of f/stop and shutter speed

This article is about "Understanding the camera **Numbers**". It is about the numerical numbers. There is of course much more to photography, and understanding the “What, Where, When, and Why” about using these settings is necessary to ever get out of Auto mode (to have any control of your pictures). The actual Numbers part starts just below. Some basic lens properties are below that.

But first, **learning exposure** is the beginning skill you need to acquire about photography, and if you don't understand about choosing camera settings yet, you’ll surely want to look into it. “Exposure” is NOT just about how dark or light the picture is. Exposure is all about using the settings to get the best picture, like by freezing the subject motion or by stopping down aperture to increase the depth of field. Situations vary, and need different techniques. Automation is very handy, since we don't have to know anything or to think about it (and we likely won’t even know what the settings used were), but that only result is whatever the automation does. Except automation has no brain, and needs human help to recognize the situation to choose better choices for the specific scene. There are many creative choices possible in photography, which does require an adjustable camera, and a little thought about we’re doing, but it’s easy, and is the first thing photographers learn. It’s fun getting it right, and becomes our automatic procedure, and is the basics for every picture we take.

Choosing WHAT to point the camera at is a different discussion, but a photo scene allows a wide choice of possible Equivalent Exposure combinations, all “correct” with regard to exposure (even our cell phone must select one). The big advantage of using our human brain is that it can immediately recognize the subject and the situation, and can choose the best choice for this particular picture (and automation cannot). Exposure does NOT mean just exposure (not too dark, not too bright). It means choosing the best available exposure combination for the scene situation. If we consciously think about what we’re doing when we actually LOOK at the scene, our brain likely “just knows” immediately what it needs.

Shown here is an example row from the EV Chart showing sample values chosen to be a normal bright direct sunlight level (called EV 15). For example one exposure combination for this EV 15 is f/4 at 1/2000 second (freezes motion). Or f/22 at 1/60 (much depth of field, but bad for motion). We can choose what is needed. All entries on this entire row are **Equivalent Exposures**. Equivalent means that all of them are a “correct” exposure (for the example light level), but we’re looking for the one best combination for our specific situation. Our brain is the best tool in photography, it sees and knows things at a glance, and it can help the camera, if we choose to use it. This is the fun in photography for many of us, to create the photo ourselves, to be the way we want it. Experiment a bit, it is the first step of understanding.

I am speaking of using camera A mode (Aperture Preferred), which likely is most commonly used, or maybe S mode (Shutter Preferred). We start by simply entering one reasonable number choice for the situation, and the camera meter selects the other number matching the exposure. So these can still be automation controlling the actual exposure, but we can choose the settings to use. When you’re aware you’re supposed to actually LOOK at the scene, you will have a pretty good idea about the situation when you first walk up. Before you click the shutter, you should have decided what f/stop, shutter speed and ISO combination is to be used, as the best choice for this subject situation. In most cases, this is not difficult, and soon we will be able to “just know” at a glance what is necessary. Your camera meter will first pick one combination with a correct exposure, but maybe another different combination is the **best choice** for this particular photo situation. You start by entering one of the numbers, usually aperture, and the camera meter determines the other one. Then you decide if those are the best numbers you can do for this situation. So we change it first if something else is better. Actually give it a try. You may not know yet how easy and satisfactory this is, but a very little experience will work wonders for you.

Exposure is determined by the combination of aperture, shutter speed and ISO. If you increase one of the three, you have to decrease the effect of one or both of the other two to keep the same Equivalent Exposure. In lessons, this combination is usually called “the Exposure Triangle”, just meaning these are the three factors of exposure (it is a good search term, but there is no actual triangle). We just take an instant to consider if this first metered combination is appropriate for the situation.

**Shutter speed**can be used to freeze motion. Is it fast or slow motion? If fast motion, a faster choice like maybe 1/1000 second would be necessary. Or 1/200 second should handle minimal motion. Just open aperture some, or increase ISO some, and see the meter match with a faster shutter speed. Slow shutter speed can allow motion to blur, but that is not a problem for a motionless scene if the camera is held steady on a tripod.**Aperture (f/stop)**can be used to control Depth of Field. Focus can be at only one distance, and Depth of Field is the zone of the extremes of the “close” and “far” distances around the focus point that both still need to be sharp. Depth of Field can be increased by stopping down more, maybe even to extremes like to f/16 or even f/22 (in this EV example, those will then match with 1/125 or 1/60 second slower shutter speed, Equivalent Exposures). Either end of the extremes of lens aperture is less sharp than the middle range. Wide open has optical aberrations and minimal Depth of Field. Stopping down aperture extremely increases Depth of Field, but adds diffraction (hurting resolution, but the greater Depth of Field is sometimes worth it). Small cameras (like cell phones with short focal length) have extreme range of Depth of Field, but it can be a concern in larger cameras.- Increasing
**ISO**is another choice allowing either or both faster shutter or stopping down more. ISO is the sensitivity to light of the sensor (“film speed” so to speak about older days). Digital cameras can simply change it with a dial, but as with film, lower ISO 100 is better quality. Larger cameras have an advantage, but at some point, higher ISO will still add noise and objectionably hurt sharpness and clarity.

A light meter can only measure the brightness of a blob of light, but the decisions about the scene situation require a human, with eyes and brain. So remember, a fully automatic camera simply doesn’t know, and couldn’t care less. For one example, if there is a bright sun background behind a subject in the shade, the meter doesn’t have a chance without help. Just give a quick thought to what you’re doing, because that’s what makes the magic happen. This hobby gives joy in the results when you know how to get it right.

You set the aperture and the camera mode A meter figures out the shutter speed. That is automation too, but then our human brain can decide which combination is best, based on concerns other than literal exposure. We are Not so much selecting specific numbers, it’s more a hunch, generally just evaluating the obvious need for More shutter speed or More Depth of Field (or the best compromise of those). To allow that, sometimes we may need to increase ISO, or make some other compromise. Sometimes “all we can do” has to be the answer (but which is still our best choice). But when no problem factors are present, then choosing an aperture two or three stops up from maximum aperture is often a good general purpose choice (for generally good sharpness and depth, but still check the shutter speed).

It is a learning experience, and you will continue learning for years, but a little experience works wonders, and most cases are obvious and easy. The camera first meters the scene, and then we might change one of the three factors to be better for some specific reason. Then the camera meter gives us the resulting exposure combination, which we can consider again, to make sure. We looked at the scene and know what it needs. The meter is certainly a big help, but do give a thought to what you’re doing before you press the shutter button. This is easy, and soon you might not always even realize you’re doing it automatically. Like checking your car’s side mirror before changing lanes. You simply just glance at it, and the extra little bit of thought can dramatically improve the results of the situation. Try this a few times, it becomes easy and automatic.

There is much online about learning Exposure.

**The Numbers**: The page below is about "What are those numbers?" There is a How Many Stops Calculator on the next following page to help with the numbers. Then the third page shows charts of the actual precise goal values of the settings the camera actually uses.

The F/stop number is about the diameter of the lens aperture, how much light it lets in (a smaller f/stop number is a larger diameter aperture allowing more light, however a larger f/stop number is up to a point, generally sharper, creating more depth of field). Shutter speed is the time duration while the shutter is open, letting light in (a faster shutter stops motion better, but which reduces the exposure, which can be offset by opening aperture or increasing ISO). ISO is the sensitivity to light, how well the light registers on the sensor (a higher ISO is more sensitive to dim light, but which often adds greater digital noise). The choice and balance of these three factors is all important to photography.

We don't actually compute motion blur or depth of field numbers for each picture, instead we learn that action pictures need a faster shutter speed (or a speedlight flash), or that some pictures need more depth of field, stopping down aperture all that is possible. These two ideas conflict about exposure however, and what is possible becomes important, meaning we still have to make all three factors of shutter speed, f/stop, and ISO combine to match the correct exposure (use of Equivalent Exposures). This becomes easy and second nature with a little experience. It does mean we have to think a bit about what we're doing, but we can usually "just know", and then we simply do what is necessary as an automatic reflex. We just know what to do (because we have experience thinking about things before).

Camera design involves math and physics, but still** the very useful purpose of f/stop numbers is the grand concept that the same f/stop, like say f/8, will be the same exposure in any lens**, regardless of focal length, or physical size of construction.

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 online, but we also still see f/stop. I learned to write f/stop, because we also write f/8, and it reminds me of the division defining it, and it surely refers to the "fractional system" originating it (in 1895).

**f/stop number = focal length / aperture diameter**

A simple rearrangement with algebra could be:

aperture diameter = focal length / f/stop number

Note that effective aperture diameter is the frontal dimension “D” in the diagram above, also called “entrance pupil” (Wikipedia). And see special cases of short and long lens diagrams below.

So f/8 is an aperture diameter literally = (focal length / 8) mm. The "aperture diameter" is the diameter of the entrance pupil as seen from in front of the lens (see above diagram). But what photographers need to know is that the purpose of using the f/stop system is so that **the same numerical f/stop on any lens will produce the same exposure**. This is because the diameter of the aperture of a 200 mm lens at f/8 is larger, with 2x the area of the aperture of a 100 mm lens also at f/8.

**aperture diameter = focal length / f/stop number**.

The idea is that f/8 is f/8, the same exposure on any lens.

OK, sure, there can be small minor exposure variations, especially in the old days in lenses without modern coatings. **Reflection losses** on uncoated glass can be 4% or more at each glass surface, but the better modern coatings can reduce this to maybe 0.1%. Lens coatings cancel the reflection, allowing the light to pass through the lens, instead of reflecting it away. Fancy lenses today may contain 12 or 16 glass groups (24 or 32 surfaces), each surface losing a slight amount of light even if with the best coatings, and so suffer more. Lenses today normally have only slight variations, but which are still important in professional movie cameras, when switching lenses on the same scene, mixing scenes shot with different lenses.

**T-stops:** So the professional cinema lenses use
T-stops, with markings which are calibrated to the actual amount of light the lens transmits, instead of the theoretical amount (a T/2 lens actually transmits the light that a perfect f/2 lens should, matching the light meter). This is important because movies might switch lenses in the same scene and keep going, which requires the continued exposures to be closely matched. But this T-stop concept is not considered necessary on still cameras, since each frame is normally stand-alone, and is used individually.

Modern lenses today use improved coatings to improve this tremendously. So today, this is relatively solved for still cameras, if the camera meters through the lens anyway, automatically accounting for any possible variance in the lens losses.

All lenses on all cameras will expose equally if set to the same f/stop (generally, but see T-stops just above). That's what f/stop is, and means, and is designed to do. A handheld light meter simply specifies f/stop, which then works for any camera and lens. A light meter may tell you to use ISO 100 f/16 at 1/100 second, which does not matter which lens or camera sensor you might use.

There were several aperture numbering systems back in the earliest days. Until about the mid-1920s, Kodak cameras still used the "U.S. system" (**U**niform **S**ystem, from Britain in 1880s) of aperture markings, with U.S. 1 starting equivalent to today’s f/4. That ancient U.S. system camera dial was marked 1, 2, 4, 8, 16, 32 (doubling U.S. number is one stop, but doubling f/stop number is two stops). The equivalent exposures are shown this way:

U.S. | 1 | 2 | 4 | 8 | 16 | 32 | 64 |
---|---|---|---|---|---|---|---|

f/stop | 4 | 5.6 | 8 | 11 | 16 | 22 | 32 |

F/stop steps are √2 apart (1.414x steps). At first thought, this old U.S. system doubling the NUMBER being one stop (numbering factors of 2x, like shutter speed and ISO do) might seem an advantage, but then different focal length lenses would give different exposure results. Two photographers standing at the same scene with different lenses probably could not use the same exposure numbers. Wikipedia shows some early history of aperture numbering, and the current f/stop system was invented in the late 1800s, but it took at least 20 years to win out world wide.

The reason to use the f/stop method of numbering aperture is so that **any two lenses** (of any different diameters or focal lengths, even if on different cameras and sensor sizes), will give the same exposure if at the same numerical f/stop. The entire idea is that f/8 exposure is f/8 exposure, in any lens or camera of any size. This also means that hand-held light meters can report a reading usable for any camera.

Another strong benefit of the f/stop numbering system we use is for Guide Numbers used for direct flash exposure. Both f/stop numbering and the Inverse Square Law distance involve full stop steps of the square root of 2. An aperture number like f/4 multiplied by √2 is f/5.656 (which we say rounded as f/5.6), which is one EV less exposure. Distance multiplied by √2 also receives one EV less light due to Inverse Square Law. This directly allows the Guide Number system (specifically, Guide Number = Distance x fstop **Number**) to conveniently and easily take Inverse Square Law into account to determine the exposure of direct light from a flash. If the Guide Number is accurate for one distance, then it can compute any other distance or f/stop number combination that also will give correct exposure of direct flash or continuous local light.

A tricky point: For two lenses at the same f/8, a lens with 3x longer focal length has an aperture diameter 3x larger. But the 3x focal length magnifies the view 3x and then crops it to 1/3 size, so only 1/3x width and 1/3x height is seen, which is 1/9 area, which can only reflect 1/9 of the total light the wider lens sees in the total area. But the longer 3x lens also has aperture 3x larger, which is 9x area, and so now admits 9x more total light (in the total area), which before was 1/9 as much, from a 1/9 area field... so the 9x times 1/9 result is the same total light in both f/8 lenses. But exposure is Not about total light.

*A qualification to prevent misunderstanding: Exposure is based on average Intensity, per unit of area. The “total light” reference here is about the sum of all of the units of area, which does depend on the sensor area.* A larger sensor area does see more photons of light, but that total area is Not exposure. A city block of land gets more total light than any one square meter on it, but the photo exposure is the same for both. Photographic Exposure is only concerned about intensity, which means an average

So, f/8 denotes (focal length / 8), which represents the aperture of the lens (effective diameter as seen through the front lens element). 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 on the dial which marked the 2x area multiples of f/stop. Today in photography, the word stop is used to mean any step of double or half value of *exposure* for f/stop or shutter speed or 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).

There are two concepts of our camera numbers. **"Nominal"** numbers are the numbers actually marked on the camera, which are just simpler approximations of what I call the **"Precise"** numbers of the oal that the camera is actually designed to use (my "precise" term refers to the intended design **goal**, which the camera implements to the best accuracy it can). The camera uses the right numbers, but the marked numbers are shown to us, “rounded” numbers simply made easier for humans.

The next two pages are all about these Nominal and Precise values, with significant detail about the numbers of f/stop, shutter speed and ISO. The large charts on the second following page show all the camera numbers (Precise and Nominal, for f/stop, shutter speed, and ISO).

The f/stop and shutter speed numerical values that we see marked on the camera are approximations called Nominal values (meaning, not literal, existing in name only), which might be rounded or truncated, but are often approximated into a friendly ballpark numbers close enough for humans. It is fully enough that photographers become familiar with only the nominal marked numbers, but any math calculation needs to use the precise values, because the camera does. For example, the nominal number we call 1/500 second is actually 1/512 second. The nominal number f/11 is technically f/11.314. Because to make each stop always be exactly a 2x exposure difference (which concept is sacred to photography), the lens and camera must actually aim for the exact precise values. I don't claim the camera mechanisms are always exactly precisely accurate to a dozen significant digits, but their intended target goal is necessarily the precise calculation of the powers of 2 for shutter speed, and f/stops are powers of √2 (and the cameras today do get pretty close). Specifically, to honor 2x stops, the precise shutter speed values use the binary sequence 1, 2, 4, 8, 16, 32, 64, 128, etc. Also 1/those for faster shutter speeds (1/2, 1/4, 1/8, 1/16, etc).

**Details**: I’m trying to put the math out of the way elsewhere, but nevertheless, this is how camera values are determined. There are large charts here of these values (already computed). And it’s not difficult, hand-held scientific calculators have a x^{y} key on them for exponents. But to answer any obvious questions to explain this method, the stops are simply sequentially numbered 0,1,2,3,4,5, etc, called Stop Number. These are the exponents to create values of the powers of 2 or √2 (to ensure 2x steps of exposure). The Stop Numbers start exponent 0 at the value 1, as at 1 second or f/1 (because any number to the exponent 0 is value 1). A negative exponent gives 1/Value (as shutter speed). The shutter speed Numbers are 2 to power of these exponents, precise powers of 2. However the f/stop Numbers are √2 to these exponents, so the f/stop Numbers increment in steps of √2, but which are still exposure steps of 2x (explained below). Fractional third and half stops simply add 1/3, 1/2, or 2/3 to the exponent (examples here). This method is easy and precise, Not difficult at all.

The precise goal value for **f/stop is (√2) ^{Stop Number}**

(which step Numbers are the powers of √2, with exposure value steps of 2x).

f/stop Nominal | f/0.5 | f/0.7 | 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Precise Goal | 0.5 | 0.707 | f/1 | 1.414 | 2 | 2.828 | 4 | 5.657 | 8 | 11.31 | 16 | 22.6 | 32 |

Stop Number | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

The precise goal value for **shutter speed is 2 ^{Stop Number}**

(which step Numbers are the powers of 2, with exposure value steps also of 2x).

S.S. Nominal | 30 | 15 | 8 | 4 | 2 | 1 | 1/2 | 1/4 | 1/8 | 1/15 | 1/30 | 1/60 | 1/125 | 1/250 | 1/500 | 1/1000 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Precise Goal | 32 | 16 | 8 | 4 | 2 | 1 | 1/2 | 1/4 | 1/8 | 1/16 | 1/32 | 1/64 | 1/128 | 1/256 | 1/512 | 1/1024 |

Stop Number | 5 | 4 | 3 | 2 | 1 | 0 | -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | -9 | -10 |

To add **fractional stops** of 1/3 or 2/3 EV, just add the EV fraction to the Stop Number exponent to compute third stops. Or add 1/2 or x/10 to add halfs or tenths.

**Examples**: f/5.6 + 2/3 EV = (√2)^{(5 + 2/3)} = 1.41421^{5.66667} = f/7.127.

1/125 second - 1/3 EV = 2^{(-(7 - 1/3))} = 2^{-0.666667}= 0.0098431 second, and 1/0.0098431 = 1/101.59 second.

If you wonder about the truth of this, just carefully time your cameras 30 second or 15 second shutter speed. It will be 32 or 16 seconds (powers of 2, which must be exactly 2.0x steps). You do want it to do 2x exposure steps from 8 seconds.

The method for **ISO** is also powers of 2 like shutter speed, but today, the specific ISO numbers are modified to make ISO 100 be a full even stop (matching the APEX method). See the math page for the longer story of the ISO numbers.

Nominal values are just various arbitrary approximations (established by convention of about 100 years), and so cannot be computed. Therefore the precise goal values cannot be computed from Nominals. The Precise goal values are computed from the Stop Number sequential order into the binary powers of two, as the goal targets that the cameras strive to achieve. The f/stops are mechanical mechanisms, but today, shutter speeds are controlled by a digital crystal clock chip.

The integer full f/stops of f/1, 2, 4, 8, 16, etc are the actual precise goals (actual powers of 2). And the shutter speed full stops in the range from 1/8 second to 8 seconds are precise values (actual values of 2). The rest of the nominals are approximations, more or less rounded some way (from say f/11.314 or 1/512 second).

**Why the numbers like √2 and f/11.314?** Shutter speed and ISO values increment EV in powers of 2 (double value is one stop), but f/stop NUMBERS increment EV in powers of √2 (double NUMBER is two stops, but one stop is still a factor of 2x exposure).

**f/stops**: The circular aperture areas for the exposures of one EV steps are exact powers of 2, but the f/stop NUMBER is a power of √2. Which is a key factor for the circular f/stop aperture because the area of a circle is Pi × r². Doubled area is 2 (Pi × r²) = Pi × (√2 × r)². So multiplying the aperture radius by √2 doubles its area and exposure. The value of √2 is 1.41421.

**f/stop Number = focal length / (effective) aperture diameter.** Full stop f/stop Numbers are integer steps of powers of √2, which numbers are proportional to the area of the circular aperture. The purpose of using this f/stop numbering definition is that then any camera and lens using f/8 is the same exposure as all other cameras at f/8 (if also at same ISO and shutter speed). The idea of EV is that ± 1 EV is exactly 1/2x or 2x the exposure, and to be precise, calculations need to use the precise setting numbers to compute EV precisely.

The numeric values f/1, f/2, f/4, f/8, f/16 are nice even precise values, but the alternating values between them are odd exponents of 1.414, not producing integers. We simply say these as rounded (in various ways) for convenience (called nominal values), but the camera is designed to aim for the precise goal values (see charts of all these values, on a following page). the full f/stops are each exactly 2x exposure, but to do that, the f/stop NUMBERS are powers of √2.

For example, the cameras and light meters are marked f/11, and we say it and think it as f/11, but f/11.314 is the necessary correct actual calculated value. This is only about 0.08 stop difference, but any difference exists only in our mind, since the camera is designed to always do it right. This is not a large difference, and most other nominal f/stop values are closer, but precise calculations should use f/11.314 instead of f/11. A few users don’t want to believe their prized camera would do such a thing, but the camera is in fact designed to do it right, and this little difference is nothing to a photographer. But the math and the design must pay a little more attention.

As another simple way to show this fact is obviously true (that f/11 is actually f/11.314), note that f/stop numbering is the sequence of √2 intervals, (which are 1.414 numeric intervals), making **every other stop number** be a multiple of 2. This sequence of 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 |

On either first or second row, each “every other” number is exactly 2x the previous. It can be handy to realize that doubling any f/stop Number (for example, f/7 to f/14) is exactly two stops. The sequence 1, 2, 4, 8, 16, 32 etc are exact precise numbers, but the others are more unwieldy, and are marked with approximate numbers. But the camera knows to work with the right values.

Shutter speed marking numbers are also similarly 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 design does use the correct value (third stop 20 seconds will be 20.16 seconds, and half stop 20 seconds will be 22.6 seconds, see standard shutter speed charts on next page). The camera design does it right, but we humans are shown easier approximated numbers. Unless we're doing calculations, we really don't care about the precise numbers. But each stop being exactly 2.0x exposure is very important to us.

The f/stop system works this way:

The focal length affects the magnification of the field of view. Doubling the focal length zooms in to half of the view width.

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 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 equal 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 exposure, on any lens on any camera**. The reading from our light meter is applicable exposure for any lens and any sensor.

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 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 equal 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/stop number = focal length / aperture diameter**

What this means is:

- A 200 mm lens at f/2.8 has an effective aperture diameter of 200 mm / 2.8 = 71 mm, which is 2.8 inches diameter. If it were f/2, the aperture would be 200/2 or 100 mm or nearly 4 inches diameter (large, heavy, expensive, but lots of light). The "aperture diameter" is the diameter of the entrance pupil as seen from in front of the lens (diagram at top of this page).
- But if at the same f/stop aperture, every lens transmits the same amount of light onto every sensor (true of any focal length, and any sensor size). A light meter reading that specifies f/4 will work at f/4 on every lens and every sensor.

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 x Pi r² = Pi (1.414 x r)² , so 1.414x radius gives one stop. √2 is 1.414. This is why f/stop numbers must increase by 1.414 to represent one stop of 2x exposure.

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.

For double area: 2 x Pi r² = Pi (1.414 x r)² , so 1.414x radius gives one stop. √2 is 1.414. This is why f/stop numbers must increase by 1.414 to represent one stop of 2x exposure.

Since

Inversely, when the diameter and area are made larger, the f/number from the ratio f/d becomes a smaller number.

Cameras today also have 1/3 stop or 1/2 stop increments. F/stop Numbers are numbered in powers of √2, which is convenient in both usage and the math. But Exposure is still powers of 2, meaning 2^{EV} = fstop^{2}/time duration. So log_{2}(fstop^{2} x shutter speed) is the EV number of the exposure, and equal EV numbers are Equivalent Exposures.

**Full f/stop numbers** advance in steps of 1.414x numeric multiples (f/1, f/1.4, f/2, f/2.8, f/4 ...) 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 times 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 exposure.

**Half f/stop numbers** advance in multiples of the square root of √2, or 1.1892x times the previous (speaking of f/stops).

Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the exposure.

Less is more. Less f/stop Number is More exposure. f/4 is more exposure than f/8.

Lens manufacturers seem to truncate numbers instead of round off. For example, f/5.6 is actually 5.66, and f/3.5 is 3.56. Except 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 easy nominal numbers to show humans. The lens and camera are designed to do it right.

Trying here for definitions instead of math, but don’t plan calculations using the fractional stops shown down here (of X times the previous step). That’s accurate, but awkward (has no base), and is just for information. To compute fractional stops, the first above method is the best way, using the Stop Number exponents.

The values of shutter speed and ISO are linear scales (square root of 2 is Not involved), 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) times 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 exposure.

The next **half-stop shutter step** is square root of 2 (1.4142) times the previous value.

Every two half-stop steps (from any point) is exactly 1.0 stop and a 2.0x change of the exposure.

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.)

Again (f/stop, shutter speed, ISO), the nominally marked numbers may not be the exact precise values, but the camera knows exactly what to do.

**ISO** is a bit tricky today. 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 farther down in the manufacturers’ data, it shows the precise values used, ISO 252 or ISO 2016. (The ISO base is considered to be 100 today, using ... 100, 200, 400, 800 instead of 1, 2, 4, 8 (otherwise, if starting at 1, ISO 100 would 101.6 and a third stop less than full stop ISO 128). This makes third stops of 252 and 2016 instead of full stops 256 and 2048, that we call 250 and 2000.) Auto ISO is probably using 1/6 stops, which are steps of the sixth root of 2. 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 ourselves, using the nominal numbers instead of the actual precise numbers. For this new ISO numbering, see the math page.

Marked | Shutter Speed | Precise |
---|---|---|

30" | 30 seconds | 32.0 sec. |

2" | 2 seconds | 2.0 sec. |

2 | 1/2 second | 0.5 sec. |

30 | 1/30 second | 1/32 sec. |

1000 | 1/1000 second | 1/1024 sec |

See full charts of these |

Shutter speed is 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 30 or 30". Just the number alone, like 30, is an implied fraction (1 over the number), meaning 1/30 second. The same number written as 30" means 30 whole seconds, not a fraction. However, these are nominal markings, and 1/30 nominal is actually precisely 1/32 second (precise values run 1,2,4,8,16,32 seconds or 1/ those values). A slow shutter is a longer duration, and a fast shutter is a shorter duration.

A flash, especially a speedlight flash is typically faster (much shorter duration) than the shutter is capable. 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.

**Aperture** is not exactly the obvious physical diameter (shown by black vertical lines in the image above), but instead is it as seen through the magnification of the front lens elements as the apparent "working" diameter D above (named **Entrance pupil**). The physical aperture diameter is designed accordingly. Stopping down (to use a larger f/stop number like f/16) is one factor increasing Depth of Field, and opening wider (to use a smaller f/stop number like f/2.8) passes more light and increases exposure. f/0.5 is considered the theoretical limit for f/stop in camera lenses to be able to still focus in air (refractive index of 1). However, in practice, more reasonable practical limits are typically f/1.4 or f/2.8, due to diameter affecting size, weight, cost and image quality.

**Focal length:** Focal Length is the distance from the focal node in the lens to the sensor plane when the subject is in focus. The Focal Length number **marked** on the lens applies when focused at infinity. Focal length changes as we focus closer (focal length normally becomes longer if front elements are extended to focus closer, or if internal focusing focuses closer). This change is relatively minor if at focus distances greater than a few feet, but at 1:1 macro, the focal length becomes 2x longer than marked, and equal to the working distance in front of the lens (specifically, to the front Principle Point H). The actual focal length is measured to the rear Principle Point, H', as shown above. Due to varying focus distance, the focal length marked for infinity is not always necessarily the exact distance to the sensor plane.

The front **Principle Point** H is where the vertex of the **Angle of View** through the Entrance Pupil is located. Same with H' and Exit Pupil. The front Angle of View from H is necessarily the **same** Angle of View at H' back to the sensor (so, the sensor size and focal length determine the numerical Angle of View). The Principle Points are on the designer's apparent planes where the subject and sensors images appear to be. Design of lens elements can move these points, and both H and H' points are often inside the lens, but in fact, they can often be moved literally outside the actual lens, either in front or behind the lens. They are moved by adding convex lens elements that converge, or concave lenses that diverge. Zoom lenses greatly complicate this.

Wikipedia can show the definition of the Principle Points (H and H') of a lens. In the classic Thin Lens model (which is a hypothetical single element lens, like a simple magnifying glass, or a pinhole), the points H and H' are at the same point in the center of the single element. In an actual multi-element camera lens, these points move apart, with more glass elements (with different index of refraction) used to correct color aberration in the lens, which relay the image between the two points.

Panoramic photos (combining multiple photos) show the most accurate perspective (of close subjects) when the panoramic camera is rotated on an axis through the H Principle Point. Panoramic articles describe ways to determine this pivot point they call Nodal point, or sometimes Perspective point.

Below are diagrams of two (50 year old) Nikon prime lenses (early Nikon F era. Zoom lenses are more complex today). The node H is the frontal node (nodes are called Principle Points), which is the convergent point of the front angle to the distant Field of View. The node H’ is the rear focal node, the convergence point of the rear angle to the sensor. A significant point is that in the wide angle lens here, you can just about see the front angle (as clearly as if it were drawn) from the H node through the largest diameter of the front elements to the distant Field of View. And in the telephoto lens, you can just about see the rear angle drawn from the H’ node through the largest diameter of the rear element (to the sensor). These two angles are equal. The intermediate elements transfer the image between these nodes (and may move to assist focus).

Telephoto - H' is moved to be slightly in front of the lens.

In **telephoto** lenses, the sensors H' point is designed slightly in front of the front lens element, because, the actual optical technical definition of "telephoto" is that the lens is made to be physically shorter than its focal length (which is a practical way to build a smaller long lenses). The sensor H' point is the focal length dimension, but in just front of the front lens element.

"Telephoto" does not actually mean the subject is at long distance, since wide angle lenses focus at infinity too. Telephoto refers to moving the H' point forward, to design a camera lens physically shorter than the focal length, which is typically used for camera long focal lengths, but telescopes use entirely different designs.

Retro-focus wide angle, H' is moved to be behind the lens.

**Wide angle** lenses are normally retro-focus (for SLR, DSLR), which means the rear node H' is designed 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 20 mm lens would block raising a mirror 24 mm tall.

Here is the standard and classic **Thin Lens Equation** used in many lens calculations:

Actual focal length can be determined by the Magnification (Wikipedia). The distance from the front nodal point to the object in the subject plane (s1), and the distance from the rear nodal point to the sensor plane (s2) are related by this Thin Lens equation (Wikipedia). If OK with a little geometry and algebra, you can see the derivation of the Thin Lens Equation at the Khan Academy.

In this equation, we can see that if the subject at s1 is at infinity, then 1/s1 is zero, so then s2 = f. This is the marked focal length that applies when focused at infinity.

If at 1:1 magnification, then 1:1 means that s1 = s2, saying that the 1:1 working distance in front of the lens is equal to the (extended at 1:1) distance to the sensor image plane, and 2s = f, so at 1:1, f becomes 2x marked focal length.

**Field of View:** The field of view of the lens is determined by the angle from the rear principle point H' (focal length) back to the sensor size dimensions (sensor size is a major factor of field of view). Then the same angle from the subject principle point H determines the field of view forward to the focus distance. Focal length and sensor size are the factors of angular field of view. This is the purpose that sensor size determines "Crop Factor", which is about the field of view seen by the same focal length on different sensor sizes.

But more is possible, and basic geometry (similar triangles) is that these ratios are equal:

Size of sensor (mm)

focal length (f, mm)

focal length (f, mm)

Size of Field of View **

Distance to Object **

Distance to Object **

****** Feet, meters, miles, km, yards, cubits or parsecs (but both are the same units)

This formula is Not accurate at macro distances or for fish eye lenses.

For Field of View, more at Field of View math.

**Closest focus distance:** In the above “Thin Lens”, the value d is the Working Distance (in front of the lens, but technically from the node, not from the glass), **but “Focus Distance” is the distance to the image sensor**. For example, B&H and Nikon report the closest focus of the 105mm VR f/2.8 macro lens to be s at 12 inches (305 mm), but at 1:1, the working distance in front of the lens is d, only 140 mm (5.5 inches). The lens length is 114 mm and the Nikon F mounting flange to sensor distance is 46 mm, plus 140 mm in front totals 300 mm, or distance s. OK, that is 300 mm vs 305 mm, but technically, the Thin Lens computes to the Principle Point in the lens (which we don’t know where it is, but it is a tiny difference except for macro work.) But the point is, focus distance and working distance are very different numbers at macro distances. There is a marking (looks like Greek Theta θ) on the camera near the rear of the top LCD to mark the sensor plane, to which Focus Distance is measured. This common "Minimum Focus Distance" specification is NOT "Working Distance" in front of the lens, which is actually measured to the H principle point, likely inside the lens (not to the filter ring).

The focal length marked on the lens applies only when focused at infinity. When focused closer, the focal length lengthens longer (some internal focusing lenses can be exceptions). Since f/stop number depends on focal length, regular lenses typically will not focus closer than around 0.2x to 0.15x magnification (altering f/stop number). The Closest Focus Distance specification also provides the Magnification at Closest Focus (like maybe 0.15x), which in turn provides the maximum internal extension, shown in the Extension Tube Formulas as Internal Extension = Focal length x Closest Magnification.

**Elements and Groups:** Complex camera lenses are designed with several individual glass elements. An element is an individual piece of optical glass, with curves (convex converges the light toward center, or concave diverges the light, seen in the picture at page top). A single element is counted as a group of one, but sometimes two elements are cemented together to combine into one group (specifically, with no air boundary between them). These doublet group elements (called achromatic doublet) each have a different Index of Refraction, to bend color wavelengths differently to correct color aberration (see Wikipedia). The wide angle lens at right just above has nine elements in seven groups. That telephoto lens has four elements in three groups.

**Magnification:** Binocular and telescope magnification numbers are a different system than cameras, being "viewing devices", and their "x power optical magnification" number is relative to the size our naked eye sees at 1x. It could be called “apparent magnification”. The angular size of the full moon is 0.5°. In binoculars with 10x magnification it appears 10x larger than the bare eye sees it, an apparent 5 degrees size. If this viewing device uses a magnifying eye piece (like binoculars and telescopes use), then the magnification is (main lens focal length / eye piece focal length). So the long focal length main objective lens magnifies, like a camera lens, and the short eyepiece magnifies that. But if eyepiece were also the same focal length as the objective lens, that is a magnification of 1, or same non-magnified size as the naked eye would see.

Camera lens magnification is a different system of numbers, being reproduction devices (the eye does not see the actual original lens image). **In cameras**, image size on the sensor can be measured (like on film, so to speak), and image magnification is computed as *object size on sensor / the actual size of the original object*. Or alternatively, in the geometry, *the focal length / subject distance* gives the same number (but focal length varies with focus distance). It could be called “actual magnification”, and is normally fractional, quite small, normally a size reduction (magnification at infinity is virtually zero). In macro work, if the image size on the sensor is the same size as the real life subject, it is 1x magnification, which is called 1:1. So 1:1 also necessarily implies the subject distance is same as the effective focal length (distance on either side of the lens is the same, both sides are measured from the focus nodes of the lens).

A telescope or binocular objective lens is normally fixed focus and fixed aperture, and any focusing is normally done by moving the rear eyepiece back and forth to find focus. But if no eyepiece lens is used (if the telescope is attached like a camera lens, called prime focus photography), then focus moves the camera sensor position, and the normal **camera lens Magnification = focal length / subject distance** applies. Or the ratio of sizes also applies. If the Moon is 3474 km diameter (basically at infinity), and if its image is 0.5 mm, that's an extreme size reduction, and not likely a meaningful number.

With a consumer camera on a telescope, it is sometimes seen as an attempt to compare it with the ratio that if a 50 mm lens is assigned a 1x magnification (but with an entirely different meaning of 1x), then a 2000 mm telescope directly attached as prime focus lens might be said to give 2000/50 = 40x magnification (relative to what a 50 mm lens sees instead of our naked eye. That's only an approximation based on a 50 mm lens being the "normal lens" if on a 35 mm film body, and only has any significance in that context Because in a different sensor size situation, 50 mm and its field of view may not have meaning to your sensor. A different sensor size would be a different situation, but still in this 2000 mm case, 2000/(your comparison lens focal length) would give a meaningful comparison size number of those two lenses. That's all the 50 mm comparison tries to do, but many fewer people use 50 mm camera lens today. Compact and cell phone camera lenses are normally about 4 mm. Use your own number there.

In a camera, if a real subject that is physically 100 mm wide is reproduced to be 10 mm wide on the sensor image (on the film, so to speak), then the magnification is 10/100 = 1:10 = 0.1x reproduction ratio (actually a reduction). Which 1:10 also happens to be the same ratio as the distances behind and in front of the lens nodes then, also necessarily 1 to 10 distances.

If at 1:1 reproduction ratio (macro), then 1:1 means equal sizes, both the real life subject and in the camera sensor image. So then at 1:1, the "working distance" in front of the lens (in front of point H) is necessarily equal to the distance behind the lens (the modified focal length, behind point H'), due to similar triangles, etc. Seems a cute fact, which aids understanding, however (today, with zoom lenses and internal focusing which shifts thing inside), we probably are not told the exact locations of H and H' (which change positions with zoom and focal length).

Continued: There is much more detail about the setting numbers on

the second page, with a calculator of f/stop, shutter speed and ISO differences, and

the third page, with charts of the nominal and precise setting numbers, and

the math page with a few hints about calculations with these numbers.