Explain How Irrational Numbers Differ from Rational Numbers
Question: Explain how irrational numbers differ from rational numbers.
Explanation:
The difference between rational and irrational numbers is mainly based on whether the number can be written in the form \[ \frac{p}{q} \] where \(p\) and \(q\) are integers and \(q \ne 0\). :contentReference[oaicite:0]{index=0}
Difference Table:
| Rational Numbers | Irrational Numbers |
|---|---|
| Can be written as \( \frac{p}{q} \) | Cannot be written as \( \frac{p}{q} \) |
| Decimal expansion is terminating or repeating | Decimal expansion is non-terminating and non-repeating |
| Examples: \( \frac{1}{2}, 0.75, 0.\overline{3} \) | Examples: \( \sqrt{2}, \pi \) |
Key Points:
- All rational numbers can be expressed as fractions.
- Irrational numbers cannot be expressed as fractions.
- Rational numbers have finite or repeating decimals, while irrational numbers have infinite non-repeating decimals. :contentReference[oaicite:1]{index=1}
Conclusion:
Thus, the main difference lies in their representation and decimal expansion. Rational numbers are predictable (terminating or repeating), whereas irrational numbers are infinite and non-repeating.