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Irrational numbers have infinite decimal expansions that neither terminate nor repeat, unlike rational numbers. The existence of irrational numbers was first discovered by ancient mathematicians, who were puzzled by the fact that some measurements, such as the diagonal of a square, could not be expressed as a fraction. Irrational numbers are characterized by their non-repeating and non-terminating decimal representations. Examples of well-known irrational numbers include π (pi), √2 (the square root of 2), and e, the base of the natural logarithm. These numbers have been studied extensively throughout history, and their properties continue to be a subject of interest in mathematics. The concept of irrationality has far-reaching implications in number theory, geometry, and other branches of mathematics. It has been proven that the set of irrational numbers is uncountably infinite, meaning that their cardinality is larger than that of the set of natural numbers. This realization has led to the development of advanced mathematical theories, such as Cantor's set theory. The study of irrational numbers has also influenced philosophical and philosophical discussions about the nature of numbers and the foundations of mathematics. Rationality and irrationality are key concepts in the understanding of numbers and provide a deeper understanding of the nature of mathematical reality. In conclusion, irrational numbers are an important and fascinating aspect of mathematics. Their discovery and study have had profound implications for various branches of mathematics and continue to inspire research and exploration.