The camera lens acts like a funnel in that it gathers light. The larger the working aperture diameter, the more light the lens will gather. The camera lens also acts like a projection lens; it projects an image of the outside world onto the surface of film or digital chip.
The focal length of the lens determines the size of the image produced. The longer the focal length, the larger the image. Suppose you take a picture of a tree using a 50mm focal length lens, the image of the tree measures 5mm in height. Now you mount a 100mm lens, the image of the tree will now measure 10mm. Now you mount a 25mm lens, the image height is now 2 ½ mm. Thus the magnification yield of the lens is a linear function.
What you likely do not know: When the focal length is changed, the image brightness also changes. Each time we double the focal length, the image brightness falls off four fold (4X reduction). This is because the now twice bigger (2X magnified) image must play over 4X more surface area. In other words, double the focal length and the image brightness reduces two f/stops (a stop is a 2X change in light energy). This image brightness change with focal length is true should we keep the working diameter of the lens aperture unchanged. The brightening and dimming of image brightness with focal length changes is chaotic. We need to avoid this at all cost. If we don’t, under and over exposure will result.
Ratio to the rescue: We need to calculate a lens speed (brightness) value that holds regardless of focal length or working aperture diameter. Fortunately a ratio is dimensionless. The focal ratio is computed by dividing the focal length by the working diameter. Thus a 50mm lens with a aperture diameter of 25mm has a focal ratio of 50 ÷ 25 = 2. We write this value a f/2. The beauty of this system is: any lens, regardless of focal length or aperture diameter, functioning at f/2, will deliver the same image brightness. The f/number system intertwines the focal length and the working diameter and the resulting focal ratio takes the chaos away.
The f/number system is based on changing the working diameter of the lens to deliver a 2X (double of half) change in light energy on film or chip. This ratio is founded of the geometry of circles. Now the amount of light that a lens transmits changes as we change the focal length or working diameter of the aperture. Factorial: If we multiply the diameter of any circle by the square root of 2 = 1.414, we compute a revised circle with twice the surface area. For photographic purposes, we can round this value to 1.4. If we apply this value to the working diameter of lens, a number set emerges.
The number set (f/numbers):
1 – 1.4 – 2 – 2.8 – 4 – 5.6 – 8 – 11 – 16 – 22 – 32
Note: each number going right is its neighbor on the left multiplied by 1.4. Each number going left is its neighbor on the right divided by 1.4. This is linear progression of the f/number system. Each value is called a full stop. The term stops is used because the aperture diameter stops some light energy and passes some light energy. The stop is a 2X change is light brightness as it plays on film or chip.
For some applications, the full stop i.e. 2X change is too course, we can further refine the f/number system into ½ stops and 1/3 stops. The multiplier for ½ stop is 1.18 and the multiple for 1/3 stop is 1.12. This is the linear progression of the f/number system.