To determine which of the following numbers is irrational, let's examine the options carefully:

1. 2.25 (with bar notation over 25): This number represents a repeating decimal, $2.252525...$, which is a rational number since it can be expressed as a fraction (in this case, $\frac{223}{99}$).

2. 2.24: This is a terminating decimal, and any terminating decimal is rational because it can be written as a fraction (in this case, $\frac{224}{100}$).

3. 2.2 (with bar notation over 2): This represents a repeating decimal, $2.222...$, which is also rational, as it can be written as a fraction $\frac{20}{9}$.

4. $\sqrt{5}$: This is an irrational number because the square root of 5 cannot be expressed as a simple fraction, and its decimal expansion is non-repeating and non-terminating.

So, the irrational number between 2.2 and 2.4 is $\sqrt{5}$.

Would you like more details or explanations?

Here are 5 related questions:

1. What makes a number irrational?
2. Can repeating decimals ever be irrational?
3. How do you convert repeating decimals into fractions?
4. What is the approximate value of $\sqrt{5}$?
5. Are all square roots irrational?

Tip: Irrational numbers cannot be written as exact fractions and their decimal expansion goes on infinitely without repeating.