In the course of this chapter and the next, we shall learn that the real numbers can be classified not only into rational and irrational numbers, but also into two other categories. One category contains the so-called algebraic numbers, i.e., those numbers which are solutions of algebraic equations with integer coefficients, and the other includes all remaining numbers and these are called transcendental numbers. This distinction will become more meaningful in what follows. We mention at once, however, that some algebraic numbers are rational and some are irrational, but all transcendental numbers are irrational.

The over-all purpose of this chapter is to devise a systematic method for determining whether or not a given algebraic number is rational. (Actually, we shall not treat the class of algebraic numbers in its greatest generality, but we shall apply our method to many examples.) But before we derive this method, we shall study some simple properties of irrational numbers.

Closure Properties

In contrast to the rational numbers which were shown to be closed under addition, subtraction, multiplication, and division (except by zero), the irrational numbers possess none of these properties. Before showing this, we prove a theorem which will enable us to manufacture infinitely many irrational numbers from one given irrational number.