Why do smaller F-stop numbers indicate larger apertures?

Also, why doesn't halving the f-number increase the aperture by a full stop?

The aperture is the opening at the front of your lens that allows light to pass though the lens and on to the film. The aperture can be opened or closed to control the amount of light entering the lens. The larger the opening, the more light will reach the film.

You will usually see the f-number, or f-stop, written as f/2.0, f/3.2, f/8.0 as examples.

The maximum aperture that a given lens may have is the f-number printed on the lens barrel. f/2.8 for example.

The f-stop is actually a ratio that describes the relationship between the focal length of the lens (f) and the size of the opening.

For example: a 100mm lens at f/2.0 would indicate an aperture diameter of 50mm (100 divided by 2), at f/4.0 the diameter would be 25mm, and at f/16 it would be 6.25mm.

A 600mm lens at f/2 would require an aperture diameter of 300mm! That is why you don't see long lenses with huge apertures - such a lens would be massive, and heavy, and super expensive.

A range of aperture openings

So, why is that doubling, or halving, the f-number doesn't change the exposure value by one stop (like ISO or shutter speed)? Or, why is it that the difference in exposure value between say f/4 and f/8 is two stops and not one?

Bear with me here...

The lens opening is fairly close to being a circle, to calculate the surface area of a circle (the size of the hole) we use the equation A = π r² - the length of the radius squared multiplied by the constant Pi.

The radius squared is the key bit here:

Using the above example of a 100mm lens at f/4 we have an aperture diameter of 25mm. The radius is half the diameter - 12.5mm. So, the surface area of the circle = π 12.5² or around 491mm².

The same lens at f/8 would have a surface area of only 123mm² making it one quarter the size of the hole at f/4... two full stops difference.