According to my old university optics course the f/number is equal to f/D' where D is the diameter of the entrance pupil of the lens and f is the focal length.
It goes on with an example, a 50mm lens with a 25mm aperture has an f/number of 2.0.
Then it goes into photographic lenses, stating that one f-stop in those is equal to a step equal to the square root of 2 (rounded), yielding the set we know of 1.0, 1.4, 1.8, 2, 2.8, 4, 5.6, 8, and so on.
Thus closing down 1 f-stop decreases the lens opening by 1 divided by the square root of 2 and the total light reaching the focal point of the lens by a factor of 2 (thus halving it). (it assumes you know that the area of a circle is defined as pi * the square of half the diameter of that circle, therefore the square root in the formula).
Hope this explains the mathematics behind it somewhat.
I wish I were flying