[Soundy]

http://blog.buitek.com/media/diag1.gif

As shown in the diagram above, the F number represents the amount of times the width of the apeture can be spanned along the focal length of the lens.
So say you have a F/4 apeture and a 50mm lens. Then the width of the apature would be 50 / 4 = 12.5mm. Using the formula of a circles area you can then do a mathematical proof that a shift of one stop is in fact double the area.

Really nicely done. The only thing I'd add is something I've seen a good deal of confusion over in rec.photo.digital: f/4 and f/8 are NOT one stop apart, but two. f/4 is FOUR times the opening of f/8 - twice the diameter must be applied in both X and Y dimensions.

A better way to visualize it is as squares instead of circles: a square a 4x4 is 16 square units, but you can't simply double the 16 and get 32 square units, because that would give a rectangle; both dimensions must be increased equally, meaning 8x8=64, or four times the area of the 4x4.

To double the area of a cicrle, you must multiply the radius or diameter by the square root of two, since you're squaring the radius to get the area (a=(pi)r^2).