A rational number can be expressed as a fraction p/q, where p and q are integers and q ≠ 0, while an irrational number cannot be expressed in this form.
Here's a detailed breakdown:
Rational Numbers
- Definition: A rational number is any number that can be written as a ratio or fraction, p/q, where both p and q are integers and q is not equal to zero.
- Form: p/q, where p and q are integers and q ≠ 0.
- Examples:
- 2/3
- -5/7
- 4 (can be written as 4/1)
- 0 (can be written as 0/1)
- 0.5 (can be written as 1/2)
- Repeating decimals like 0.333... (can be written as 1/3) and 0.142857142857... (can be written as 1/7)
- Decimal Representation: Rational numbers have either terminating decimal expansions (like 0.5) or repeating decimal expansions (like 0.333...).
Irrational Numbers
- Definition: An irrational number is a number that cannot be expressed as a fraction p/q, where p and q are integers.
- Form: Cannot be expressed in the form p/q, where p and q are integers.
- Examples:
- √2 (square root of 2)
- √3 (square root of 3)
- π (pi)
- e (Euler's number)
- Decimal Representation: Irrational numbers have non-terminating and non-repeating decimal expansions.
Key Differences Summarized
| Feature | Rational Number | Irrational Number |
|---|---|---|
| Fractional Form | Can be expressed as p/q (p and q are integers, q≠0) | Cannot be expressed as p/q (p and q are integers) |
| Decimal Expansion | Terminating or repeating | Non-terminating and non-repeating |
| Example | 1/2, 5, 0.75, 0.333... | √2, π, e |
In essence, the ability to represent a number as a simple fraction of two integers distinguishes rational numbers from irrational numbers. This difference also manifests in their decimal representations: rational numbers terminate or repeat, while irrational numbers continue infinitely without repeating.
