Find the Opposite Number: Determining the Additive Inverse of -8

Convert $-8$ into its reciprocal form:

To determine the opposite number of $-8$ according to the instructions, we need to find its reciprocal.

• Step 1: Recall that the reciprocal of a number $a$ is defined as $\frac{1}{a}$.
• Step 2: Applying this definition, the reciprocal of $-8$ is $\frac{1}{-8}$.
• Step 3: Simplify $\frac{1}{-8}$ to $-\frac{1}{8}$.

The result is $-\frac{1}{8}$. Comparing this with the provided choices, we see that choice 3, $-\frac{1}{8}$, is correct.

Therefore, the opposite number in the context of this problem is $\boxed{-\frac{1}{8}}$, making the correct answer:

$-\frac{1}{8}$