I'm familiar with the maths behind flash usage (guide numbers, etc.). Here's a more practical question that doesn't yet seem to have an answer on photo.se:
Consider a subject, outdoors, on a regular "sunny 16" day (EV 15). Assume you've rated your film @100. For some reason, you want to seperate your subject from the background by underexposing the latter by, say, two stops. Say, f/11 & 1/800 rather than f/16 & 1/100. Now in order to compensate for that underexposure, you want to use a flash on your subject to get it back to EV 15.
What guide number would the flash have to have for that purpose?
GN = d * f
If the subject were at a distance of 2.5m, a GN 40 flash would provide enough light for a f/16 & 1/100 exposure (in a setting that's completely dark otherwise). So using that flash on our subject would exactly double the amount of light it receives. Am I correct assuming that this in turn means we've been able to compensate for one of the two stops (EV 14), but won't be able to compensate for the remaining one?
Answer
You are correct, light is additive, so if the flash provides the same amount of illumination to the subject as the sun, and if no flash spills onto the background, then the background will be one stop darker than the subject.
To underexpose the background by 2 stops would require a flash with three times the power, guide number 70.
It's worth noting that guide numbers are useful for comparing flash brightness, but are not always useful for calculating actual light output at certain distances because modern hot-shoe flashes have optics to change the width of the beam, effectively focusing the light at further distances so the inverse square law on which the guide number is based doesn't necessarily apply.
However, from experience, I've never used a hot-shoe flash that was capable of underexposing the background by 2 stops on a bright sunny day at any sensible distance. Very high powered units have been produced in the past, but seem less popular now as high ISO performance is so much better with digital than it was with film, so people can always up the ISO (this doesn't help underexpose the background, but that's a rare use case).
The second problem with underexposing the background by 2 stops comes from the flash sync speed. Most cameras can't shoot the flash at full power with shutter speeds faster than 1/250s or 1/200s. So you wouldn't be able to shoot at f/11 & 1/800s, as per your example. You'd have to use f/22 & 1/200s, at which point diffraction can be a problem.
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