larger. But all judgments as to relative size run into the same difficulty that we
encounter with calibrated measuring instruments: limited precision. There is a
point at which one simply cannot say with certainty that the two quantities are
either equal or unequal. Docs this limitation have any practical significance? What
is its theoretical significance? Show how it could give a wrong value for the greatest
common measure even when the greatest common measure exists. How could it
ever show that two quantities have no common measure?

7.3. The remainders in the Euclidean algorithm play an essential role in finding the
greatest common divisor. The greatest common divisor of 488 and 24 is 8, so that
the fraction 24/488 can be reduced to 3/61. The Euclidean algorithm generates
two quotients, 20 and 3 (in order of generation). What is their relation to the two
numbers? Observe the relation
1 _ 3^

If you find the greatest common divisor of 23 and 56 (which is 1) this way, you will
generate the quotients 2, 2, 3, 3. Verify that

23 _ 1

(^2) + -l
-3-
This expression is called the continued fraction representation representation of
23/56. Formulate a general rule for finding the continued fraction representation
of a proper fraction.