# Understanding the Camera Numbers -Aperture f-stop, Shutter Speed, ISO

This article is about "Understanding the camera Numbers". It is NOT a primer about using those numbers to take photos (for that, see for example).

## Where the Numbers Come From

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 exactly the obvious physical diameter (shown by black vertical lines below), but instead is the apparent "working" diameter D 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.

OK, yes, there can be slight minor variations, especially in the old days in lenses without modern coatings (losing a lot of light), and even today in fancy lenses containing 15 or 20 glass elements (30 or 40 surfaces), losing a slight amount of light if with best coatings (lens coatings allow the light to pass through the lens, instead of reflecting it away). There are normally only slight variations, but which are important in professional movie cameras, when switching lenses on the same scene. So the professional movie lenses use T-stops, with markings which are calibrated to the actual amount of light the lens transmits, instead of the theoretical amount (a T2 lens actually transmits the light that a perfect f/2 lens should, matching the light meter). But improved modern lens coatings largely improve this today. Fancy zooms with many elements suffer a little more, which the coatings are to help it. But it is minor and relatively unimportant for still cameras, since the camera meters through the lens anyway, automatically accounting for any possible variance in the lens losses.

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 retro-focus, 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 f-stop or fstop? The lens manufacturers properly write f/8. The internet changes things, and the term f-stop has become very popular on-line, but we also still see f/stop. I learned to write f/stop, because we also write f/8, to be remindful of the division defining it:

f/stop number = focal length / aperture diameter

f/8 is an aperture diameter, literally = focal length / 8.

Why do all lenses expose equally if all are set to the same f/stop? For two lenses at the same f/8, the lens with 3x longer focal length has an aperture diameter 3x larger. Tricky, but the 3x focal length magnifies the subject 3x, and so it sees 1/3x width and 1/3x height, which is 1/9 area, which only reflects 1/9 the light the wider lens sees. But the longer 3x lens also has aperture 3x larger, which is 9x area, and so now admits 9x more light, which before was 1/9 as much, from a 1/9 area field... so the 9x times 1/9 result is the same exposure in both f/8 lenses. Another argument is the Inverse Square Law over the 3x longer focal length is 1/9 the light, when the image reproduction reaches the sensor plane (which is just repeating the first explanation again). This is why the f/stop system is used. It's good stuff.

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.

### Rounding

There are two concepts of camera numbers here. "Nominal" numbers are the numbers actually marked on the camera, which are just convenient approximations of what I call the "Precise" numbers that the camera actually uses. The camera knows to actually do it right, but the marked numbers are made easier for humans. Much more below.

Nominal f/stop numerical values might be rounded, but are often truncated, or even approximated.

The cameras and light meters are marked f/11, and we say it as f/11, but f/11.31 is the necessary correct actual calculated value. This is only about 0.08 stop difference, and any difference exists 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/11.31 instead of f/11.

To make this fact be obviously true, note that f/stop numbering is the sequence of √ 2 intervals, (which is 1.414) - making every other stop number be a multiple of 2. These sequence progressions, when arranged into rows of every other doubled aperture values, are:

 f/ 1 2 4 8 16 32 f/ 1.414 2.828 5.657 11.314 22.627

It may be handy to realize that doubling any f/stop number (for example, f/5 to f/10) is exactly two stops.

Shutter speed marking numbers are also approximated. For example, the camera nominal markings show 1/20 second and 1/10 second (and 10 and 20 seconds) to be both third stop values and half stop values. But the same value cannot be both values, 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, see standard shutter speed charts below). The camera does it right, but we humans are frequently shown easier approximated numbers.

## Calculator to Compare Difference of any two f/stop values

It is not obvious in f/stop math that the difference between f/4 and f/5 is 2/3 stop, so the calculator purpose is to help. The calculator does NOT just use camera nominal values, and then fudge by presuming to round off to thirds. It instead uses the same precise values the camera uses internally. Where it is a factor (Option 3), it shows both the precise result, and the precise difference between it and the precise value of the nearest nominal value it also shows.

• Option 1 uses the precise f/stop values that the camera also uses (better accuracy than the nominal marked numbers). A negative result means the second value is stopped down more, positive is stopped down less.
f/stops: Green is Full stops, Blue is Third stops, Red is Half stops. If your camera uses third stops, then you should generally avoid the red half stop values, and vice versa if you bull-headedly insist on using half stops (thirds are more precise exposures. Some bigger cameras can be set either way.)
• Option 2 can be any numbers. Not sure of any use for that, since the marked numbers we can select on the camera are nominal numbers (hence the precise choice is more accurate). Any NaN result means entry was Not a Number.
• Option 3 range is very large, but not quite infinite:
f/stop Selection range is f/0.5 to f/128
f/stop Result range here, -1 EV under f/0.5 (f/0.35), to +3 EV above f/128 (f/360)

Of course 0.333 EV is 1/3 stop, and 0.667 EV is 2/3 stop.

• Option 4 tenth stop values are detailed in the chart just below, but this option is the difference from tenth stop values to a nominal value we can set on the camera.
• Or, there is also an Exposure calculator

## Calculator, Stops Difference

1. Precise values (full, third, half stops)

from   to

2. Any numeric values

from f/   to   f/

3. Add ± EV to a f/stop value

4. Tenth stops to Nominal stop values

from plus   to

## Tenth Stop Usage

DSLR cameras today can be set to use full stops, half stops, or third stops. I cannot imagine any reason not to choose third stops, more precision of adjustment can only be a good thing. When the light meter reads a certain value, the camera can be set to the nearest third stop. The error is not more than 1/6 stop.

But 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):

f/8 plus 7/10 stop

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, the equivalent value of f/8 plus 7/10 stop is simply two third-clicks past f/8, or one third-click below f/11 (easy to set). The camera dial will indicate f/10 there, but we can instead meter and work in tenth-stop 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 the nearest third stop. 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 values 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 (in use, that is really big).

## F/stop Table, in Tenth-Stops

 StopNumber 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 1.8 1.87 1.93 2 2.0 2.07 2.14 2.22 2.3 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 5.1 5.28 5.46 5 5.66 5.86 6.06 6.28 6.5 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 13.5 13.9 14.4 14.9 15.5 8 16.0 16.6 17.1 17.8 18.4 19 19.7 20.4 21.1 21.9 9 22.6 23.4 24.3 25.1 26 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 53.8 55.7 57.7 59.7 61.8 12 64.0 66.3 68.6 71 73.5 76.1 78.8 81.6 84.4 87.4

Notes: (√ 2 is 1.414).   f/stop = 1.414 (stop number + fraction)

e.g., two tenth stops past f/11 (stop number 7) is 1.414 7.2 = f/12.126
Or 1/3 stop past f/11 is 1.414 7.333 = f/12.698
Stop number -1 is f/0.7, and stop number -2 is f/0.5.

If interested, here is a one page printable PDF file of this tenths chart.

## How the Stops are Numbered (Origin of these numbers)

### f/stop

Two purposes of f/stop nomenclature:
• 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. Prior to about the mid-1920s, Kodak used the "U.S. system" (Uniform System, from Britain) 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 a quite different setting on a different camera or lens.
• 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 wide view, and concentrates that light onto the camera sensor area.
A long lens (telephoto) gathers less light from a smaller view, onto the same sensor area.
But fstop = focal length / aperture diameter equalizes these, a larger aperture in a longer lens, giving constant exposure at equal f/stop numbers. Exposure is about Illumination per unit of scene area, which stays the same. That's why we bother with f/stop numbers, the benefit is great. f/8 is always f/8, on any lens. Our light meter works the same for any 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.

See the note under the tenth stop table above for a formula computing f/stop for partial stops.

Full f/stop numbers advance in 1.414x numeric multiples. From any f/stop number, in all cases, double or half of that number is two stops (for example, f/10.2 is two stops above f/5.1). Every second stop is the doubled f/number. Or one stop is x1.414 (or /1.414 which is x0.707).

Third f/stop numbers advance in multiples of the cube root of √ 2, or 1.12246x the previous (speaking of f/stops).
Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

Half f/stop numbers advance in multiples of the square root of √ 2, or 1.1892x the previous (speaking of f/stops).
Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

Less (f/stop 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. But we see the same f/1.2 marking for the half stop (f/1.189) and third stop (f/1.260). Point is, the markings are just nominal numbers to show us humans. The lens and camera know to try to do it right.

### Shutter speed and ISO

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 third-stop shutter step is cube root of 2 (1.26992) greater than the previous value (but for f/stop, see above)
Every three third-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

The next half-stop shutter step is square root of 2 (1.41421) greater than the previous value.
Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the light.

The next full-stop 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 not be precise, but the camera knows exactly what to do. For example, set ISO 250 or ISO 2000 in the Nikon DSLR camera. Then near the top of the Exif data will show the ISO 250 or 2000 values, but further down in the manufacturers data, it shows the precise values used, ISO 252 or ISO 2016. (The ISO base appears to be 100 instead of 1... 100, 200, 400, 800 instead of 1, 2, 4, 8. This makes third stops of 252 and 2016 instead of full stops 256 and 2048 - which we call 250 and 2000.) Auto ISO is probably using 1/6 stops, but which will still be the precise sixth root of 2 steps. The numbers we see are just convenient nominal numbers, which the number really does not much matter to us humans. We just want one stop to always be a 2x light value. The point here is that the camera typically uses numbers a little different than the numbers we see. The only time that actually matters is if we try calculating ourself, using the nominal numbers instead of the actual precise numbers.

 Shutter Speed Marked As: 30 seconds 30" 4 seconds 4" 1/4 second 4 1/30 second 30 1/4000 second 4000

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 either 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.

## Camera settings are marked with Nominal values

Values of 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are a special system, each exactly double the previous. That is how our number system works, and it is how our camera work. All of of those values are exact multiples, but many of these on the camera are "marked" with rounded nominal values (easy approximations, for example 1/64 is marked 1/60, and 1/1024 is marked 1/1000). The camera uses the exact values internally, so that full, half and third stops always actually are exact half or third stops. But also for example, both third stop and half stop systems have a shutter speed marked 1/10, 1/20, 10, and 20 seconds (both halves and thirds cannot be the same, which is a worse case error of about 13%, but the only error is the marked nominal value). The camera knows to use the exact precise value.

ISO sensitivity is the equivalent of film speed. Digital sensitivity is a gain factor, results are multiplied after the digital sensor native sensitivity does what it does (i.e., around ISO 100), 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, as a full stop, but if we divide 100 by 2 a few times, we end up at 1.5625 or 0.78125, etc - instead of 1. So the ISO speed specification ought to start at base value "1" too, and could also advance as 1,2,4,8,16, etc. And Wikipedia does show the ISO and old ASA specs starting at 1. That would make ISO 128 be the full stop (called 125). And ISO 100 would be a third stop, 2/3 stop above ISO 64. Which is all relative, we humans don't really need to care what the exact number is.

And we do tend to favor ISO 100 being the base, and the cameras mark 100, 200, 400, 800 as nice even full stops, and I did it here too (to match the cameras). There is no real difference either way, and the camera does it right. This works perfectly fine, and technically, full stops are not special to us. All that matters is that all third stop clicks are exactly one third stop (exactly cube root of 2 apart numerically, so that three thirds are 1.0 full stop). One second and f/1 are convenient on those scales, but for ISO, it really doesn't matter so much 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 (3rd and 4th long table below), the camera's normal rounded marked values, and the theoretical computed value goals that are actually used.

Each "Theoretical Actual" shutter speed (4th long table 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 third-stops, or two half-stops, 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).

 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

The other "Shutter Speed" chart is the camera's normal marked numbers for same shutter speeds. These marked values take liberties to show nice rounded or even approximate values for convenient human use, numbers we won't struggle with. This is just a marking, which does not precisely 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). Some cameras may actually do 30 seconds, but many do the correct 32 seconds. The basis of "stops" in photography is that one stop is 2x the light, so there is dignity in cameras honoring the 1, 2, 4, 8, 16, 32 numbers.

As an example of nominal settings, users using a Nikon cameras interval timer to record multiple 30 second shots (star trails, etc) may have a problem, if they set the interval timer to 31 second intervals, so it can fit the 30 second shutters. This sounds very reasonable, but this cannot work, because the camera 30 second setting actually does 32 second exposures (because the sequence 1,2 , 4, 8, 16, 32 seconds must each be 2x full stops). So remember that your interval timer requires 33 second intervals for a so-called 30 second shutter setting. The difference between nominal and precise does exist. The difference in 30 and 32 seconds is only about a 1/10 stop, not very important to us, but when doing math, the numbers come out right for 32 seconds. In the old days, camera shutters stopped at one second, so it was not an issue.

Camera brands may vary. The 2x stop concept does seem sacred, but some brands are said to honor the 30 second nominal markings instead. Nikon DSLR do 32 seconds, which makes the stops be correct. A Canon compact does 16 seconds for 15. And a Sekonic meter reading tenth stops will show exactly two 2.0 EV difference between 8 seconds and "30" seconds, which has to be computing for 32 seconds, which IMO, is the expected right thing to do. Cameras actually doing 30 seconds is hard for me to believe, but you can easily verify which way your camera works by timing your shutter yourself.

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 half-stop 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 precise numbers instead of the nominal markings shown to humans.

So the actual shutter speed sequences 1/2, 1/4, 1/8 second do not suddenly shift to 1/15, 1/30, 1/60, and then suddenly shift again to 1/125, 1/250, 1/500 second. The camera does it right, and only the markings change, thought more helpful for mere humans. 64 may seem a nice round number today, but it was not always the case. :) This nomenclature was adopted maybe 100 years ago, before the computer era, and before the light meter era, and before mechanical shutters could be very accurate. But if invented today, we would probably have no issue with the real 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 numbers (however the third stop markings, like 1/323 or 1/406 second may still look odd to us, see 4th long table below). But we are used to this old system now, and it is convenient for humans. Nominal does have a certain beauty, and it serves our purpose. The exact 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.

## Computing Precise Values

So (unless humans are doing precise math calculations), the point is NOT that there is a marking discrepancy, but is instead that we need not be concerned about it. The camera knows to do the right thing, and it is a rather neat system. But if interested in computing precise values, there are two methods (results should be the same).

• Method 1: Repeatedly multiplying or dividing subsequent stops by a constant for a sequence of third stop intervals. Multiply to advance one direction, divide to go the opposite direction.

For f/stops, start multiplying with third stop steps, which (for f/stop) is cube root of √ 2, or 1.12246x. Each third is this factor greater than the previous. Start with f/1, 2, 4, 8, 16, etc (power of 2, and nominal is same value as precise). Half f/stop numbers advance in multiples of the 4th root of 2, or 1.1892x.

f/8   (a starting point, of 1, 2, 4, 8, 16, etc)
8 x 1.12246 = f/8.9797
x 1.12246 = f/10.079
x 1.12246 = f/11.314
x 1.12246 = f/12.699

For shutter speed fractions, start dividing at 1 second with cube root of 2 or 1.26992. Each third is this factor less than the previous. Multiply to advance non-fractional seconds, greater than 1. Half-stop shutter steps are square root of 2 (1.41421).

1/64 second   (a starting point, of 1, 2, 4, 8, 16, 32, etc)
(1/64) / 1.26992 = 0.01240157 = 1/81
/ 1.26992 = 0.00984313 = 1/102
/ 1.26992 = 0.0078125     1/128
/ 1.26992 = 0.00620079 = 1/161
/ 1.26992 = 0.00492157 = 1/203 second

For ISO, multiply by 1.26992 (cube root of 2 for thirds, linear like shutter speed), but today, start Not at 1, but at 50, or 100/128 (0.78125) will make ISO 50, 100, 200, 400, 800, etc all become Full stops at exactly x.000 values. I think the ISO base used to be ISO 1, however digital cameras are using ISO 100 as a full stop instead of a third stop. This is a 1/3 stop offset.

Each three thirds exactly hits each full stop (if using sufficient significant digits for precision).

• Method 2: Math exponents allowing computing any one specific step independently, any or all:  ShutterSpeed Stop NumberConcept f/stops 2 seconds +1 -1 f/0.7 1 second 0 0 f/1 1/2 second -1 1 f/1.4 1/4 second -2 2 f/2 1/8 second -3 3 f/2.8 1/16 second -4 4 f/4 1/32 second -5 5 f/5.6 1/64 second -6 6 f/8 1/128 second -7 7 f/11 1/256 second -8 8 f/16 ...   Fractions use a negative exponent

The precise values are "the stop number" (0,1,2,3,4... see chart at right) raised to a power of square root of 2 for f/stops, or a power of 2 for shutter speed.

f/stops are steps of √ 2 from f/1 (stop 0 is f/1, and the square root of 2 is 1.4142).

f/stop = 1.4142 (stop number + fraction)

e.g., 2/10 stops above f/2.8 (stop number 3) is 1.4142 3.2 = f/3.031
Or 1/3 stop above f/11 is 1.414 7.333 = f/12.698
Or 1/2 stop below f/11 is 1.414 6.5 = f/9.514

Shutter speed is steps of 2 from 1 second   (stop 0 is 1 second).

Shutter Speed = 2 (stop number + fraction)

Note that the interval for f/stops is 1.414, and the interval for shutter speed stops is 2 (increment factor to the next stop).

Most shutter speeds are fractional, which is a negative exponent, but above 1 second is a positive exponent.
1/100 second is 6 2/3 stops "below" 1 second, or 2 -6.667 = 0.00984 second, which we call 0.01 or 1/100.
1/125 second is 7 stops "below" 1 second, or 2 -7 = 0.00781 second, which we call 1/125 (but 1/0.00781 = 128).
The nominal shutter speed 1/1000 second is 2 (-10) = 0.0009766, and the reciprocal is 2 (+10) = 1024 (for 1/1024 seconds).

Stop Number 0 produces the value 1... for f/1 or 1 second. Any number to the exponent of 0 is 1. Incrementing the exponent increments the value in multiples of the number. So exponents for third stops add 0.333 or 0.667, and half stops add 0.5 fractions. Using 2 for shutter speed, then:
2 ^ (stop number) = full stop
2 ^ (stop number + 0.1) = tenth stop
2 ^ (stop number + 0.333) = third stop
2 ^ (stop number + 0.5) = half stop
2 ^ (stop number + 0.667) = 2/3 stop
2 ^ (stop number + 1) = Next full stop
Negative (stop number + fraction) computes 1/(same numeric value) - e.g., for f/stops, f/0.7 is f/(1/1.4)

For ISO, I think you want to use the first method and cube root of 2, because ISO 100 as a full stop is actually used today (digital). If instead using ISO 1 as a base, it creates a 1/3 stop offset, numbers become approximate. The cameras are NOT using ISO 1 as a numbering base now. The ISO chart below shows low values to make this point.

Precision:   In any method, use at least as many significant digits in all values, as in the result precision desired. That means results of four digit shutter speeds like 1/2048 seconds need to calculate with at least four significant digits everywhere. If seeking four significant digit precision in the results, then use 0.3333 instead of 0.33 (I like to use 1/3 so Javascript will compute it precisely). To be precise for ISO, use at least as many digits as the ISO result will have.

Calculation results that show all of these precise values are presented here.

If interested, here is a one page printable PDF file of these next four charts.

## Tables for Aperture F/stop, ISO, Shutter SpeedValues in Full, Third and Half stops

 Aperture f/stop Full Thirds Half 1.0 1.1 1.2 1.2 1.4 1.6 1.7 1.8 2 2.2 2.4 2.5 2.8 3.2 3.3 3.5 4 4.5 4.8 5.0 5.6 6.3 6.7 7.1 8 9.0 9.5 10.1 11 12.7 13 14.3 16 18.0 19 20.2 22 25.4 27 28.5 32 35.9 38 40.3 45 50.8 54 57.0 64 71.8 76 80.6 90
 ISO Stops Full Thirds Half 0.75 1 1.12 1.25 1.5 2 2.25 2.5 3 4 4.5 5 6 8 9 10 12 16 18 20 25 32 35 40 50 64 70 80 100 125 140 160 200 250 280 320 400 500 560 640 800 1000 1100 1250 1600 2000 2200 2500 3200 4000 4400 5000 6400 8000 8800 10000 12800 16000 17600 20000 25600 32000 35200 40000 51200 64000 70400 80000 102400
 Nominal Shutter Speed Stops Full Thirds Half 1/8000 1/6400 1/6000 1/5000 1/4000 1/3200 1/3000 1/2500 1/2000 1/1600 1/1500 1/1250 1/1000 1/800 1/750 1/640 1/500 1/400 1/350 1/320 1/250 1/200 1/180 1/160 1/125 1/100 1/90 1/80 1/60 1/50 1/45 1/40 1/30 1/25 1/20 1/20 1/15 1/13 1/10 1/10 1/8 1/6 1/6 1/5 1/4 1/3 1/3 1/2.5 1/2 1/1.6 1/1.5 1/1.3 1 sec 1.3 sec 1.5 sec 1.6 sec 2 sec 2.5 sec 3 sec 3 sec 4 sec 5 sec 6 sec 6 sec 8 sec 10 sec 10 sec 13 sec 15 sec 20 sec 20 sec 25 sec 30 sec
 Theoretical Actual Shutter Speed Third Stops Half Stops 1/sec Seconds Seconds 1/sec 8192 0.0001221 6502 0.0001538 0.0001726 5793 5161 0.0001938 4096 0.0002441 3251 0.0003076 0.0003453 2896 2580 0.0003875 2048 0.0004883 1625 0.0006152 0.0006905 1448 1290 0.0007751 1024 0.0009766 813 0.0012304 0.0013811 724 645 0.0015502 512 0.0019531 406 0.0024608 0.0027621 362 323 0.0031004 256 0.0039063 203 0.0049216 0.0055243 181 161 0.0062008 128 0.0078125 102 0.0098431 0.0110485 91 80.6 0.0124016 64 0.0156250 50.8 0.0196863 0.0220971 45 40.3 0.0248031 32 0.0312500 25.4 0.0393725 0.0441942 22.6 20.2 0.0496063 16 0.0625000 12.7 0.0787451 0.0883883 11.3 10.1 0.0992126 8 0.1250000 6.3 0.1574901 0.1767767 5.7 5.0 0.1984251 4 0.2500000 3.2 0.3149803 0.3535534 2.8 2.5 0.3968503 2 0.5000000 1.6 0.6299605 0.7071068 1.4 1.3 0.7937005 1 1 sec 1.2599 1.4141 1.5874 2 sec 2.5198 2.8284 3.1748 4 sec 5.0397 5.6569 6.3496 8 sec 10.0794 11.3137 12.6992 16 sec 20.1587 22.6274 25.3984 32 sec

Combining multiple lights   (techie stuff, a use of f/stop)

You can see the Ganged Flash Calculator, or it works as described below:

Two lights illuminating same subject 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 equal 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:

1 light:   f/8 x square root (1) = f/8     one light
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
8 lights: f/8 x square root (8) = f/22     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:

If 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)

If two lights at Guide Numbers 60 and 120 :  (two stops difference)

Square root of (60² + 120²) = GN 134   (about 1/3 stop more than brightest)

All of the formulas are exactly the same concept, since Guide Number = f-stop * distance (distance being a constant in the equations here).

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