Mathematical_Recreations-Kraitchik-2e

CHAPTER FOUR
ARITHMETICO-GEOMETRICAL QUESTIONS
IN MANY geometrical problems we must find geometrical elements
whose measures are integers. Nearly all these questions
lead back to the Pythagorean relation connecting the
lengths of the sides of a right triangle: In any right triangle
the square on the hypotenuse is equal to the sum of the squares on
the other two sides. We shall examine this question from the
beginning.
1. The algebraic expression of the Pythagorean relation
is X2 + y2 = Z2, where z is the length of the hypotenuse and x,
yare the lengths of the legs of the right triangle. If we were
to examine the algebraic relation completely we should have
to admit irrational and even imaginary values of x, y, and z.
However, the most interesting and difficult problems arise
when the numbers are required to be rational. The relation
is trivial if any of the numbers is zero; and since only their
squares appear in the relation the numbers may just as well
be considered to be positive. Finally, if any of the numbers
are fractions the relation connecting the set of numbers is
equivalent to a relation connecting integers obtained from
the original set by clearing the equation of fractions. As a
result of all these considerations we shall give the following
definition:
A Pythagorean number-triple (x, y, z) or, more briefly, a
right triangle (x, y, z) is a set of three positive integers connected
by the relation X2 + y2 = Z2. In this notation z will always
refer to the number whose square is the sum of the squares
95