1. **State the problem:** We need to calculate the division of 13 by 55, i.e., find the value of $\frac{13}{55}$.\n\n2. **Formula used:** Division of two numbers $a$ and $b$ (with $b \neq 0$) is given by $\frac{a}{b}$. Here, $a=13$ and $b=55$.\n\n3. **Simplify the fraction:** Check if 13 and 55 have any common factors other than 1. Since 13 is a prime number and does not divide 55 evenly, the fraction is already in simplest form.\n\n4. **Convert to decimal:** Perform the division $13 \div 55$.\n\n5. **Calculation:** $$13 \div 55 = 0.2363636\ldots$$ (repeating decimal)\n\n6. **Final answer:** The exact value is $\frac{13}{55}$ or approximately $0.2364$ when rounded to four decimal places.