Where to stop when doing decimal division?

Question

We know that $\frac{1}{8} = 0.125$ via calculator; however, if I didn't have access to a calculator and wanted to find this via long division, why would I stop at 3 decimal places? Why not 2 or 4?

For context, I was working on a linear approximation problem for $\sqrt{15}$. From there, I got $L(X) = 4 - \frac{1}{8}$. In class, we're told not to use calculators, and I don't know what $1\over8$ is just by looking at it. So, I was wondering how do I find out what is $1\over8$.

EDIT: My bad, I see why we stop at 0 now.

Answer 1

As $\; 8 = 2 \times 2 \times 2$, and I suppose you know that $\frac 12 = 0.5$, then $$ \frac 18 = \frac 12 \times \frac 12 \times \frac 12 = 0.5 \times 0.5 \times 0.5 = 0.125\,. $$

Edit
Long division:

$$ \begin{array}{r} 0.125\phantom{)} \\ 8{\overline{\smash{\big)}\,1\phantom{).000}}}\\ \underline{-~\phantom{(}0\phantom{.000)}}\\ 10\phantom{).00}\\ \underline{-~\phantom{(}8\phantom{.00)}}\\ 20\phantom{.0)}\\ \underline{-~\phantom{(}16\phantom{.0)}}\\ 40\phantom{.)}\\ \underline{-~\phantom{(}40\phantom{.)}}\\ \text{Stop at zero}\rightarrow \qquad 0\phantom{.)}\\ \end{array} $$

Answer 2

I'm not sure that you can know what the decimal value of a fraction is "just by looking at it." Over time you get to remember values that come up regularly. And you can use values that you know to find values you don't: if asked to find $1/16$ now you would know just to divide $0.125$ by $2$. Another answer to the question of when to stop doing long division is when you have enough digits for whatever you're doing. If your linear approximation problem requires 4 significant digits of accuracy, stop dividing after 4 (or possibly 5) digits.

Answer 3

If I understood your question, you are searching for a method to make a division without a calculator. Similar to Miscellaneous answer, in the case of $1\over8$ this is what I would do:

You can write the rational like this: $$\frac{1}{8} = \frac{1}{2\cdot2\cdot2} = \frac{\frac{\frac{1}{2}}{2}}{2}$$

This means that you need to take the mid part of a number three times, so

$$\frac{1}{2} = 0.5$$ $$\frac{0.5}{2} = 0.25$$ $$\frac{0.25}{2} = 0.125$$