1. Introduction
Real numbers are broadly classified into rational numbers and irrational numbers. Both types appear on the number line, but they differ in how they are expressed, their decimal expansions, and their mathematical properties. The table below provides a clear comparison.
2. Comparison Table
| Property | Rational Numbers | Irrational Numbers |
|---|---|---|
| Definition | Can be expressed as \(\dfrac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\). | Cannot be expressed as \(\dfrac{p}{q}\) with integers \(p, q\). |
| Decimal Expansion | Terminating or repeating decimals. | Non-terminating, non-repeating decimals. |
| Examples | \(\dfrac{1}{2}, -3, 0, 4.75, \dfrac{5}{8}\) | \(\pi, \sqrt{2}, e, \sqrt{3}\) |
| Countability | Countable (can be listed in a sequence). | Uncountable (cannot be listed exhaustively). |
| Representation on Number Line | Dense; between any two rationals lies another rational. | Also dense; between any two real numbers, irrational numbers exist. |
| Arithmetic Operations | Closed under addition, subtraction, multiplication, and division (except division by zero). | Not closed—operations may give rational or irrational results. |
| Symbol | Denoted by \(\mathbb{Q}\). | No standard symbol; described as \(\mathbb{R} \setminus \mathbb{Q}\). |
3. Summary
Rational and irrational numbers together form the set of real numbers. Rational numbers include fractions and numbers with predictable decimal patterns, while irrationals represent quantities with infinite, unpredictable decimal expansions. Understanding the differences helps in identifying any real number quickly.