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| [[Image:Aperture diagram.svg|right|thumb|320px|Diagram of decreasing apertures, that is, increasing f-numbers, in one-stop increments; each aperture has half the light gathering area of the previous one.]]
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| In [[optics]], the '''f-number''' (sometimes called '''focal ratio''', '''f-ratio''', '''f-stop''', or '''relative aperture'''<ref>Smith, Warren ''Modern Lens Design'' 2005 McGraw-Hill</ref>) of an optical system is the [[ratio]] of the [[photographic lens|lens's]] [[focal length]] to the diameter of the [[entrance pupil]].<ref name="ReferenceA">Smith, Warren ''Modern Optical Engineering, 4th Ed.'' 2007 McGraw-Hill Professional</ref> It is a [[dimensionless number]] that is a quantitative measure of [[lens speed]], and an important concept in [[photography]].
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| == Notation ==
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| The f-number ''N'' is given by
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| :<math>N = \frac{f}{D} \ </math>
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| where <math>f</math> is the [[focal length]], and <math>D</math> is the diameter of the entrance pupil (''effective aperture''). It is customary to write f-numbers preceded by {{f/}},<ref name="ReferenceA"/> which forms a mathematical expression of the entrance pupil diameter in terms of <span style="font-style:italic;font-family:Trebuchet MS,Candara,Georgia,Calibri,Corbel,serif">f</span> and ''N''. For example, if a lens's focal length is 10 mm and its entrance pupil diameter is 5 mm, the f-number is 2 and the aperture size would be expressed as {{f/}}2.
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| Ignoring differences in light transmission efficiency, a lens with a greater f-number projects darker images. The brightness of the projected image ([[illuminance]]) relative to the brightness of the scene in the lens's field of view ([[luminance]]) decreases with the square of the f-number. Doubling the f-number decreases the relative brightness by a factor of four. To maintain the same [[Exposure (photography)|photographic exposure]] when doubling the f-number, the [[shutter speed|exposure time]] would need to be four times as long.
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| Most lenses have an adjustable [[diaphragm (optics)|diaphragm]], which changes the size of the [[aperture stop]] and thus the entrance pupil size. The entrance pupil diameter is not necessarily equal to the aperture stop diameter, because of the magnifying effect of lens elements in front of the aperture.
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| A 100 mm focal length {{f/}}4 lens has an entrance pupil diameter of 25 mm. A 200 mm focal length {{f/}}4 lens has an entrance pupil diameter of 50 mm. The 200 mm lens's entrance pupil has four times the area of the 100 mm lens's entrance pupil, and thus collects four times as much light from each object in the lens's field of view. But compared to the 100 mm lens, the 200 mm lens projects an image of each object twice as high and twice as wide, covering four times the area, and so both lenses produce the same illuminance at the focal plane when imaging a scene of a given luminance.
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| A [[F-number#T-stop|T-stop]] is an f-number adjusted to account for light transmission efficiency. | |
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| == Stops, f-stop conventions, and exposure ==
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| [[Image:Canon 7 with 50mm f0.95 IMG 0374.JPG|thumb|A [[Canon 7]] mounted with a 50 mm lens capable of an exceptional {{f/}}0.95]]
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| [[Image:lens aperture side.jpg|thumb|A 35 mm lens set to {{f/}}11, as indicated by the white dot above the f-stop scale on the aperture ring. This lens has an aperture range of {{f/}}2.0 to {{f/}}22]]
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| The word ''stop'' is sometimes confusing due to its multiple meanings. A stop can be a physical object: an opaque part of an optical system that blocks certain rays. The ''[[aperture stop]]'' is the aperture setting that limits the brightness of the image by restricting the input pupil size, while a ''field stop'' is a stop intended to cut out light that would be outside the desired field of view and might cause flare or other problems if not stopped.
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| In photography, stops are also a ''unit'' used to quantify ratios of light or exposure, with each added stop meaning a factor of two, and each subtracted stop meaning a factor of one-half. The one-stop unit is also known as the EV ([[exposure value]]) unit. On a camera, the aperture setting is usually adjusted in discrete steps, known as '''''f-stops'''''. Each "'''stop'''" is marked with its corresponding f-number, and represents a halving of the light intensity from the previous stop. This corresponds to a decrease of the pupil and aperture diameters by a factor of 1/<math>\scriptstyle \sqrt{2}</math> or about 0.7071, and hence a halving of the area of the pupil.
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| Modern lenses use a standard f-stop scale, which is an approximately [[geometric sequence]] of numbers that corresponds to the sequence of the [[exponentiation|powers]] of the [[square root of 2]]: {{f/}}1, {{f/}}1.4, {{f/}}2, {{f/}}2.8, {{f/}}4, {{f/}}5.6, {{f/}}8, {{f/}}11, {{f/}}16, {{f/}}22, {{f/}}32, {{f/}}45, {{f/}}64, {{f/}}90, {{f/}}128, etc. Each element in the sequence is one stop lower than the element to its left, and one stop higher than the element to its right. The values of the ratios are rounded off to these particular conventional numbers, to make them easier to remember and write down.
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| The sequence above is obtained by approximating the following exact geometric sequence:
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| : {{f/}}1 = <math>\frac{f/1}{(\sqrt{2})^0} </math>, {{f/}}1.4 = <math>\frac{f/1}{(\sqrt{2})^1} </math>, {{f/}}2 = <math> \frac{f/1}{(\sqrt{2})^2} </math>, {{f/}}2.8 = <math> \frac{f/1}{(\sqrt{2})^3} </math> ...
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| In the same way as one f-stop corresponds to a factor of two in light intensity, [[shutter speed]]s are arranged so that each setting differs in duration by a factor of approximately two from its neighbour. Opening up a lens by one stop allows twice as much light to fall on the film in a given period of time. Therefore to have the same exposure at this larger aperture as at the previous aperture, the shutter would be opened for half as long (i.e., twice the speed). The film will respond equally to these equal amounts of light, since it has the property of ''[[Reciprocity (photography)|reciprocity]]''. This is less true for extremely long or short exposures, where we have [[reciprocity failure]]. Aperture, shutter speed, and film sensitivity are linked: for constant scene brightness, doubling the aperture area (one stop), halving the shutter speed (doubling the time open), or using a film twice as sensitive, has the same effect on the exposed image. For all practical purposes extreme accuracy is not required (mechanical shutter speeds were notoriously inaccurate as wear and lubrication varied, with no effect on exposure). It is not significant that aperture areas and shutter speeds do not vary by a factor of precisely two.
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| Photographers sometimes express other [[Exposure (photography)|exposure]] ratios in terms of 'stops'. Ignoring the f-number markings, the f-stops make a logarithmic scale of exposure intensity. Given this interpretation, one can then think of taking a half-step along this scale, to make an exposure difference of "half a stop".
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| === Fractional stops ===
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| Most old cameras had a continuously variable aperture scale, with each full stop marked. Click-stopped aperture came into common use in the 1960s; the aperture scale usually had a click stop at every whole and half stop.
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| On modern cameras, especially when aperture is set on the camera body, f-number is often divided more finely than steps of one stop. Steps of one-third stop (1/3 EV) are the most common, since this matches the ISO system of [[film speed]]s. Half-stop steps are used on some cameras. Usually the full stops are marked, and the intermediate positions are clicked. As an example, the aperture that is one-third stop smaller than {{f/}}2.8 is {{f/}}3.2, two-thirds smaller is {{f/}}3.5, and one whole stop smaller is {{f/}}4. The next few f-stops in this sequence are
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| :{{f/}}4.5, {{f/}}5, {{f/}}5.6, {{f/}}6.3, {{f/}}7.1, {{f/}}8, etc.
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| To calculate the steps in a full stop (1 EV) one could use
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| : 2<sup>0×0.5</sup>, 2<sup>1×0.5</sup>, 2<sup>2×0.5</sup>, 2<sup>3×0.5</sup>, 2<sup>4×0.5</sup> etc.
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| The steps in a half stop (1/2 EV) series would be
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| : 2<sup>0/2×0.5</sup>, 2<sup>1/2×0.5</sup>, 2<sup>2/2×0.5</sup>, 2<sup>3/2×0.5</sup>, 2<sup>4/2×0.5</sup> etc.
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| The steps in a third stop (1/3 EV) series would be
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| : 2<sup>0/3×0.5</sup>, 2<sup>1/3×0.5</sup>, 2<sup>2/3×0.5</sup>, 2<sup>3/3×0.5</sup>, 2<sup>4/3×0.5</sup> etc.
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| As in the earlier DIN and ASA film-speed standards, the ISO speed is defined only in one-third stop increments, and shutter speeds of digital cameras are commonly on the same scale in reciprocal seconds. A portion of the ISO range is the sequence
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| : ... 16/13°, 20/14°, 25/15°, 32/16°, 40/17°, 50/18°, 64/19°, 80/20°, 100/21°, 125/22°...
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| while shutter speeds in reciprocal seconds have a few conventional differences in their numbers (1/15, 1/30, and 1/60 second instead of 1/16, 1/32, and 1/64).
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| In practice the maximum aperture of a lens is often not an [[integer|integral]] power of <math>\scriptstyle \sqrt{2}</math> (i.e., <math>\scriptstyle \sqrt{2}</math> to the power of a whole number), in which case it is usually a half or third stop above or below an integral power of <math>\scriptstyle \sqrt{2}</math>.
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| Modern electronically-controlled interchangeable lenses, such as those used for SLR cameras, have f-stops specified internally in 1/8-stop increments, so the cameras' 1/3-stop settings are approximated by the nearest 1/8-stop setting in the lens.
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| ==== Standard full-stop f-number scale ====
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| Including [[APEX system|aperture value]] AV:
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| {{f/}}No. = <math>\sqrt{(2^{AV})}</math>
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| {|class="wikitable" style="text-align:center"
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| !AV
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| | −1 || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
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| |-
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| |-bgcolor="#CCFFCD"
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| !{{f/|No.}}
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| | 0.7 || 1.0 || 1.4 || 2 || 2.8 || 4 || 5.6 || 8 || 11 || 16 || 22 || 32 || 45 || 64 || 90 || 128 || 180 || 256
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| |}
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| ==== Typical one-half-stop f-number scale ====
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| {|class="wikitable" style="text-align:center"
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| |-bgcolor="#FFFFCC"
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| !{{f/|No.}}
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| |style="background:#CCFFCC;"| 0.70 || 0.84 ||style="background:#CCFFCC;"| 1.0 || 1.2 ||style="background:#CCFFCC;"| 1.4 || 1.7 ||style="background:#CCFFCC;"| 2 || 2.4 ||style="background:#CCFFCC;"| 2.8 || 3.3 ||style="background:#CCFFCC;"| 4 || 4.8 ||style="background:#CCFFCC;"| 5.6 || 6.7 ||style="background:#CCFFCC;"| 8 || 9.5 ||style="background:#CCFFCC;"| 11 || 13 ||style="background:#CCFFCC;"| 16 || 19 ||style="background:#CCFFCC;"| 22 || 27 ||style="background:#CCFFCC;"| 32
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| |}
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| ==== Typical one-third-stop f-number scale ====
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| {|class="wikitable" style="text-align:center"
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| |-bgcolor="#e5d1cb"
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| !{{f/|No.}}
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| |style="background:#CCFFCC;"| 0.7 || 0.8 || 0.9 ||style="background:#CCFFCC;"| 1.0 || 1.1 || 1.2 ||style="background:#CCFFCC;"| 1.4 || 1.6 || 1.8 ||style="background:#CCFFCC;"| 2 || 2.2 || 2.5 ||style="background:#CCFFCC;"| 2.8 || 3.2 || 3.5 ||style="background:#CCFFCC;"| 4 || 4.5 || 5.0 ||style="background:#CCFFCC;"| 5.6 || 6.3 || 7.1 ||style="background:#CCFFCC;"| 8 || 9 || 10 || style="background:#CCFFCC;"|11 || 13 || 14 ||style="background:#CCFFCC;"| 16 || 18 || 20 ||style="background:#CCFFCC;"| 22
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| |}
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| ====Typical one-quarter-stop f-number scale====
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| {|class="wikitable" style="text-align:center"
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| |-bgcolor="#5D8AA8"
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| !{{f/|No.}}
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| |style="background:#CCFFCC;"| 1.0 || 1.1 || 1.2 || 1.3 ||style="background:#CCFFCC;"| 1.4 || 1.5 || 1.7 || 1.8 ||style="background:#CCFFCC;"| 2 || 2.2 || 2.4 || 2.6 ||style="background:#CCFFCC;"| 2.8 || 3.1 || 3.4 || 3.7 ||style="background:#CCFFCC;"| 4 || 4.4 || 4.8 || 5.2 ||style="background:#CCFFCC;"| 5.6 || 6.2 || 6.7 || 7.3 ||style="background:#CCFFCC;"| 8 || 8.7 || 9.5 || 10 || style="background:#CCFFCC;"|11 || 12 || 14 || 15 ||style="background:#CCFFCC;"| 16 || 17 || 19 || 21 ||style="background:#CCFFCC;"| 22
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| |}
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| Sometimes the same number is included on several scales; for example, {{f/}}1.2 may be used in either a half-stop<ref>
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| {{cite book
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| | url = http://books.google.com/?id=YjAzP4i1oFcC&pg=PA136&lpg=PA136
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| | title = Set lighting technician's handbook: film lighting equipment, practice, and electrical distribution
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| | author = Harry C. Box
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| | edition = 3rd
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| | publisher = Focal Press
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| | year = 2003
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| | isbn = 978-0-240-80495-8
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| }}</ref>
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| or a one-third-stop system;<ref>
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| {{cite book
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| | url = http://books.google.com/?id=DvYMl-s1_9YC&pg=PA19
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| | title = Underwater photography
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| | author = Paul Kay
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| | publisher = Guild of Master Craftsman
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| | year = 2003
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| | isbn = 978-1-86108-322-7
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| }}</ref>
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| sometimes {{f/}}1.3 and {{f/}}3.2 and other differences are used for the one-third stop scale.<ref>
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| {{cite book
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| | url = http://books.google.com/?id=IWkpoJKM_ucC&pg=PA145&lpg=PA145
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| | title = Manual for cinematographers
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| | author = David W. Samuelson
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| | edition = 2nd
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| | publisher = Focal Press
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| | year = 1998
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| | isbn = 978-0-240-51480-2
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| }}</ref>
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| === T-stop ===
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| A '''T-stop''' (for Transmission-stops) is an f-number adjusted to account for light transmission efficiency (''[[transmittance]]''). A lens with a T-stop of ''N'' projects an image of the same brightness as an ideal lens with 100% transmittance and an f-number of ''N''. For example, an {{f/}}2.0 lens with transmittance of 75% has a T-stop of 2.3. Since real lenses have transmittances of less than 100%, a lens's T-stop is always greater than its f-number.<ref>[http://www.dxomark.com/index.php/About/In-depth-measurements/Measurements/Light-transmission Light transmission], DxOMark</ref>
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| Lens transmittances of 60%–90% are typical,<ref>[http://forums.dpreview.com/forums/post/33785655 Marianne Oelund, "Lens T-stops", dpreview.com, 2009]</ref> so T-stops are sometimes used instead of f-numbers to more accurately determine exposure, particularly when using external [[light meter]]s.<ref>[[Eastman Kodak]], [http://www.kodak.com/US/en/motion/support/h2/intro01P.shtml "H2: Kodak Motion Picture Camera Films"], November 2000 revision. Retrieved 2 September 2007.</ref> T-stops are often used in cinematography, where many images are seen in rapid succession and even small changes in exposure will be noticeable. Cinema camera lenses are typically calibrated in T-stops instead of F-numbers. In still photography, without the need for rigorous consistency of all lenses and cameras used, slight differences in exposure are less important.
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| === Sunny 16 rule ===
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| An example of the use of f-numbers in photography is the ''[[sunny 16 rule]]'': an approximately correct exposure will be obtained on a sunny day by using an aperture of {{f/}}16 and the shutter speed closest to the reciprocal of the ISO speed of the film; for example, using ISO 200 film, an aperture of {{f/}}16 and a shutter speed of 1/200 second. The f-number may then be adjusted downwards for situations with lower light. Selecting a lower f-number is "opening up" the lens. Selecting a higher f-number is "closing" or "stopping down" the lens.
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| == Effects on image sharpness ==
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| [[Image:Jonquil flowers merged.jpg|thumb|400px|Comparison of {{f/}}32 (top-left corner) and {{f/}}5 (bottom-right corner)]]
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| [[File:Blumen im Sommer.jpg|thumb|400px|Shallow focus with a wide open lens]]
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| [[Depth of field]] increases with f-number, as illustrated in the image here. This means that photographs taken with a low f-number will tend to have subjects at one distance in focus, with the rest of the image (nearer and farther elements) out of focus. This is frequently used for [[nature photography]] and [[portrait photography|portraiture]] because background blur ([[bokeh]]) can be aesthetically pleasing and puts the viewer's focus on the main subject in the foreground. The [[depth of field]] of an image produced at a given f-number is dependent on other parameters as well, including the [[focal length]], the subject distance, and the [[film format|format]] of the film or sensor used to capture the image. Depth of field can be described as depending on just angle of view, subject distance, and [[entrance pupil]] diameter (as in [[Moritz von Rohr|von Rohr's method]]). As a result, smaller formats will have a deeper field than larger formats at the same f-number for the same distance of focus and same [[angle of view]] since a smaller format requires a shorter focal length (wider angle lens) to produce the same angle of view, and depth of field increases with shorter focal lengths. Therefore, reduced–depth-of-field effects will require smaller f-numbers (and thus larger apertures and so potentially more complex optics) when using small-format cameras than when using larger-format cameras.
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| Image sharpness is related to f/number through two different optical effects, [[aberration]], due to imperfect lens design, and [[diffraction]] which is due to the wave nature of light.<ref>{{cite book | title = Basic Photography | author = Michael John Langford | isbn = 0-240-51592-7 | year = 2000 | publisher = [[Focal Press]]}}</ref> The blur optimal f-stop varies with the lens design. For modern standard lenses having 6 or 7 elements, the [[sharpness (visual)|sharpest]] image is often obtained around {{f/}}5.6–{{f/}}8, while for older standard lenses having only 4 elements ([[Zeiss Tessar|Tessar formula]]) stopping to {{f/}}11 will give the sharpest image. The larger number of elements in modern lenses allow the designer to compensate for aberrations, allowing the lens to give better pictures at lower f-numbers. Even if aberration is minimized by using the best lenses, [[diffraction]] creates some spreading of the rays causing defocus. To offset that use the largest lens opening diameter possible (not the f/ number itself).
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| Light falloff is also sensitive to f-stop. Many wide-angle lenses will show a significant light falloff ([[vignetting]]) at the edges for large apertures.
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| [[Photojournalist]]s have a saying, "{{f/}}8 and be there", meaning that being on the scene is more important than worrying about technical details. Practically, {{f/}}8 allows adequate depth of field and sufficient lens speed for a decent base exposure in most daylight situations.<ref>{{cite book|last=Levy|first=Michael|title=Selecting and Using Classic Cameras: A User's Guide to Evaluating Features, Condition & Usability of Classic Cameras|publisher=Amherst Media, Inc|year=2001|page=163|isbn=978-1-58428-054-5}}</ref>
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| == Human eye ==
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| Computing the f-number of the [[human eye]] involves computing the physical aperture and focal length of the eye. The pupil can be as large as 6–7 mm wide open, which translates into the maximum physical aperture.
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| The f-number of the human eye varies from about {{f/}}8.3 in a very brightly lit place to about {{f/}}2.1 in the dark.<ref>{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd|publisher=[[Addison Wesley]]|isbn=0-201-11609-X}} Sect. 5.7.1</ref> The presented maximum f-number has been questioned,<ref>[http://www.clarkvision.com/imagedetail/eye-resolution.html Clarkvision Photography – Resolution of the Human Eye]</ref> as it seems to only match the focal length that assumes ''outgoing'' light rays.{{clarify|date=December 2010}} According to the ''incoming'' rays of light (what we actually see), the focal length of the eye is a bit longer, resulting in minimum f-number of {{f/}}3.2.
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| Note that computing the focal length requires that the light-refracting properties of the liquids in the eye are taken into account. Treating the eye as an ordinary air-filled camera and lens results in a different focal length, thus yielding an incorrect f-number.
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| Toxic substances and [[poison]]s (like [[atropine]]) can significantly reduce the range of aperture. Pharmaceutical products such as eye drops may also cause similar side-effects. [[Tropicamide]] and [[phenylephrine]] are used in medicine as mydriatics to dilate pupils for retinal and lens examination. These medications take effect in about 30–45 mins after instillation and last for about 8 hours. Atropine is also used in such a way but its effects can last up to 2 weeks, along with the mydriatic effect; it produces cycloplegia (a condition in which the crystalline lens of the eye cannot accommodate to focus near objects). This effect goes away after 8 hours. Other medications offer the contrary effect. Pilocarpine is a miotic (induces miosis); it can make a pupil as small as 1 mm in diameter depending on the person and their ocular characteristics. Such drops are used in certain [[glaucoma]] patients to prevent acute glaucoma attacks.
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| == Focal ratio in telescopes ==
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| [[Image:Focal ratio.svg|right|thumb|250px|Diagram of the [[focal ratio]] of a simple optical system where <math>f</math> is the [[focal length]] and <math>D</math> is the diameter of the [[Objective (optics)|objective]]]]
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| In astronomy, the f-number is commonly referred to as the ''focal ratio'' (or ''f-ratio'') notated as <math>N</math>. It is still defined as the [[focal length]] <math>f</math> of an [[Objective (optics)|objective]] divided by its diameter <math>D</math> or by the diameter of an [[aperture]] stop in the system.
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| <math>N = \frac fD \quad \xrightarrow {\times D} \quad f = ND</math> | |
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| Even though the principles of focal ratio are always the same, the application to which the principle is put can differ. In [[photography]] the focal ratio varies the focal-plane illuminance (or optical power per unit area in the image) and is used to control variables such as [[depth of field]]. When using an [[optical telescope]] in astronomy, there is no depth of field issue, and the brightness of stellar point sources in terms of total optical power (not divided by area) is a function of absolute aperture area only, independent of focal length. The focal length controls the [[Field of view#Astronomy|field of view]] of the instrument and the scale of the image that is presented at the focal plane to an [[eyepiece]], film plate, or [[Charge-coupled device|CCD]].
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| For example, the [[Southern Astrophysical Research Telescope|SOAR]] 4m telescope has a small field of view (~f/16) which is useful for stellar studies. The [[LSST]] 8.4m telescope, which will cover the entire sky every 3 days has a very large field of view. Its short 10.3 meter focal length (f/1.2) is made possible by an error correction system which includes secondary and tertiary mirrors, a three element refractive system and active mounting and optics.<ref name=RefDesign>{{Cite journal |title=LSST Reference Design |publisher=LSST Corporation |author1=Charles F. Claver |author2=et al. |date=19 March 2007 |url=http://lsst.org/files/docs/LSST-RefDesign.pdf |pages=45–50 |accessdate=10 January 2011 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| == Working f-number ==<!-- This section is linked from [[Numerical aperture]] -->
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| The f-number accurately describes the light-gathering ability of a lens only for objects an infinite distance away.<ref name="Greivenkamp">{{cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | isbn=0-8194-5294-7 }} p. 29.</ref> This limitation is typically ignored in photography, where objects are usually not extremely close to the camera, relative to the distance between the lens and the film. In [[optical design]], an alternative is often needed for systems where the object is not far from the lens. In these cases the '''working f-number''' is used. A practical example of this is, that when focusing closer, the lens' effective aperture becomes smaller, from e.g. f/22 to f/45, thus affecting the exposure.
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| The working f-number ''N<sub>w</sub>'' is given by
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| :<math>N_w \equiv {1 \over 2 \mathrm{NA}_i} \approx (1+|m|)N</math> ,
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| where ''N'' is the uncorrected f-number, NA<sub>''i''</sub> is the image-space [[numerical aperture]] of the lens, and <math>|m|</math> is the [[absolute value]] of lens's [[magnification]] for an object a particular distance away.<ref name="Greivenkamp"/> In photography, the working f-number is described as the f-number corrected for lens extensions by a [[bellows factor]]. This is of particular importance in [[macro photography]].
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| == History ==
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| The system of f-numbers for specifying relative apertures evolved in the late nineteenth century, in competition with several other systems of aperture notation.
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| === Origins of relative aperture ===
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| In 1867, Sutton and Dawson defined "apertal ratio" as essentially the reciprocal of the modern f-number:<ref name="Sutton">Thomas Sutton and George Dawson, ''A Dictionary of Photography'', London: Sampson Low, Son & Marston, 1867, (p. 122).</ref>
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| <blockquote>In every lens there is, corresponding to a given apertal ratio (that is, the ratio of the diameter of the stop to the focal length), a certain distance of a near object from it, between which and infinity all objects are in equally good focus. For instance, in a single view lens of 6 inch focus, with a 1/4 in. stop (apertal ratio one-twenty-fourth), all objects situated at distances lying between 20 feet from the lens and an infinite distance from it (a fixed star, for instance) are in equally good focus. Twenty feet is therefore called the 'focal range' of the lens when this stop is used. The focal range is consequently the distance of the nearest object, which will be in good focus when the ground glass is adjusted for an extremely distant object. In the same lens, the focal range will depend upon the size of the diaphragm used, while in different lenses having the same apertal ratio the focal ranges will be greater as the focal length of the lens is increased. The terms 'apertal ratio' and 'focal range' have not come into general use, but it is very desirable that they should, in order to prevent ambiguity and circumlocution when treating of the properties of photographic lenses.</blockquote>
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| In 1874, [[John Henry Dallmeyer]] called the ratio <math>1/N</math> the "intensity ratio" of a lens:<ref name="Dallmeyer">John Henry Dallmeyer, ''Photographic Lenses: On Their Choice and Use – Special Edition Edited for American Photographers'', pamphlet, 1874.</ref>
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| <blockquote>The ''rapidity'' of a lens depends upon the relation or ratio of the aperture to the equivalent focus. To ascertain this, divide the ''equivalent focus'' by the diameter of the actual ''working aperture'' of the lens in question; and note down the quotient as the denominator with 1, or unity, for the numerator. Thus to find the ratio of a lens of 2 inches diameter and 6 inches focus, divide the focus by the aperture, or 6 divided by 2 equals 3; i.e., 1/3 is the intensity ratio.</blockquote>
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| Although he did not yet have access to [[Ernst Abbe]]'s theory of stops and pupils,<ref>{{Cite journal | url = http://books.google.com/?id=-r6LPy-nWPwC&pg=RA3-PA537&dq=theory-of-stops | title = The principles and methods of geometrical optics: Especially as applied to the theory of optical instruments | author1 = Southall | first1 = James Powell Cocke | year = 1910}}</ref> which was made widely available by [[Siegfried Czapski]] in 1893,<ref name="Czapski">Siegfried Czapski, ''Theorie der optischen Instrumente, nach Abbe,'' Breslau: Trewendt, 1893.</ref> Dallmeyer knew that his ''working aperture'' was not the same as the physical diameter of the aperture stop:<ref name="Dallmeyer"/>
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| <blockquote>It must be observed, however, that in order to find the real ''intensity ratio'', the diameter of the actual working aperture must be ascertained. This is easily accomplished in the case of single lenses, or for double combination lenses used with the full opening, these merely requiring the application of a pair of compasses or rule; but when double or triple-combination lenses are used, with stops inserted ''between'' the combinations, it is somewhat more troublesome; for it is obvious that in this case the diameter of the stop employed is not the measure of the actual pencil of light transmitted by the front combination. To ascertain this, focus for a distant object, remove the focusing screen and replace it by the collodion slide, having previously inserted a piece of cardboard in place of the prepared plate. Make a small round hole in the centre of the cardboard with a piercer, and now remove to a darkened room; apply a candle close to the hole, and observe the illuminated patch visible upon the front combination; the diameter of this circle, carefully measured, is the actual working aperture of the lens in question for the particular stop employed.</blockquote>
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| This point is further emphasized by Czapski in 1893.<ref name="Czapski"/> According to an English review of his book, in 1894, "The necessity of clearly distinguishing between effective aperture and diameter of physical stop is strongly insisted upon."<ref>Henry Crew, "Theory of Optical Instruments by Dr. Czapski," in ''Astronomy and Astro-physics'' XIII pp. 241–243, 1894.</ref>
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| J. H. Dallmeyer's son, [[Thomas Rudolphus Dallmeyer]], inventor of the telephoto lens, followed the ''intensity ratio'' terminology in 1899.<ref>Thomas R. Dallmeyer, ''Telephotography: An elementary treatise on the construction and application of the telephotographic lens'', London: Heinemann, 1899.</ref>
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| === Aperture numbering systems ===
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| [[File:No1-A Autographic Kodak Jr.jpg | thumb | right | A 1922 Kodak with aperture marked in U.S. stops. An f-number conversion chart has been added by the user.]]
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| At the same time, there were a number of aperture numbering systems designed with the goal of making exposure times vary in direct or inverse proportion with the aperture, rather than with the square of the f-number or inverse square of the apertal ratio or intensity ratio. But these systems all involved some arbitrary constant, as opposed to the simple ratio of focal length and diameter.
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| For example, the ''Uniform System'' (U.S.) of apertures was adopted as a standard by the [[Royal Photographic Society|Photographic Society of Great Britain]] in the 1880s. Bothamley in 1891 said "The stops of all the best makers are now arranged according to this system."<ref>C. H. Bothamley, ''Ilford Manual of Photography'', London: Britannia Works Co. Ltd., 1891.</ref> U.S. 16 is the same aperture as {{f/}}16, but apertures that are larger or smaller by a full stop use doubling or halving of the U.S. number, for example {{f/}}11 is U.S. 8 and {{f/}}8 is U.S. 4. The exposure time required is directly proportional to the U.S. number. [[Eastman Kodak]] used U.S. stops on many of their cameras at least in the 1920s.
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| By 1895, Hodges contradicts Bothamley, saying that the f-number system has taken over: "This is called the f/x system, and the diaphragms of all modern lenses of good construction are so marked."<ref>John A. Hodges, ''Photographic Lenses: How to Choose, and How to Use'', Bradford: Percy Lund & Co., 1895.</ref>
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| Here is the situation as seen in 1899:
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| [[File:Diaphragm Numbers.gif|center|823 px]]
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| Piper in 1901<ref>C. Welborne Piper, ''A First Book of the Lens: An Elementary Treatise on the Action and Use of the Photographic Lens'', London: Hazell, Watson, and Viney, Ltd., 1901.</ref> discusses five different systems of aperture marking: the old and new [[Carl Zeiss|Zeiss]] systems based on actual intensity (proportional to reciprocal square of the f-number); and the U.S., C.I., and Dallmeyer systems based on exposure (proportional to square of the f-number). He calls the f-number the "ratio number," "aperture ratio number," and "ratio aperture." He calls expressions like {{f/}}8 the "fractional diameter" of the aperture, even though it is literally equal to the "absolute diameter" which he distinguishes as a different term. He also sometimes uses expressions like "an aperture of f 8" without the division indicated by the slash.
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| Beck and Andrews in 1902 talk about the Royal Photographic Society standard of {{f/}}4, {{f/}}5.6, {{f/}}8, {{f/}}11.3, etc.<ref>Conrad Beck and Herbert Andrews, ''Photographic Lenses: A Simple Treatise'', second edition, London: R. & J. Beck Ltd., c. 1902.</ref> The R.P.S. had changed their name and moved off of the U.S. system some time between 1895 and 1902.
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| === Typographical standardization ===
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| By 1920, the term ''f-number'' appeared in books both as ''F number'' and ''f/number''. In modern publications, the forms ''f-number'' and ''f number'' are more common, though the earlier forms, as well as ''F-number'' are still found in a few books; not uncommonly, the initial lower-case ''f'' in ''f-number'' or ''f/number'' is set in a hooked italic form: <span style="font-style:italic;font-family:Trebuchet MS,Candara,Calibri,Corbel,serif">f</span>, or <span style="font-style:italic;font-family:Georgia,serif">f</span>.<ref>[http://books.google.com/books?as_q=lens+aperture&num=50&as_epq=f-number Google search]</ref> Notations for f-numbers were also quite variable in the early part of the twentieth century. They were sometimes written with a capital F,<ref>{{cite book| url=http://books.google.com/?id=ypakouuKvwYC&pg=RA2-PA61| format=Google| accessdate=12 March 2007| title=Airplane Photography| last=Ives| first=Herbert Eugene| publisher=J. B. Lippincott| location= Philadelphia| year=1920| pages=61}}</ref> sometimes with a dot (period) instead of a slash,<ref>{{cite book| url=http://books.google.com/?id=V7MCVGREPfkC&q=aperture+lens+uniform-system+date:0-1930| accessdate=12 March 2007| title=The Fundamentals of Photography| first=Charles Edward Kenneth| last=Mees| publisher=Eastman Kodak| year=1920| pages=28}}</ref> and sometimes set as a vertical fraction.<ref>{{cite book| url=http://books.google.com/?id=AN6d4zTjquwC&pg=PA83| format=Google| accessdate=12 March 2007| title=Photography for Students of Physics and Chemistry| first=Louis| last=Derr| location=London| publisher=Macmillan| year=1906| pages=83}}</ref>
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| The 1961 [[American National Standards Institute|ASA]] standard PH2.12-1961 ''American Standard General-Purpose Photographic Exposure Meters (Photoelectric Type)'' specifies that "The symbol for relative apertures shall be <span style="font-style:italic;font-family:Georgia,serif">f/</span> or <span style="font-style:italic;font-family:Georgia,serif">f </span>: followed by the effective <span style="font-style:italic;font-family:Georgia,serif">f</span>-number." Note that they show the hooked italic <span style="font-style:italic;font-family:Georgia,serif">f </span> not only in the symbol, but also in the term ''f-number'', which today is more commonly set in an ordinary non-italic face.
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| == See also ==
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| {{Portal|Physics|Film}}
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| * [[Circle of confusion]]
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| * [[Group f/64]]
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| * [[Photographic lens design]]
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| * [[Pinhole camera]]
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| * [[movie camera]]
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| * [[Photography]]
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| == References ==
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| {{Reflist|2}}
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| == External links ==
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| {{Commons category|F-number}}
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| * [http://fcalc.net/manual/fnumbers.html f Number arithmetic]
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| * [http://www.largeformatphotography.info/fstop.html Large format photography—how to select the f-stop]
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| {{photography subject}}
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| {{Use dmy dates|date=December 2010}}
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| {{DEFAULTSORT:F-Number}}
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| [[Category:Optics]]
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| [[Category:Science of photography]]
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| [[Category:Dimensionless numbers]]
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