The Set Of Irrational Numbers: A Key Component Of The Real Number System

Abstract

This article examines the set of irrational numbers, a critical component of the real number
system characterized by numbers that cannot be expressed as fractions of integers. Represented as \(
\mathbb{R} \setminus \mathbb{Q} \), irrational numbers include notable examples such as \(\pi\) and
\(\sqrt{2}\). The paper explores the defining properties of irrational numbers, including their non-terminating,
non-repeating decimal expansions, their density on the real number line, and their uncountable infinity.
Historical developments, from the Pythagoreans' discovery of irrational numbers to modern mathematical
advancements, are discussed. The article also highlights the importance of irrational numbers in various
mathematical and scientific fields, including geometry, calculus, and physics. By examining their theoretical
and practical implications, the article underscores the fundamental role of irrational numbers in understanding
and applying the real number system.