Find an Irrational Number Between Given Decimals
Question: Find one irrational number between \(0.2101\) and \(0.\overline{2}\) (i.e., \(0.222…\)).
Concept Used:
An irrational number is a non-terminating, non-repeating decimal. :contentReference[oaicite:0]{index=0}
Solution:
We are given:
\[ 0.2101 < \ ? \ < 0.2222... \]
We need a number between these which is non-terminating and non-repeating.
Consider the number:
\[ 0.21221222122221… \]
This number:
- Is clearly greater than \(0.2101\)
- Is less than \(0.2222…\)
- Does not repeat in any fixed pattern
Hence, it is an irrational number.
Final Answer:
\[ \boxed{0.21221222122221…} \]
Conclusion:
Since irrational numbers are infinite and non-repeating, we can construct infinitely many such numbers between any two real numbers.