2. Converting $\frac{1}{2}$ to a decimal equals 0.5, which is a finite decimal; converting $\frac{1}{11}$ to a decimal equals $0.090 \cdots$, briefly written as 0.09, which is a pure repeating decimal; converting $\frac{1}{6}$ to a decimal equals $0.1666 \cdots \cdots$, briefly written as 0.16, which is a mixed repeating decimal. Now, converting 2004 fractions $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{2005}$ to decimals, the question is: how many of them are pure repeating decimals?
2. 【Solution】If the prime factors of the denominator only contain 2 and 5, the fraction can be converted to a finite decimal. If the prime factors of the denominator do not contain 2 and 5, the fraction can be converted to a pure repeating decimal after the finite decimal part. Therefore, this problem is actually asking how many numbers from 2 to 2005, among the 2004 numbers, do not contain the prime factors 2 or 5. Among these 2004 numbers, there are $2004 \div 2=1002$ numbers that contain the prime factor 2, $2005 \div 5=401$ numbers that contain the prime factor 5, and $2000 \div 10=200$ numbers that contain both 2 and 5. Therefore, the numbers that can be converted to a pure repeating decimal are $2004-1002-401+200=801$ numbers.