In a camera, the f-stop number is the ratio of the length of the lens to the diameter of the aperture. Higher f-stop numbers give you a greater depth of field. At the extreme end, a pinhole camera will give you almost total depth of field.

On your fancier cameras with zoom lenses, the hyperbolae on the barrel of the lens correspond to different f-stop settings, and are to be used in conjunction with the focus to tell you the depth-of-field of a specific f-stop at that zoom level.

At its simplest, the f-stop is the focal length divided by the diameter of the aperture. A 50mm lens (normal lens) with an aperture of 50mm would have a maximum f-stop of f/1, while a 50mm lens with an aperture of 25mm would have a maximum f/stop of f/2. Many point and shoot cameras have a fixed f/stop of f/16 - only 3 1/8mm across (this helps in computing the sunny sixteen for those who use it). Likewise, a 200mm lens with a 50mm aperture is f/4.

The larger the aperture, the more glass that is necessary in the lens itself and the more costly the lens. For comparison - a Nikon 50mm f/1.8 costs on the order of \$70, while the f/1.4 is about \$200 and the f/1.2 is over \$400 (the 50mm f/1.2 Noct is \$1,500!). I have never seen a lens with f/1.0.

update: Canon does make a 50mm/1.0 lens: It costs \$2500, and is huge, heavy and a solid cube of optical glass. The image quality is lower than a 50mm/1.4 or a 50mm/1.8... but if you absolutely must open up that extra stop for very dim light or very small depth of field this is the lens to use.
http://www.photo.net/photo/canon/canon-50-1.0
Bistromath pointed out this lens and link to me

The term 'stop' may have come from older cameras where the aperture was selected by turning a wheel with various sized holes in it. Each stop would let in twice as much light as the previous one. This can be seen today in some older twin lens reflex cameras. The phrase "open up N stops" or "stop down N stops" means to change the aperture to allow 2^N times more (or less respectively) light.

The sequence of f-stops (including minor stops) is often confusing:
1, 1.2, 1.4, 1.7, 2, 2.4, 2.8, 3.4, 4, 4.8, 5.6, 6.8, 8, 9.6, 11, 13.6, 16, 19, 22, 27, 32, 38, 45, 54, 64 and 90. (You rarely seen above f/32 on a 35mm lens)

The key to this sequence is realizing that this to the diameter of the aperture and each increment is a doubling (or half of a doubling) of the area (thus allowing twice as much light in).

Take the maximum aperture - this has a unit area of '1'. Half that amount would have a unit area of 1/2, and half that again would have a unit area of 1/4 and so on.

Working out the actual math for a 50mm lens. f/1 would have an area of: (50mm/2)2 * π = 1963.5mm2 Half this area would be 981.75mm2 and would have a diameter of 35.35mm which has a ratio of 1/1.4 times the original diameter. Thus, f/1.4 lets in half as much light as f/1. The f/1.2 has 3/4 the area 1472.6mm2 and a diameter of 43.3mm which again has a ratio to the original 50mm diameter of 1/1.2.

The quick way to find the half stop between two stops is half each stop and sum them. Between 1 and 1.4 is .5 + .7 = 1.2

The 'f' of 'f-stop' is claimed to have been created by Ansel Adams:

"...Willard (Van Dyke) remembered that he proposed "U.S.256," the old system name for f/64. He said that Ansel responded, "U.S. 256 is not good, it sounds like a highway." Willard continued, "He then took a pencil and made a a curving 'f' followed by the dot and 64. The graphics were beautiful and that was that." At first, it was written "Group f.64" in the style of the old aperture notation, but that was soon updated to the new notation with its slash, "Group f/64." To those familiar with Ansel's handwriting, the f in the Group f/64 exhibition invitation appears nearly identical to his own typical, very musical looking f."
(from Ansel Adams, A Biography by Mary Street Alinder)
You will note that the term was originally "Group f.64" which likely was shorted to "f.64". Across telegrams, this would be read as "f stop sixty four".

http://www.photo.net/bboard/q-and-a-fetch-msg?msg_id=000oVh
http://www.photo.net/photo/optics/lensFAQ
http://www.photographytips.com/page.cfm/380

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