Real Number System

The real number system includes all numbers used for counting, measuring, and comparing quantities. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers expand this set by including zero and negative whole numbers: ..., –3, –2, –1, 0, 1, 2, .... Rational numbers are those that can be expressed as the ratio of two integers m/n, where n≠0. This includes fractions like 1/2​, integers such as 46, and decimals that terminate, such as 0.17, or those that repeat, like 0.317171717. A repeating decimal can be rewritten as a fraction using algebraic techniques.

Irrational numbers cannot be written as a ratio of integers. Their decimal expansions go on forever without repeating. Examples include √2, √5, 𝜋, e, and 3/𝜋². These often arise from geometric problems, such as finding the diagonal of a square. Every real number corresponds to a point on the number line. This continuous representation allows rational and irrational numbers to coexist within a single mathematical framework, enabling precise calculations and measurements in both theoretical and applied mathematics.