What we are actually after is a way to compare the image brightness that a camera system will cause to play on film or image sensor. Now the lens acts like a funnel in that the larger its working diameter, the brighter the image will be. Additionally, the focal length is the projection distance from lens to film or image sensor. The longer the focal length the more the image will be magnified. The more the magnification, the dimmer will be the projected image. We need a universal way to compare one camera lens with another to get a handle on image brightness. This comparison needs to be universal. In other words, a method that works regardless of the working diameter and the focal length.
If we can find such a universal method, we can compare a tiny camera to a giant astronomical camera had make common comparisons as to image brightness.
We have such a method! Ratio to the rescue: A ratio is dimensionless. We divide the focal length by the working diameter and get a value we call the focal ratio. Such as, what is the focal ratio of a lens 400 feet in focal length with a working diameter of 40 feet? Answered 400 ÷ 40 = 10 written as f/10. We can compare this lash-up to a system that is 40 inches in focal length with a working diameter of 4 inches. Thus 40 ÷ 4 = 10 written as f/10. We can compare both to a miniature camera 40mm focal length 4mm working diameter thus 40 ÷ 10 = 10 written f/10
What I am trying to say: The focal ratio is universal, it is void of dimension. Any lens system operating at the same focal ratio delivers the same image brightness when imaging the same object illuminated the same way.
Now we abbreviate focal ratio at f/# or as f-number or aperture f-#. All this means the same, we just write the ratio a different way.
The f-number set is by tradition a change (delta) of 2X. Meaning a doubling of halving of the amount of light allowed to play on film or digital sensor. Since the lens is a circular figure, we fall back on the geometry of circles. If you multiply the diameter of any circle by the sq. root of 2 = 1.4, you calculate a revised circle with twice the surface area. If you divide the diameter by 1.4, you have calculated a revised circle with half the surface area.
Using the 1.4 factor, the f-number set is:
1 – 1.4 – 2 – 2.8 -4 -5.6 – 8 – 11 – 16 -22 – 32
This is a set with 2X change.
We can calculate sets with finer increments of change.