Stupid Question about stops

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I'm having an issue with finding a definitive answer about stops. When you say that, for instance, using a ND 0.6 cuts 2 stops of light, is this referring to the f-stops on the lense or is it a messurment of light. Does a stop equal a certain amount of foot candles or does it simpley means that I can drop down two f-stops? Also, is f2.4 to f2.8 be considered a half a stop and not another stop till I get to f3.4 and what about when I get up to f4.0? Is that a full stop or not till f4.8? I understand what f-stops do for light and depth of field, I understand measurement of light in Kelvin, foot candles and Lux's but I've never found a complete clear answer about stops.
 
The f number is the focal legth of the lens divided by the diameter of the pupil. So, the larger the f number, the smaller the diameter, and, therefore, less light gets through.

Considering that the pupil is two-dimensional, to let twice as much light through, the diameter must increase by a square root of two (which is about 1.4) and the f number will decrease by that much (i.e., it will be divided by square root of two).

Conversely, to let only half as much light through, you need to make the diameter smaller by the square root of two, and the f number increases by the same amount (is multiplied by it).

One f-stop is then the square root of two, or roughly 1.4, where each stop is the previous one multiplied by that number and allows half as much light through as the previous f-stop.

So, staring with a theoretical f-stop of 1, we go like this:

Code:
f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64, f/90, f/128, etc.

In casual talk we just skip the f/, so we just get a progression like this:

Code:
1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, 90, 128, etc.
 

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