1.4.7 ** Summation
$$
S=[\lg 2]+[\lg 3]+\cdots+[\lg 2008]+\left[\lg \frac{1}{2}\right]+\left[\lg \frac{1}{3}\right]+\cdots+\left[\lg \frac{1}{2008}\right]
$$

Find the value of the sum. Here $[x]$ denotes the greatest integer not exceeding $x$.
Notice that, for any real number $x$, we have
$$
[x]+[-x]=\left\{\begin{array}{cl}
0, & x \in \mathbf{Z}, \\
-1, & x \notin \mathbf{Z}
\end{array}\right.
$$

For $m \in\{2,3, \cdots, 2008\}$, only when $m=10,100,1000$, $\lg m \in \mathbf{Z}$. Therefore,
$$
S=\sum_{m=2}^{2008}\left([\lg m]+\left[\lg \frac{1}{m}\right]\right)=\sum_{m=2}^{2008}([\lg m]+[-\lg m])=-2004 .
$$

Thus, the value of the required sum is -2004.