Chapter 13. Problems Leading to Algebra
Algebra suffers from a motivational problem. Examples of the useless artificiality
of most algebraic problems abound in every textbook ever written on the subject.
Here, for example, is a problem from Girolamo Cardano's book Ars magna (1545):
Two men go into business together and have an unknown capital.
Their gain is equal to the cube of the tenth part of their capital.
If they had made three ducats less, they would have gained an
amount exactly equal to their capital. What was the capital and
their profit? [Quoted by Pesic, (2003), pp. 30-31]If reading this problem makes you want to suggest, "Let's just ask them what
their capital and profit were," you are to be congratulated on your astuteness. The
second statement of the problem, in particular, marks the entire scenario as an airy
flight of fancy. Where in the world would anyone get this kind of information?
What data banks is it kept in? How could anyone know this relationship between
capital and profit without knowing what the capital and profit were? One of the
hardest questions to answer in teaching either algebra or its history is "What is it
/or?" Although some interesting algebra problems can be generated from geometric
figures, it is not clear that these problems are interesting as geometry. Reading the
famous treatises on algebra, we might conclude that it is pursued for amusement
by people who like puzzles.^1 That answer is not very satisfying, however, and we
shall be on the alert for better motivations as we study the relevant documents.
1. Egypt
Although arithmetic and geometry fill up most of the Egyptian papyri, there are
some problems in them that can be considered algebra. These problems tend to
be what we now classify as linear problems, since they involve the implicit use of
direct proportion. The concept of proportion is the key to the problems based on
the "rule of false position." Problem 24 of the Ahmose Papyrus, for example, asks
for the quantity that yields 19 when its seventh part is added to it. The author
notes that if the quantity were 7 (the "false [sup]position"), it would yield 8 when
its seventh part is added to it. Therefore, the correct quantity will be obtained
by performing the same operations on the number 7 that yield 19 when performed
on the number 8. The Egyptian format for such computations is well adapted for
handling problems of this sort. The key to the solution seems to be, implicitly, the(^1) In one episode of a popular American situation comedy series during the 1980s, a young police-
woman was working undercover, pretending to be a high-school student. While studying algebra
with a classmate, she encountered a problem akin to the following. "Johnny is one-third as old as
his father; in 15 years he will be half as old. How old are Johnny and his father?" Her response—a
triumph of common sense over a mindless educational system—was: "Do we know these people?"
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