Rational Numbers: A Comprehensive Guide
📐 What is a Rational Number?
Formal Definition: A rational number is any number that can be expressed as the quotient (or fraction) of two integers, written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
The word "rational" comes from "ratio" — because every rational number is a ratio of two integers.
The set of rational numbers is denoted by the symbol $\mathbb{Q}$:
$$\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z},\; q \neq 0 \right\}$$
🔢 Which Numbers Are Rational?
✅ Rational Numbers Include:
| Type | Example | As $\frac{p}{q}$ |
|---|---|---|
| Integers | $5$ | $\frac{5}{1}$ |
| Negative integers | $-3$ | $\frac{-3}{1}$ |
| Zero | $0$ | $\frac{0}{1}$ |
| Proper fractions | $\frac{3}{4}$ | $\frac{3}{4}$ |
| Mixed numbers | $2\frac{1}{3}$ | $\frac{7}{3}$ |
| Terminating decimals | $0.75$ | $\frac{3}{4}$ |
| Repeating decimals | $0.\overline{3}$ | $\frac{1}{3}$ |
❌ NOT Rational (Irrational):
| Number | Why Not? |
|---|---|
| $\sqrt{2} \approx 1.41421356...$ | Non-repeating, non-terminating |
| $\pi \approx 3.14159265...$ | Non-repeating, non-terminating |
| $e \approx 2.71828...$ | Non-repeating, non-terminating |
| $\sqrt{3} \approx 1.73205...$ | Non-repeating, non-terminating |
Key Insight: Irrational numbers have decimal expansions that go on forever without any repeating pattern.