Define an Irrational Number
Question: Define an irrational number.
Definition:
An irrational number is a number that cannot be expressed in the form \[ \frac{p}{q} \] where \(p\) and \(q\) are integers and \(q \ne 0\).
Its decimal expansion is non-terminating and non-repeating.
Examples:
- \(\sqrt{2}\)
- \(\sqrt{3}\)
- \(\pi\)
These numbers cannot be written as a simple fraction and their decimal form never ends or repeats.
Key Points:
- Cannot be written as \( \frac{p}{q} \)
- Decimal expansion is non-terminating
- Decimal expansion is non-repeating
Conclusion:
Irrational numbers are real numbers that cannot be represented as fractions and have infinite non-repeating decimal expansions.