8. The set of positive real numbers $X$ satisfies: $x \in X$ if and only if $x+\{x\}^{2}$ is an integer (where, $\{x\}$ denotes the fractional part of $x$). If the numbers in $X$ are arranged in strictly increasing order, then the sum of the first 100 terms is $\qquad$ .
8. $2475+25 \sqrt{5}$.

Obviously, every positive integer $x$ belongs to $X$, and we denote them in strictly increasing order as $a_{1}, a_{2}, \cdots$.

Next, consider the elements in $X$ that are not positive integers, and we denote them in strictly increasing order as $b_{1}, b_{2}, \cdots$.

Since $x+\{x\}^{2}$ is an integer if and only if $\{x\}+\{x\}^{2}$ is an integer, and $0<\{x\}<1$, we have:
$$
\{x\}+\{x\}^{2}=1 \text {. }
$$

Solving this, we get $\{x\}=\frac{\sqrt{5}-1}{2}$.
Thus, $b_{k}=k-1+\frac{\sqrt{5}-1}{2}, a_{k}=k(k=1,2, \cdots)$, and the first 100 terms of $X$ arranged in ascending order are:
$b_{1}, a_{1}, b_{2}, a_{2}, \cdots, b_{50}, a_{50}$.
Therefore, $S_{100}=\sum_{k=1}^{50}\left(b_{k}+a_{k}\right)=2475+25 \sqrt{5}$.