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The number for lens f/stop in photography (for example, f/8) is the ratio of lens focal length divided by the effective lens aperture. Aperture is not the obvious physical diameter (shown by black vertical lines below), but instead is the apparent "working" diameter as seen through the magnification of the front lens element.
That is math and physics, but still, the very useful purpose of f/stop numbers is the grand concept that f/8 will the same exposure in any lens, regardless of focal length, or physical size of construction.
And while we are momentarily distracted, the marked Focal Length is when focused at infinity, but it changes as we focus closer (focal length normally becomes longer if front element is extended, but internal focus lenses probably become shorter). The actual focal length is measured to the rear Principle Point, H', as shown above. The Principle Point is the designer's apparent plane where the image appears to be. Design of curved lens elements can move this point, and this H' point is in fact often literally outside the actual lens, either in front or behind. In telephoto lenses, this H' point is always slightly in front of the front lens element, because, the actual optical technical definition of "telephoto" is that the lens is physically shorter than its focal length (which is a practical way to build the long lenses that show distant objects enlarged). Wide angle lenses are often retrofocus, which means the rear node H' is well behind the rear element. This allows the short lens to be mounted well forward, leaving space for the SLR camera mirror to be raised. Otherwise for example, an 18 mm lens would block raising a mirror 24 mm tall. FWIW, regarding this H' distance, the ratio of subject size to image size is called Magnification, and when equal sizes (at 1:1 reproduction ratio), then also equal distances  the distance in front of the lens is necessarily equal to the distance behind the lens, at 1:1 size (similar triangles, etc.) Seems a cute fact, which aids understanding.
Is f/stop written f/stop or fstop? The lens manufacturers properly write f/8. But the term fstop has become popular online, so we see both. Maybe it is just me, but my habit is still to write f/stop, because we write f/8, to be remindful of the division defining it:
f/stop number = focal length / aperture diameter
At the same f/8, a lens with 3x longer focal length has an aperture diameter 3x larger. So, f/8 denotes (focal length / 8), which represents the aperture of the lens, and this exposure value can be compared with other lenses in this way. A series of multiple f/stop steps is designed, called "stops". Stop originally denoted the notched detent which marked the 2x area multiples. Today in photography, the word stop is used to mean any step of double or half value of exposure, also in regard to shutter speed and ISO. Each full stop towards larger f/stop numbers gives half the light exposure of the previous step (called stopping down, which also increases depth of field).
The tables below are the computed f/stop numbers of the camera aperture. The first table is the fractional steps in tenth stops. These charts show the actual numbers, and the relationships, and one purpose could be to aid determining span in stops between two values.
Stop Number  f/stop  +0.1  +0.2  +0.3  +0.4  +0.5  +0.6  +0.7  +0.8  +0.9 
0  1.0  1.04  1.07  1.11  1.15  1.19  1.23  1.27  1.32  1.37 
1  1.41  1.46  1.52  1.57  1.62  1.68  1.74  1.80  1.87  1.93 
2  2.0  2.07  2.14  2.22  2.30  2.38  2.46  2.55  2.64  2.73 
3  2.83  2.93  3.03  3.14  3.25  3.36  3.48  3.61  3.73  3.86 
4  4.0  4.14  4.29  4.44  4.59  4.76  4.92  5.10  5.28  5.46 
5  5.66  5.86  6.06  6.28  6.50  6.73  6.96  7.21  7.46  7.73 
6  8.0  8.28  8.57  8.88  9.19  9.51  9.85  10.2  10.6  10.9 
7  11.31  11.7  12.1  12.6  13.0  13.5  13.9  14.4  14.9  15.5 
8  16.0  16.6  17.1  17.8  18.4  19.0  19.7  20.4  21.1  21.9 
9  22.6  23.4  24.3  25.1  26.0  26.9  27.9  28.8  29.9  30.9 
10  32.0  33.1  34.3  35.5  36.8  38.1  39.4  40.8  42.2  43.7 
11  45.3  46.9  48.5  50.2  52.0  53.8  55.7  57.7  59.7  61.8 
12  64.0  66.3  68.6  71.0  73.5  76.1  78.8  81.6  84.4  87.4 
Notes: f/stop = √ 2 ^(stop number + fraction). A third stop past f/11 (stop number 7) is √ 2 ^7.3333 = f/12.6992.
Rounding: (values are more often truncated, or even approximated).
The cameras and light meters are marked f/11, and we say it as f/11, but f/11.3 is the correct actual calculated value. This is less than 1/10 stop difference, and the caemra definitely knows to do it right anyway. Most other values are closer, but Guide Number calculations for speedlights can use f/11.3 instead of f/11.
To make this fact be obvious, note that progressions, when arranged into rows of everyother doubled aperture values, are also defined as:
f/  1  2  4  8  16  32  64  
f/  1.41  2.83  5.66  11.31  22.62  45.25  (√ 2 is 1.414) 
Marked shutter speed numbers are also approximated. For example, the standard shutter speed chart below shows 1/20 second and 1/10 second (and 10 and 20 seconds) to be both third stop values and half stop values. Same value cannot be both, and the camera does compute the actual value closer (half stop 20 seconds will be 22.6 seconds, and full stop 30 seconds will be 32 seconds), but we humans are frequently shown easier rounded or even "equivalents". No big deal, except if you are setting the interval timer for 30 second shutters, you have to set a 33 second interval, or it will skip every other one.
Usage: If you set your light meter to read in tenth stops, the format of the result value shown is (for example):
This is NOT f/8.7. It is 7/10 of the way between f/8 and f/11  or about f/10, but read as f/8 plus 7/10 stop.
By definition, both of these equivalent values are simply two thirdclicks past f/8, or one thirdclick below f/11 (easy to set). The camera dial will indicate f/10 there, but we can instead meter and work in tenthstop differences from full stops.
Fractions: 1/10 is 0.1 stop. The fraction 1/3 stop is 0.33 stops, and 2/3 stop is 0.67 stops, so a reading around 0.3 is one third stop, and one around 0.7 is two third stops. The lens can only be set to third stops, so just pick the nearest third stop: 0, 1/3, 2/3, or 1 stop.
There would seem no point of 1/10 stop meter readings for daylight, since we can only set the camera to third stops. However there are two good reasons to use tenth stops for multiple flash. One is for greater precision in adjusting the power level of individual flash units  the actual difference between two lights could be controlled more closely. But the overwhelming advantage is when pondering fill level for that lighting ratio  how much is 1.3 stop less than f/10? It is about f/6.3, but who knows that? But if we read this as f/5.6 plus 3/10 stop vs. f/8 plus 6/10 stop, then we easily know 1.3 stops in our heads, immediately.
You can count the tenths between any two f/stop numbers in the chart, to determine the span between them in stops. But here is a calculator that may be easier to use.
Two purposes:
1. The reason to invent and use the f/stop method of numbering aperture is so that any two lenses (of any different sizes or focal lengths), will give the same exposure if at the same numerical f/stop. The entire idea is that f/4 exposure is f/4 exposure, in any lens.
2. Values of f/stop intervals are selected to provide useful standard steps, called stops, each stop providing 2x or /2 of the previous step's light. An aperture with double or half area will pass double or half the light. Cameras today also have 1/3 stop and 1/2 stop increments.
f/stop = focal length / aperture diameter.
The focal length factor is about the magnification of the field of view. Illumination per unit of area.
A short lens (wide angle) gathers a lot of light from a very wide view, and concentrates all of that light onto the camera sensor area.
A long lens (telephoto) gathers much less light from a much smaller view, onto same sensor area. Less light collected from a smaller area.
But fstop = focal length / aperture diameter equalizes these, giving constant exposure at equal f/stop numbers. That's why we bother with f/stop numbers.
Aperture is circular, and the area of a circle is defined as Pi r². Double area is twice the light, or one stop.
For double area: 2 Pi r² = Pi (1.414 x r)² , so 1.414x radius gives one stop. √ 2 is 1.414.
Since f/stop = focal length / aperture diameter, then f/stop numbers increase in 1.414x steps (or 1/1.414 is 0.707x decreasing steps).
Inversely, when the diameter and area are made larger, the f/number from the ratio f/d becomes a smaller number.
Full f/stop numbers advance in 1.414x multiples. From any f/stop number, in all cases, double or half of that number is two stops (for example, f/6.2 is two stops above f/3.1). Every second stop is the doubled f/number. Or one stop is x1.414 or /1.414.
Third f/stop numbers advance in multiples of the cube root of √ 2 , or 1.12246x the previous. Half stops are the square root of 1.414 apart.
Lens manufacturers seem to truncate instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is probably 3.56 (if it is a third stop). Less (number) is More (light).
The values of shutter speed and ISO are linear scales, meaning that 2x the number is a 2x difference, and 2x is one stop. But half and third stop numbers are not simply 1.5x or 1.33x, that's wrong. Because  two half stops computed 1.5 x 1.5 comes out 2.25 more, not 2.0. And three third stops computed 1.33 x 1.33 x 1.33 comes out 2.37 more, not 2.0. So instead, square root and cube root steps are the proper values.
The next thirdstop increment is cube root of 2 (1.26) greater than the previous value.
Every three thirdstop steps (from any point) is exactly a 2.0x change of the light.
The next halfstop increment is square root of 2 (1.414) greater than the previous value.
Every two halfstop steps (from any point) is exactly a 2.0x change of the light.
The next fullstop value is 2x greater than the previous value. Doubling any numeric value is one stop (speaking of shutter speed or ISO, but 2x number is two stops for f/stops, see above.)
ISO speed was a film sensitivity concept. Digital speed is a gain factor, multiplied after the digital sensor base rate does what it does, but the same ISO numbering scheme is used, still an apparent "sensitivity" indication.
Shutter Speed  Marked As: 
30 seconds  30" 
2 seconds  2" 
1/2 second  2 
1/30 second  30 
1/1000 second  1000 
Shutter speed is of course the time duration when the shutter is open, exposing the sensor or film to the light from the aperture. On many cameras, numerical values for shutter speed are marked on the camera using two methods with different meanings  for example, marked as 4 or 4". Just the number alone, like 4, is an implied fraction (1 over the number), meaning 1/4 second. The same number written 4" means four whole seconds, not a fraction. A slow shutter is a longer duration, and a fast shutter is a shorter duration.
A flash, especially a speedlight flash, is typically a much shorter duration than the shutter. The flash simply must occur while the shutter is open (sync), but the faster flash exposure is not affected by the slower shutter speed. Keeping the shutter open longer does increase the continuous ambient light seen, but shutter speed does not change what the fast flash does.
Related subjects:
Calculate Difference of Any Two Exposures How many stops?
EV  Exposure Value and Chart
Field of View Calculator for lens and sensor
Values of 1,2,4,8,16,32 are special, exact stop values, each exactly double the previous. The few shutter speeds and f/stops and ISO of those values or multiples are exact values, but many/most other values are marked with rounded nominal values (easy approximations). The camera uses the exact equivalents internally, so that full, half and third stops always actually work. For one example, ISO 1000 and 2000 are exactly double, but which is not enough. To be exactly 1/3 stop greater than full stops of ISO 800 and 1600, those exact values can only be ISO 1007.937 and 2015.874 (instead of 1000 and 2000). Which only matters in precise human calculations, since the camera always uses the correct values.
You've seen f/stops above, and there are two shutter speed charts below, the camera's normal rounded marked values, and the theoretical computed value goals that are actually used.
Each "Theoretical Actual" shutter speed at right below is just the simple progression (starting at one second), showing third stop times in sequential multiples of cube root of 2 (1.2599), and half stop times as sequential multiples of square root of 2 (1.4142). This insures that every interval of three thirdstops, or two halfstops, is exactly 2x or 1/2x value (i.e., exactly one stop). These are the "Actual" shutter speed goals the camera uses. I am not implying practical accuracy is within a microsecond, my goal was merely to show four significant digits for 1/1000 to 1/8000 second values, and to show that any and every third value of third stops is exactly double or half value (one stop).
The other "Shutter Speed" chart is the camera's normal marked numbers for same shutter speeds. These marked values take liberties to show even or round values for convenient human use. This is just a marking, which does not affect what the shutter does. It is difficult to verify the fast numbers, but at the 30 second end, we can easily measure and confirm the camera shutter in fact does use the computed theoretical numbers (32 seconds actual instead of the marked 30 seconds). It must do that, because the basis of the system is that one stop is 2x the light. The markings only have about 2% or 6% numeric discrepancy. Which is a tiny difference, not more than 1/10 stop, but f/stop and shutter can combine to add (the shutter halfstop markings of 10 and 20 are 13%, near 2/10 stop).
Do realize of course, that any such error is Not real, it exists only in our own minds, since the camera is designed to use the right numbers instead of the markings for humans.
Nominal  Precise  stop 
1/125  1/128  Full 
1/100  1/102  Third 
1/90  1/91  Half 
1/80  1/80.6  Third 
1/60  1/64  Full 
In years past, after learning about stops being 2x the light, I used to wonder how the literal shutter speed sequence 1/2, 1/4, 1/8 second, could suddenly shift to 1/15, 1/30, 1/60, and then suddenly shift again to 1/125, 1/250, 1/500 second? But of course, nothing actually changes, except only the helpful markings for humans. 64 may seem a nice round number today, but it was not always the case. :) This nomenclature was adopted 100 years ago, before the computer era, but if invented today, I think we would have no issue using the real 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 numbers. But we are used to this old system now, and it is convenient. And the case of third or half stops is harder to solve, see example shutter speeds at right. Nominal does have a certain beauty, and it serves our purpose. The markings we see are not very important, the important need is for each full stop (and each three third stops) to be exactly 2x the light from previous stop  easy work for today's crystal timed shutter. So (unless doing precise human calculations), the point is NOT that there is a marking discrepancy, but is instead that we need not be concerned about it. The shutter does the right thing, and it is a rather neat system.
Tables for Aperture F/stop, ISO, Shutter Speed Values  in Full, Third and Half stops 





Combining multiple lights (techie stuff, a use of f/stop)
Two lights will add to be brighter than the brightest. A light meter is a good way to meter multiple lights, but the math is like this:
We are assuming lights are ganged to light the scene area the same way.
Multiple Equal lights ganged = (one lights exposure fstop) x square root (of number of equal lights)
Assuming each light is f/8:
2 lights: f/8 x square root (2) = f/11.3 one stop brighter  remember, f/11 is actually f/11.3
3 lights: f/8 x square root (3) = f/13.85 1.58 stops brighter than one
4 lights: f/8 x square root (4) = f/16 2 stops brighter than one
5 lights: f/8 x square root (5) = f/17.9 2.3 stops brighter than one
Diminishing returns. We must double the number of lights to gain one stop.
The same formula is used for combined Guide Number of N equal flashes. GN of N equal lights = GN of one x square root (N).
Unequal lights ganged:
Two lights at f/8 and f/4 : (two stops difference)
Square root of (8² + 4²) = f/8.9 (about 1/3 stop more than brightest)
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