Definition And Notation

The Set of Real Numbers

Definition and Notation

The set of real numbers, denoted by R, encompasses all rational and irrational numbers. Rational numbers can be expressed as fractions of integers (e.g., 1/2, -3/4), while irrational numbers cannot be expressed in such a way (e.g., √2, π).

The Cartesian Product R²

R² represents the Cartesian product of the set R with itself, forming the set of all ordered pairs of real numbers. For example, (1, 2), (-3, 5) are members of R².

Composition and Properties

The set R includes all the points on the number line, including well-known constants like π, √2, and √3. It is constructed by combining the rational numbers (ℚ) and the irrational numbers (I).

Despite the abundance of rational numbers, their set is countable, meaning they can be listed in a specific order. On the other hand, the set of irrational numbers is uncountable, indicating that no such ordering is possible.