Irrational Numbers

Irrational numbers are a set of numbers that cannot be expressed as the ratio of two integers. This means they cannot be written in the form p/q, where p and q are integers and q ≠ 0.

Definition

Irrational numbers are all the real numbers that are not rational. They cannot be expressed as a simple fraction, and their decimal representations are non-repeating and non-terminating.

Some of the most famous irrational numbers include √2 (the square root of 2), π (pi), and e (the base of the natural logarithm).

Properties of Irrational Numbers

Irrational numbers have several important properties:

  1. Non-Repeating, Non-Terminating Decimals: The decimal representation of an irrational number neither terminates nor repeats.

  2. Closure Property (with respect to multiplication and addition): The sum or product of a rational number and an irrational number is irrational. However, the sum or product of two irrational numbers may be rational or irrational.

  3. Dense on the Number Line: Between any two irrational numbers, there are infinitely many other irrational numbers.

Relationship to Rational Numbers

Irrational numbers, together with rational numbers, form the set of real numbers. While every rational number can be written as a simple fraction or a terminating or repeating decimal, irrational numbers cannot.

Applications

Irrational numbers are used in a variety of mathematical contexts, such as geometry (π is used in formulas related to circles), trigonometry, calculus (e is used as the base of the natural logarithm), and complex number theory. They also form the basis for the continuity of the real numbers, which is a fundamental concept in analysis and calculus.