Find Three Irrational Numbers Between \( \frac{5}{7} \) and \( \frac{9}{11} \)
Solution (Direct Method):
\[ \frac{5}{7} = 0.714285… \quad \text{and} \quad \frac{9}{11} = 0.818181… \]
So, \[ 0.714… < x < 0.818... \]
Choose non-terminating, non-repeating decimals between them:
\[ 0.721001000100001…, \quad 0.752002000200002…, \quad 0.801003000300003… \]
All these numbers:
- Lie between the given numbers
- Are non-terminating and non-repeating
- Hence, irrational
Final Answer:
\[ 0.721001000100001…, \quad 0.752002000200002…, \quad 0.801003000300003… \]