Do Significant Figures apply with numbers without units? Say I need to divide a number with a unit by a number without a unit, e.g.: $\frac{16.4\: \text{meters}}{2}$
Following the rules on significant figures, will the correct answer be $8.2$m (since $2$ has no unit), or $8$ m (since $2$ has only one significant figure)?
 A: It depends on the type of number. If the dimensionless number arises from pure math or counting, then it essentially has infinite precision and doesn't limit the number of digits you report. This is the case for the $2$ relating radius and diameter, or the $1/2$ in the kinetic energy formula, or the $N!$ you often see in statistical mechanics.
$\pi$ is the same, but if you only plug in finitely many digits into your calculator, your are limited by that. Thus $3.14$ has $3$ significant figures, even if you are approximating the exact value of of $\pi$.
Other dimensionless numbers are indeed measured, and thus have limited precision as well. For example, the fine-structure constant or the g-factor related to the electron gyromagnetic ratio are all measured and have uncertainties.
In general the question of having units vs. being dimensionless is orthogonal to the question of being exact vs. having uncertainty.
A: Think of it in terms of percentage error.
If a reading is $100 \pm 2$ metres this is $100 \pm 2\%$.
Now divide the 100 metres by 5 to give 20 metres.  
The percentage error does not change so the value is $20 \pm 2\% = 20.0 \pm 0.4$ metres.
If you are not given a percentage error then you would have to judge it from the given value.
110 might have the second 1 as significant so approximately $10\%$?  Whereas 111 might indicate a $1\%$ error?
