i really didn't want to open that can of worms..

here is what i meant about flash: the flash guide number is distance * f-number. convenient, eh?

. let's see why sqrt (as mentioned by matt above) comes in:

the f numbers are meant to designate 1ev step distances between them, as matt pointed out. they are also meant to have the same meaning regarding how much light goes in across focal lengths (so f2.8 means as much light for a 50mm and a 200mm), so the term of relative aperture was born: focal length/aperture diameter. to get 1ev between them, you need to have double/half the light going through when moving up/down one stop, this means, physically, that the aperture _area_ needs to be twice as big/small, but the area is a function of the _square_ of the diameter (pi*(d/2)^2), which means that you end up with these strange square root "leftovers" as f-numbers. they look strange, but are surprisingly convenient.

now back to the guide number: the light falling on the subject coming from the flash will depend on (the inverse of) the square of the distance, the light going back to the sensor will depend on the area of the open aperture, which is described by the square of the f-number, so for one aperture step up, we will have twice the light going through to the sensor/film, and thus we will be able to have sqrt(2)*distnace by using the same flash, when using aperture f-numbers, we already have numbers dimensionally suitable to put against the simple distances (instead of needing to work with squared distances), because, though adimensional, f-numbers are, like distance to subject itself, linear in nature (not areal, so to speak, so, you could say, already sqrt'ed)

2.8*3m=8.4=4*2.1m

and this is why the guidenumber formula works, it might not sound like a big deal these days, with pttl and all this crap, but "back in the days", it was very usefull, and even these days, strobists i think still use this formula to understand and ballance their more complex lighting schemes. so, while the convention seems terribly silly and meaningless, it's actually quite brilliant and well thought over, and has the only downfall of the funky-looking numbers.