Why Read This Book? - Index of

110 Chapter 3 Sets and Their Properties
down onto the imaginary paper using techniques of construction. How long is
it? Some whole multiple of a (possibly very short) segment that can be constructed
by reciprocating a segment of positive integer length n. In modern
language, what number is represented by the length of a segment? It is always
a rational one.
Now suppose we are given two segments of arbitrary lengths. If both of them
can be visualized in this way, what sort of relationship must exist between these
two segments? In the same way that we would find a common denominator of two
fractions m/n and p/q, they would say that there is a single segment, perhaps very
short, that can be attached to itself a finite number of times to produce each of
the two given segments. In our language, 1/nq can be added to itself mq times to
produce m/n and np times to produce p/q. This assumed relationship between two
segments is called commensurability. The Greeks believed that all constructible
segments are commensurable, which is equivalent to our believing that all real
numbers are rational.
There was a problem with this, and the Greeks eventually figured it out. Take
a segment of length one, then construct another segment of length one at one
endpoint and at a right angle to this first segment. Then sketch the hypotenuse.
This hypotenuse line segment is obviously constructible. We just described how
to do it easily. Furthermore, by the Pythagorean theorem, its length d satisfies
d 2 = 2. The problem is that it is not commensurable with some other constructible
segments. That is, d is not rational. This is the amazing result that the Pythagoreans
discovered, and it created no small crisis. Given that the theorem traditionally
named after Pythagoras was already known, it boiled down to this fact: Since “the
sum of the [areas of the] squares on the legs of a right triangle is equal to the [area
of the] square on the hypotenuse” (see Fig. 3.12), then, as we would write it, √ 2isa
constructible length. Therefore, √ 2 is commensurable with 1, or, in our language,
√ 2 must be writable in the form p/q, where p and q are integers and q is nonzero.
The traditional proof that this is impossible is attributed to Aristotle. And, by the
way, it is the one you will discover in the following exercise, with a few hints.
c
Figure 3.12 A sketch of the Pythagorean theorem.
c
a
a
b
b