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). Some basic lens properties are at page bottom too.
The Camera Numbers: This page 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.
Camera design involves math and physics, but still the very useful purpose of the f/stop numbering system is the grand concept that the same f/stop, like say f/8, will be the same exposure in any lens or camera, regardless of focal length, or physical lens size or construction.
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.
Is f/stop written f/stop or f-stop or fstop? The lens manufacturers properly write f/8. The internet changes things, and the other terms are also seen 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 this "fractional system" originating it (in 1895).
A larger f/stop Number (f/32) is a smaller aperture which admits less light.
A smaller f/stop Number (f/2.8) is a larger aperture which admits more light.
The f/stop numbers are fractions of the focal length. The nomenclature is f/ denoting focal length and the division.
Note that the "effective" aperture diameter is the frontal dimension “D” in the diagram above, called “entrance pupil” (Wikipedia). That's the size of the "effective" aperture opening as seen from the front of the lens. The optical lens elements usually magnify its apparent size somewhat. The physical aperture inside the lens is designed to that frontal definition, which itself will measure differently to produce the correct entrance pupil. The maximum aperture generally uses the full diameter of the glass lens, and stopped down is a smaller diameter. In a 200 mm lens, a f/2.8 aperture is 70.7 mm diameter (2.78 inches), and f/32 aperture is 6.25 mm diameter (1/4 inch). See special cases of telephoto and wide angle lens diagrams below.
So f/8 is an aperture diameter dimensioned as literally a fraction of the focal length. A 50 mm lens with a 25 mm opening diameter is a fast f/2 lens. But a 25 mm aperture on a 200 mm lens is f/8, much slower, four stops slower. However (the main point), if the same f/stop number were used on both lenses, then both are the same exposure, in any lens on any camera. So what photographers need to know is that the purpose of using the f/stop numbering system is so that the same numerical f/stop on any lens will produce the same exposure, on any camera. And note that a hand-held light meter produces exposure settings that work on any camera.
The overwhelming idea is that f/8 is always f/8, and is the same exposure on any lens on any camera.
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, 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.
But modern lenses generally solve this for still cameras today. At worst for still cameras, it's just another tiny variable. And today, most cameras meter from behind the lens anyway, automatically accounting for any 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.
There were several aperture numbering systems back in the earliest days. Until about the mid-1920s, Kodak cameras still used the "U.S. system" (Uniform System, 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 EV stop, but doubling f/stop number is two stops). Said another way, U.S. had 2x steps for what we call 1 EV, which must have seemed reasonable, and f/stop has √2 steps for 1 EV, which is trickier. The equivalent exposures are compared this way:
f/stop number = focal length / aperture diameter
U.S. number = (f/stop number)² / 16 (some shared physics, but surely coincidental, since U.S. came first)
There were a few different early systems, but in this old U.S. system, doubling the NUMBER was one stop (2x numbers were exposure factors of 2x for 1 EV), so it would seem an advantage was that this made aperture numbers be like shutter speed and ISO, in that double value is one EV stop in all three systems. But f/stop steps are √2 apart (1.414x steps), so doubling f/stop numbers is two stops.
Wikipedia shows a few early historical versions of aperture numbering, until the current f/stop system was invented in the late 1800s (some define it as 1895, but the beginning was more smaller steps), but then it took at least 20 years to win out world wide. The reason the f/stop method of numbering aperture was the final design is because then any two lenses at the same f/stop, even of any different focal lengths or diameters, even if on different cameras and sensor sizes, will give the same exposure.
Therefore f/stops also mean that hand-held light meter readings are usable in any camera. The entire idea is that f/8 exposure is f/8 exposure, anywhere, in any lens or camera of any size.
Another strong benefit of the f/stop numbering system is for Guide Numbers used to determine direct manual flash exposure. For a correct flash exposure, Guide Number = Distance from flash x fstop Number. Both f/stop numbers and the Inverse Square Law distance involve 1 EV steps in units of the square root of 2. The beauty is that since both factors are √2 units, the Guide Number system conveniently and easily takes Inverse Square Law into account, which is a great simplification. Knowing either distance or f/stop, Guide Number easily computes the other. The Guide Numbers are usually printed in the flash unit manual, and then the equation can compute any distance or f/stop number combination that will give correct exposure of direct flash. For direct continuous local light too, we can compute Guide Number from any one correct exposure, and then it works for any distance (for that same light situation).
There are two concepts of our camera numbers. "Nominal" numbers are the numbers actually marked on the camera, which are just simpler rounded approximations of what I call the "Precise" numbers of the Goal 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 two numbers are pretty close, but the precision goals are used to ensure that 1 EV is precisely 2.000x the light. But the marked numbers are shown to us, “rounded” numbers simply made easier for humans. We specify the Nominals, but the camera knows to use the proper values. Because unless we're doing calculations, humans don't really care about the precision of the numbers. However each stop being exactly 2x exposure is important to us, so the "precise" numbers are those that do that job.
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, including third, half, and even sixth and tenth stops).
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 (and the Windows CALC) have a yx key 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. The Stop Numbers are simply 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 in 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 shown next). 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, (= 1.414) with exposure value steps of 2x).
Examples: (√2)4 = f/4. (√2)5 = f/5.657
|Full stop f/stops|
The precise goal value for shutter speed is 2Stop Number
(which step Numbers are the powers of 2, with exposure value steps also of 2x).
Examples: 24 = 16 seconds. 2-4 = 0.0625 seconds, and 1/0.0625 is 16, or 1/16 second
|Full stop Shutter Speeds|
To add fractional stops of 1/3 or 2/3 EV, just add or subtract 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 + 0.667) = 1.414215.66667 = f/7.127.
1/125 second - 1/3 EV = 2(-(7 - 1/3)) = 2-6.666667= 0.0098431 second,
and 0.0098431 = 1/101.594 second, a third stop called Nominal 1/100 second.
If you wonder about the truth of this Nominal vs. Precise business, just carefully time your cameras 30 second or 15 second shutter speed. It will be 32 or 16 seconds like the charts (powers of 2, like 1, 2, 4, 8, 16, 32, 64, 128 ... in order to be exactly 2.0x steps).
The method for ISO is also powers of 2 like shutter speed, but today, the specific ISO numbers are shifted a bit to make ISO 100 be a full even stop (matching the APEX method), instead of ISO 101.594 which the 100 used to be a third stop less than 128. 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, f/2, f/4, f/8, f/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:
|Precise Design Goals of full f/stops|
In both rows for full f/stop numbers, each every other number is exactly 2x the previous. From row to row, each number is 1.414x (√2) the previous, and each is exactly 1 EV from 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 but are the necessary 2x values which are the Precise Design Goals, but are marked with approximate numbers called Nominal values. 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 precision of the numbers. But each stop being exactly 2x exposure is important to us. We specify the Nominals, and the camera knows to use the proper values.
The f/stop system works this way:
f/stop number = focal length / aperture diameter
What this means is:
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 2EV = fstop2/time duration. So log2(fstop2 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, in that less f/stop Number is More exposure. f/4 is 8x more exposure than f/11 (2x 3 times).
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.
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.
|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 sensor at 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 from the focused point back through the Entrance Pupil is located (literally the angle from H through the entrance pupil to determine the subject plane at F. Same with H' and Exit Pupil back to fill the sensor at F'.) 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 F and F' are on the designer's apparent planes where the subject and sensor 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 concave lens elements that converge, or convex lenses that diverge. Zoom lenses greatly complicate this.
Wikipedia shows 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 and internally focusing 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).
The external angle and the rear angle are always equal angles, because the image is what will fit on the sensor. The lens projects what its Focal Length sees onto the sensor, and Field of View is what the sensor size can capture.
In a more typical lens (neither telephoto or wide-angle), both the H and H' nodes are normally more or less near each other and more or less near the aperture diaphragm. Like for example, in the lens image at the top of this page. The H node covers the dimensions of the field, and the H1 node covers the dimensions of the sensor.
The Thin Lens model puts H and H' at the same point in the center of a one element lens (like an ordinary magnifying glass).
This image is an example showing how Field of View is calculated with the Thin Lens model. It makes calculations much simpler, the same for every lens, instead of all the various actual thick camera lenses (which would be very difficult). They will work the same. For Field of View math, more at Field of View math and at Math.
Then 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. That is the focal length that is marked on the lens body, which applies when focused at infinity. Focusing closer typically moves the front elements forward, which moves H' forward, so then the focal length is a bit longer (a lens with internal focusing instead moves the central elements). Then at 1:1 magnification (when the real life subject and its image on the sensor are the same size in mm), then the angles and distances are the same in front and back of the lens ... 1.1, and the focal length is typically 2x what is marked on the lens body.
If focused at 1:1 magnification, then the 1:1 means that s1 = s2, saying that the 1:1 working distance in front of the lens is equal to the (focal length extended at 1:1) distance to the sensor image plane, and 2s = f, so at 1:1, f becomes 2x the marked focal length.
Basic geometry (similar triangles) is that these ratios are equal:
** Feet, meters, miles, km, yards, cubits or parsecs (but both must be the same units).
If you know three of these values, you can solve the equation for the fourth.
Angular Field of View: In the same diagram above, Trigonometry allows the Field of View of the lens to be calculated from the angle from the rear principle point H' (focal length) back to the sensor size dimensions. The angle at the sensor is all that the sensor can capture and show, so the distant field can only be the same angle (Equal Angles). If all else is the same, a longer lens zooms to show a more narrow field angle and a magnified subject. The Field of View angle is determined by the sensor size and focal length, but a shorter focal length or a larger sensor or a larger subject distance allow a wider dimensional distance of field of view. Sensor size is a major factor of Field of View. Cameras with tiny sensors must use a much shorter focal length lens to see a comparable extent of the field. The 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 (a Field of View calculator). That is the reason that sensor diagonal size determines Crop Factor, which is about the same (equivalent) Field of View seen by a different focal length on a different sensor size (typically compared to 35 mm film diagonal size).
This formula is Not accurate at macro distances or for fish eye lenses. At macro distances, the focal length is longer than marked on the lens. Macro math typically uses resulting image size instead of distance.
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 in complex lenses, not from the glass), and “Focus Distance” is typically 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 again, macro math typically uses resulting image size instead of distance, and technically, the Thin Lens computes to the Principle Point in the lens (which we normally 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 (an O with a horizontal line through it) 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 overall, the camera lens is normally a fractional size reduction (the 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 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 the normal camera lens Magnification = focal length / subject distance applies. Or the ratio of sizes also applies. If the Moon is 3474 km diameter (and basically at infinity), and if its image is 1 mm, the reproduction is 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.