There are actually two different sign conventions in optics. Without any convention it's hard to develop any universal scientific statement (like formula) so let's make up one here:
$F$ – lens focal length,
$d$ – lens to object distance,
$s$ – lens to image distance.
1. Cartesian sign convention
Small lens formula:
$$\frac1s-\frac1d=\frac1F$$
Interpretation: "curvature after - curvature before = curvature added". This convention is widely accepted by professional opticians as it employs idea of "positive = co-propagating with left-to-right ray direction".
2. "Classic" scholar sign convention
Small lens formula:
$$\frac1s+\frac1d=\frac1F$$
$s>0$ and $d>0$ if the image and objects are real and both are negative if both are virtual.
$$\pm\frac1s\pm\frac1d=\pm\frac1F$$
Depending on the type of lens and object position:
- Convex lens $+1/F$, Concave lens $-1/F$;
- Real image $+1/s$, virtual image $-1/s$;
- Diverging ray fan (real object) $+1/d$, converging ray fan (virtual object) $-1/d$.
Sources:
The longitudinal lens formula and sign conventions
Правила знаков (в оптике) on Russian Wikipedia
As I've taught it in EE fundamentals class, as one goes 'round the loop, the voltage variable is added if one comes to the positive labelled terminal first, subtracted if one comes to the negative labelled terminal first, and the terms must sum to zero.
For example consider the following simple circuit:
Before summing 'round the loop, we must choose a reference polarity for the circuit element voltages. Let's label the "top" terminal of the battery and the resistor as positive.
Then, going 'round the loop clockwise, the KVL sum is:
$$V_R - V_{BAT} = 0 \rightarrow V_R = V_{BAT} $$
Which is, for the chosen reference polarities, correct.
I think it's far easier to remember one convention for any circuit element rather than one convention for "voltage rises" and another for "voltage drops".
As an aside, I'd like to comment on the essential arbitrariness of the reference polarity. I could have chosen, for example, to label the "bottom" terminal of the resistor as positive. Let's call that voltage variable $V'_R$. The KVL sum is then:
$-V'_R - V_{BAT} = 0 \rightarrow V'_R = - V_{BAT}$
Although this looks like a different result than before, the physical result is identical.
A "picture" might be helpful. When we label one terminal or the other of a circuit element with a positive sign, we're essentially choosing where to put the red (positive) lead of our voltmeter to measure the voltage variable.
In other words $V_R$ is the voltage measured by placing the red lead on the "top" terminal and $V'_R$ is the voltage measured by placing the red lead on the "bottom" terminal (it should go without saying that the black lead goes on the other terminal of the resistor).
As you know, reversing the leads of the voltmeter changes the sign of the reading but not the magnitude.
But this is precisely the result we get mathematically as we have by inspection:
$$V'_R = -V_R$$
Best Answer
In starting it is futile, but if you use many mirrors, you will get mad.
Actually, you have derived your formula in a particular case, say when image is behind the mirror. To extrapolate it to other cases, you say that you could have used opposite sign of image distance.
But this will create different formulas for different cases as distance is always positive. To avoid a large number of these formulas, we united and decided that positive is in this direction and negative in that one. Now, all of our formulas agree and we live happily together