Real Numbers: Rational and Irrational Numbers

Real Numbers

Rational Numbers

• A rational number is a real number that can be expressed as the ratio of two integers, i.e., p/q, where p and q are integers and q ≠ 0.
• Examples: 3/4, 22/7, 1/2
• Rational numbers can be expressed as finite decimals or recurring decimals.
• Rational numbers are either terminating or repeating.
• The set of rational numbers is denoted by Q.

Irrational Numbers

• An irrational number is a real number that cannot be expressed as the ratio of two integers, i.e., it is not a rational number.
• Examples: π, e, √2, √3
• Irrational numbers have infinite non-repeating decimals.
• Irrational numbers are non-terminating and non-repeating.
• The set of irrational numbers is denoted by Q' (or sometimes by R\Q).

Key Properties

• The sum of two rational numbers is always rational.
• The sum of a rational number and an irrational number is always irrational.
• The product of two rational numbers is always rational.
• The product of a rational number and an irrational number is always irrational.
• The set of real numbers is the union of the set of rational numbers and the set of irrational numbers.