All About f-Stops, What They Are And What They Do – OTD’s explanation is the simplest I’ve come across yet:
The f: in the term stands for fraction. A f:2 setting would have a diameter 1/2 the focal length of the lens, a f:8 setting would have a diameter 1/8 the focal length of the lens. Thus in a 50mm lens the f:2 opening would be 25mm in diameter and at f:8 the opening would be 6.25mm in diameter.
Graystar adds the concept of geometry to the explanation, along with a diagram.
Graystar also explains what doubling or halving the amount of light (equivalent to a full f-stop) means:
Take a 50mm lens. At f/2 we have a 25mm diameter aperture. At f/2.8 we have a 17.86mm diameter aperture. If you were to calculate the area of these two diameters you’ll that one is roughly double or halve of the other.
So for a 50mm lens we have…
f/1 = 1963.5 mm^2
f/1.4 = 1001.8 mm^2
f/2 = 490.8 mm^2
f/2.8 = 250.5 mm^2
f/4 = 122.7 mm^2
and so on.The F-Numbers have been rounded to make them easier to manage.
dradam simplifies it further: “The standard fstop scale runs: 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22″, and provides the mathematical basis of the series:
Yes, this is a mathematical series. If you want it really spelled out it is 2^(n/2) where n starts at 0. So yes, each stop is 1.414 (or square root 2) times higher than the previous. And yes, if we square the whole thing we get 2^n, the series you list above.
If you dont feel like multiplying by 1.414 you can just try and remember this trick (it’s always what I’ve done). Just remember 1 and 1.4, every subsequent number is double the number two before it.
1, 1.4, 2 (twice 1), 2.8 (twice 1.4), 4 (twice 2), and so on.
One more by dradam, on how Pi cancels out in the formula F# = Focal length/diameter:
Given a focal length, going up and down one stop goes as doubling or halving the inverse of the area which follows from multiplying or dividing by the inverse of the inverse of the squre root of 2 (A=pi*r^2 therefore r = (A/pi)^(1/2)). Pi is simply a constant and as A = 2A or A/2 you end up with a series of the powers of root two. This is where you get 1, 1.4, 2, 2.8 etc.