The brightness of the image projected by the lens is a function of its working diameter (aperture) and focal length. The focal ratio (f/#) takes both into account. Any lens functioning at the same f/# as another, delivers the same image brightness.
The camera lens operates like a funnel. The larger its working diameter, the more light it will capture. We turn to the geometry of circles to explain how this works. Before we do, consider that giant astronomical telescopes are fitted with a lens many meters in diameter. This allows them to gather light that is super feeble.
Now the camera lens is fitted with an adjustable opening we call an iris. The camera iris mimics the human eye in that the colored portion involuntary changes the diameter of the pupil to accommodate for scene brightness. By the way, the pigmented iris is named for the Greek goddess of the rainbow.
Actually, it is the surface area of the opening that does the deed. As we open up the aperture, the diameter of the entrance increases as does the surface area. The factor, diameter to surface area is 1.4. Multiply any circle diameter by 1.4 and you construct a revised diameter that doubles the surface area. Thus the number set: 1 – 1.4 – 2 – 2.8 – 4 – 5.6 – 8 – 11 – 16 – 22 – 32. Note each value going right is its neighbor multiplied by 1.4 and then rounded. This is the mysterious f/number set in full stop increments. Each full stop doubles or halves the surface area (capture area).
Now the rest of the story: While the working diameter is the key to image brightness, there is another equally influencing factor. This is focal length. The focal length reveals the power of the camera lens to magnify. Longer focal lengths yield a larger image but at a price. Each doubting of the focal length decreases image brightness by ¼. In other words, a 50mm delivers say 100 units of light; a 100mm only delivers 25 units of light. This is because the higher magnification causes the same amount of light to be spread out over 4x more surface area. The camera baffles (blocks) the larger image only presenting the central portion to film or digital chip.
Consider that the two factors of working diameter (aperture diameter) and focal length comingle to yield the final brightness factor of the lens. This is chaotic because there is a hodgepodge of different lens diameters and focal lengths. There are a zillion combinations, each likely to deliver a different image brightness. We need a system that is universal, one that takes into account the two key factors and allows us to predict image brightness regardless of focal length or aperture diameter.
Focal ratio to the rescue: We need a ratio because a ratio is dimensionless. The f/number system (focal ratio) resolves this difficulty. We divide the focal length by working diameter (aperture). This is the focal ratio. Any lens operating at the same focal ratio (f/#) as any other, regardless of focal length or working diameter, delivers the same image brightness when imaging the same scene. Different systems have been tried, but we keep returning to the focal ratio (f/#) to set our cameras.