This article is about the theory of camera setting "numbers", but it is NOT about using those camera settings. If any mystery about using the settings to take better pictures, I would suggest starting with this inexpensive book for beginners: Bryan Peterson's Understanding Exposure, 3rd Edition: How to Shoot Great Photographs with Any Camera (an easy fun read, and surely the best $15 a beginner can spend on photography. It is in the public libraries here).
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 or fstop? The lens manufacturers properly write f/8. The term fstop has become popular online, so we see both. But I learned to 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.
Handheld light meters typically can also be set to read tenth stops (for metering multiple flash). If you set your light meter to read in tenth stops, the format of the result value we see 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 one and a third stop less than f/10? It is about f/6.3, but who knows that? But if we read these two as f/5.6 plus 3/10 stop vs. f/8 plus 6/10 stop, then we easily know 1.3 stops difference, in our heads, immediately (that seems big).
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/12.699.
The cameras and light meters are marked f/11, and we say it as f/11, but f/11.3 is the necessary correct actual calculated value. This is less than 1/10 stop difference, and only in our mind, since the camera will 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 the 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".
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.
Other calculators
Calculate Difference of Any Two Exposures How many stops?
Field of View Calculator for lens and sensor
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.
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.
Exposure is about Illumination per unit of area.
But fstop = focal length / aperture diameter equalizes these scenes, giving constant exposure at equal f/stop numbers. f/8 is always f/8, on any lens. That's why we bother with f/stop numbers, the benefit is great. Prior to about the mid1920s, Kodak used the "U.S. System" of aperture markings, where stops were just numbered 1,2,3,4... without regard to focal length... so the correct exposure on one camera was likely quite different on a different camera or lens.
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. Less (number) is More (light).
Lens manufacturers seem to truncate instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is 3.56. We see the same f/1.2 marking for the half stop (f/1.189) and third stop (f/1.260).
The values of shutter speed and ISO are linear scales, meaning that 2x the number is a 2x difference, and 2x is one stop. The very important thing to the definition of our exposure system is that any span of three third stop steps (or any two half stop steps) must come out exactly 1.0 stop of 2.0x exposure difference. To force this, cube root (and square root) steps are the proper values to create and number step intervals.
The next thirdstop increment is cube root of 2 (1.26992) greater than the previous value. (but for f/stop, see above)
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 (1.41421) 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.)
The nominally marked numbers may miss this slightly, but the camera knows what to do.
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.
Values of 1,2,4,8,16,32,64,128,256,512,1024 are special, each exactly double the previous. That is how our number system works. The few shutter speeds and f/stops and ISO of those values or multiples are exact values, but many/most values are marked with rounded nominal values (easy approximations, even 1/64 is marked 1/60, and 1/1024 is marked 1/1000). The camera uses the exact equivalents internally, so that full, half and third stops always actually are exact half or third stops. For example, both third stop and half stop systems have a shutter speed marked 1/20 second, but the camera knows thirds is 1/20.2 and half is 1/22.6.
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, still seems the same to us. We think of ISO 100 as a base, certainly as a full stop, but if we divide 100 by 2 a few times, we end up at 3.125, 1.5625, 0.78125, etc. A messy numbering system, at least for humans.
Technically, the ISO speed specification must start at base value "1" too, and must advance as 1,2,4,8,16, etc, same as f/stop and shutter speed. And it does, Wikipedia shows the ISO and old ASA specs starting at 1. So technically, this makes ISO 128 be the full stop (called 125). And ISO 100 is actually 101.6, a third stop. But the cameras still marks 100, 200, 400, 800 as nice even full stops, and I did it here too (to match the cameras). But there is not much difference either way, and the camera does it right. Technically, when every stop is an exact numerical third stop, full stops are not special to us. So long as all third stop clicks are exactly one third stop (exactly cube root of 2 apart numerically), relationships don't matter where we start the markings. The exact number only matters to humans in precise calculations, for example if we compute EV or a third stop difference. But the camera knows, and always uses the correct values. It is all relative to the user, so long as every three third stop clicks add up to be exactly one full stop at 2x.
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.
As an example of nominal settings, users might plan to use the cameras interval timer to record multiple 30 second shots. They set the interval timer to 31 second intervals, so it can fit. Sounds reasonable, but this cannot work, because the camera 30 second setting actually does 32 second exposures (the sequence 1,2,4,8,16,32 seconds must be 2x full stops). You can time your shutter yourself. So the interval timer requires 33 second intervals for a socalled 30 second shutter setting. The difference between nominal and precise does exist.
Most markings have no more than about 2% or 6% numeric discrepancy. Which is a tiny difference, not more than 1/10 stop, but f/stop and shutter together 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 nominal 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 marking 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)
Menu of the other Photo and Flash pages here