What is the Difference Between Irrational and Rational Numbers?

The main difference between rational and irrational numbers lies in their representation and decimal expansion properties. Here are the key differences:

1. Rational Numbers: These are real numbers that can be represented in the form of a ratio of two integers, say P/Q, where Q is not equal to zero. Examples of rational numbers include integers (e.g., 2, -5), fractions (e.g., 3/4, 5/6), and repeating decimals (e.g., 1/3 = 0.333…). The decimal expansion of rational numbers is either terminating or recurring.
2. Irrational Numbers: These numbers cannot be expressed as a ratio of two integers. Irrational numbers are represented in decimal form, but they do not have a repeating or terminating decimal expansion. Examples of irrational numbers include π (Pi) = 3.14159…, Euler’s Number (e) = 2.71828…, and square roots of non-perfect squares (e.g., √2, √3).

To determine whether a number is rational or irrational, check if the number can be written as a fraction. If it can, it is rational; if not, it is irrational.

Comparative Table: Irrational vs Rational Numbers

The difference between rational and irrational numbers can be summarized in the following table:

Rational numbers include integers, fractions, and decimals that either terminate or have a repeating pattern. Examples of rational numbers are 5, 10, 3/4, and 0.8. On the other hand, irrational numbers cannot be expressed as a simple fraction, and their decimal representations are non-repeating and non-terminating. An example of an irrational number is √2.