1. **State the problem:** Calculate the value of $4 - 2 \frac{5}{9}$. 2. **Convert mixed number to improper fraction:** $2 \frac{5}{9} = \frac{2 \times 9 + 5}{9} = \frac{18 + 5}{9} = \frac{23}{9}$. 3. **Convert whole number to fraction with same denominator:** $4 = \frac{4 \times 9}{9} = \frac{36}{9}$. 4. **Subtract fractions:** $$\frac{36}{9} - \frac{23}{9} = \frac{36 - 23}{9} = \frac{13}{9}.$$ 5. **Convert back to mixed number:** $\frac{13}{9} = 1 \frac{4}{9}$. **Final answer:** $1 \frac{4}{9}$