| f/1 | 1.4 | 2 | 2.8 | 4 | 5.6 | 8 | 11 | 16 | 22 | 32 |
At first this may look like a random series of numbers, but there’s actually a method to this madness, and memorizing this series can help you make quick adjustments to exposure when switching between f-numbers.
What makes this series so special?
The f-numbers in this geometric series are known as whole stops. And, the thing that makes them special is that each number in the series transmits exactly twice or one-half the amount of light of the neighboring f-number.
For example, f/5.6 transmits exactly twice as much light as f/8. This means that if you’re shooting at f/5.6 and want to switch to f/8, but still keep the same exposure, then you’ll have to make your shutter speed exactly two times slower (because at f/8 you’ll be transmitting two times less light). If you were shooting 1/100 sec at f/5.6, then at f/8 you should shoot at 1/50 sec.
Having this f-number series memorized will allow you to switch between f-numbers and quickly calculate the new exposure.
Your camera probably also has intermediate f-numbers between these whole stops (usually in increments of one-half or one-third stops). So, memorizing the whole stop sequence will also help you calculate the exposures for these intermediate stops when switching between f-numbers.
Where do those strange numbers come from, anyway?
The f-numbers are calculated as a ratio between focal length and aperture diameter:
f-number = (focal length) / (aperture diameter)
For example, if you’re shooting with a 100mm lens (the focal length), and the aperture diameter is set to 25mm, then the f-number will be 4 (100/25).
At first you might think that doubling the aperture diameter would allow twice as much light to be transmitted, but it doesn’t work this way because doubling the aperture diameter would more than double the surface area of the aperture.
It’s the surface area that needs to double when you want to transmit twice as much light, and to double the surface area of a circle, you have to multiply its diameter by the square root of 2 (which is about 1.414). And, this is why the whole stops are incremented by a factor of 1.414.
2.8 = 2.0 * 1.414
4 = 2.8 * 1.414
5.6 = 4 * 1.414
…and so on…
Confusing enough?
I realize this is probably the first time I’ve brought up math in a post, so let me know if any of this makes no sense, and I’ll be happy to clarify as much as I can in the comments 🙂
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About the Author: Steve Berardi is a naturalist, photographer, computer scientist, and founder of PhotoNaturalist. You can usually find him hiking in the beautiful mountains and deserts of Southern California.
More math of photography posts, please! Do you have any mnemonic devices to remember that sequence of numbers?
Thank you for explaining that so succinctly, Steven. I’ve been wanting to play with the maths of photography, and this was a gentle introduction.
All the best,
Edie
Here’s a sort of mnemonic device I came up with: you can split the sequence into two sequences that interweave. Look at every other f/stop and half of them are very simple–just start with one and double the number as you go; 1, 2, 4, 8, 16, 32. However, between each of these simple to remember numbers is a hard to remember number. The hard ones are not as bad as they seem at first if you just remember the first one, 1.4, then double that as you go with a little rounding. 1,4 doubled is 2.8, 2.8 doubled is 5.6, doubled and rounded off is 11, 11 doubled is 22.
It takes a little bit of mental gymnastics to weave the two series together, but with practice it gets easier. All you need to remember as an arbitrary number is 1.4 and you can work the rest out as needed.
@Edie – I’ll be happy to write more about math–it happens to be one of my favorite subjects 🙂 For memorizing this series, I think Tim makes a great suggestion and that’s what I did at first too. Soon enough though, you’ll be able to memorize this series and recall it instantly. Although, I’m not too good with numbers past f/16.. but I rarely use anything smaller than that anyway.
@Tim – thanks for providing that great tip! I use the same trick for those large f-numbers that I don’t use often.
Great post! Just what I needed. A short, straight to the point explanation of these numbers. Thanks!
Wonderful Article..I was thinking from where these numbers came from…and now I got the answer…Expecting more maths explanations of photography…
Also, Tim Smith’s comment also worth reading….nice. Thanks for sharing.
F-numbers
Good article covering just the right amount of info – enough to convey a concept, but not overwhelming. I appreciate the math as an explanation for photographic techniques. I may let the camera calculate some of this, but it is good to know for special occasions or “emergencies.” Thanks.
great explanation. short and sweet. no less no more… keep posting..thanks,
santosh gaikwad
This is a good explanation of “how,” but I am sorry to say this, but this still doesn’t give me the “why.”
I usually shoot on aperture priority, sometimes on shutter priority, and I make liberal use of exposure compensation. If, for some reason, I need to shoot on manual, I first put my camera in aperture or shutter priority, get the exposure right there, and then I move to manual and put in those same settings. Or I just fiddle to get the exposure right in the in-camera light meter.
I did memorize this stuff at one point, but because I so rarely have any need for it, I no longer have it memorized. I’m stsill not sure why I should.
@Diedre–Great question. I find thinking in full stops valuable, whether they are shutter, aperture, or iso, because once you get comfortable with it, working in full manual mode cuts through a huge amount of camera complexity. Suddenly you are able to quickly and compentently operate any camera that has manual mode; with digital cameras that means most advanced cameras, and with film that means thousands of great older classics.
I find aperture or shutter priority to be beneficial when there is simply no time to make fiddly adjustments — the shots are coming too quickly and I have to respond instantly. Without that time pressure however, I am much more likely to get the shot I have in mind (out of focus background or background included, motion blur or stopping fast motion) if I shoot in full manual. Motion blur, camera shake, depth of field, ISO noise can all make for very different results.
When shooting in one of the priority modes exposure is semi-automated, but this doesn’t mean that these automated changes can’t profoundly affect the nature of the image. With aperture priority you can easily find yourself shooting at shutter speeds that are too slow for hand-held, and in shutter priority you might not want full depth of field, or extremely shallow depth of field depending on the image you have in mind. You determine the setting of your priority manually, but you may not want all possible adjustments to the automated setting.
Exposure compensation can be great, but it might be important, depending on the situation, to know what is happening to the image when you dial in that compensation. Are you changing the depth of field? How much? Have you just lost the ability to prevent a blurring of the leaves in the breeze? Knowing the major f/stops allows you to make appropriate adjustment to aperture, shutter speed, or iso to compensate for automated adjustments that take the image out of the realm you have in mind.
When it somes to setting f/stops I tend to think in terms of three regions. Shallow depth of field being the open stops, maximum depth of field be the smallest stops (highest numbers), and greatest sharpness being the middle region (f/8 or f/11, generally). I know this is a very broad brush, but I find it useful for just that reason. In the open and middle f/stops focus can be critical, but in the smaller stops it is much more forgiving due to the greater depth of field. If I am working in shutter priority and move into the shade I can suddenly find myself with minimal depth of field — if that happens I will probably want to know it.
On the other hand, it can be very liberating to shoot on P, or with a Holga and enjoy the surprises! Being comfortable with full manual stops just gives me greater ease with precise control when I need it.
I apologize for this delayed response, but I’ve been camping in the desert the last couple of days, so I’m just now getting back to a computer 🙂
@Deirdre – I think Tim answered your question pretty well, so I’ll just add one little thing: knowing this f-number series can help you think quickly about the different options you have when making a photograph. For example, let’s say you’re photographing hummingbirds and shooting handheld: you try shooting at f/5.6 and 1/1000 sec but then realize that f/5.6 isn’t getting you enough depth of field, so you want to shoot at f/8 but then you know that this will change your exposure to 1/500 sec (which is a little too slow for handheld shots and a long lens), so to keep that 1/1000 sec shutter you know you have to make up for that stop somewhere else now (like ISO).
@Tim – thanks so much for that great response!
For me, the underlying key to exposure is knowing that there are only three things to adjust; f/stop, shutter speed, and ISO. The combination of the three makes for the chosen exposure. If you can think of each of them in full stops (doubling and halving the light) then it is easier correct for an adjustment you make to any one of the three because you want to change the nature of the picture, by then making the best compensating adjustment to bring the total exposure back to what you want.
Add to this understanding the metering mode (spot, center-weighted, matrix) and focus and that is pretty much it for shooting without flash–except for large format type camera movements. All the other menus, buttons, wheels etc. become redundant. My DSLR has over a hundred settings menus — that menus, each with multiple options. Very few of them add any value over the five basic settings (f/stop, shutter speed, iso, metering mode, focus.)
Thank you, Steve, for letting me know there were responses to my questions, and thank you Tim and Steve for your thoughtful responses.
I still believe I can do all that using aperture and shutter priorities. They don’t prevent me from changing the ISO, and if one uses aperture priority and uses exposure compensation, that only changes the shutter speed, and vice versa.
I certainly couldn’t get the results I get without understanding the exposure triangle and how the f/stops work, how my camera’s meter works, and even a general knowledge about the sunny 16 rule, but I’m still not convinced actually memorizing the f-number series is that important for me or for most people using digital cameras these days.
Being able to pick up older cameras with ease is the most compelling reason for memorizing the f/stops I’ve read here. If I were using an older camera, I expect I would carry my digital camera too and use it basically as a light meter. I don’t own a macro lens and have tried some macros reversing my 50mm lens, and I need to manually meter for these, so it is similar to using an older camera. I usually just take a few photos with the lens on the right way to get an idea of the settings I want, keep the sunny 16 rule in mind, and go from there, checking my histogram until I get it right. Other times I’ve found I need to use manual exposure are for panoramas and situations where I want a series of photos with the same exposure, for long-exposure night shots, and for the rare occasions when I’m using my speedlight (and there is where I get confused, and knowing the f-number series by heart here might be very helpful).
I am a relative novice at photography and self-taught (I read a lot), and I admit it’s very possible I’m missing something. I like to bake bread, and to me baking with a bread machine is just not getting the full effect,even though I have more control over the bread than if I bought it in the store and even if in the end it tastes pretty much the same as what I would make fully from scratch. Perhaps using my camera in full manual mode will become like that to me, but for now aperture and priority modes are working fine.
When I learned the relationship between f-stop and shutter speed about 10000 years ago, it was not clear why a “larger” f-stop as f11 let in half the light at f8. The f-stop value is really a reciprocal as 1/1, 1/1.4, 1/2, 1/4, 1/5.6, 1/8, 1/11, 1/16, 1/22, and 1/32. Clearly these numbers are decreasing, but they are not decreasing by a factor of 2. What gives? Write out a list of numbers that are twice as large as the last one as: 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. Then take the square root of each one. Now we see the numbers in the f-stop values: 1, 1.4, 2, 4, 5.6, 8, 11, 16, and 22. As you will note when you took the square root, the number was not always exactly the f-stop value we see on a lens, but close enough. While it is not necessary to understand how the numbers are calculated to use f-stops, I always want to understand as much as I can about any subject I am interested in. Clearly it is much easier for a lens manufacturer to use 1, 1.4, 2, 4, etc. instead of 1/1, 1/2, 1/4, 1/5.6. Talk about confusing plus the problem of printing 1/5.6 instead of 5.6 on the lens–what a mess that would be. Anyone should be able to memorize a sequence of just 9 numbers and know that each one lets in half as much as the number before it or twice as much if you go in the other direction.
Shutter speeds are even easier to remember as they are also reciprocals and are exactly half the speed before them and allow twice as much light: 1/1000, 1/500, 1/250, 1/125, 1/60, 1/30, 1/15. 1/8 (close enough to 1/7.5), 1/4, 1/2, and 1/1 which equals a shutter speed of 1 second.
Just learning these numbers and the relationships is not enough. One should test each lens to find the sharpest f-stop. Tests are not that important for wide-angle lenses but become more important as the size of the lens increases. For instance a 600mm lens is probably sharpest at around 5.6, while a 50mm lens is probably sharpest around f11. I have never hand-held many lenses as I always use a tripod, but the old rule of thumb says don’t try to hand-hold unless the shutter speed is about the same as the lens. For example, for a 200mm lens, the minimum shutter speed for hand-holding is 1/250. There are other little “gotcha’s” like 1/15 of a second is the absolute worst shutter speed to use when using a big telephoto. You could talk to 10 pro nature photographers and half would probably say not to use f22 as diffraction effects will soften the image while the others would say f22 is fine as the diffraction effect is so minor. Again, there is a reason to test each lens to find the sharpest f-stop.
One doesn’t have to go on a $5000 photo trip to perform the tests. Tape a piece of newspaper to a wall and shoot it with the camera on a tripod at every f-stop for the test. Before you go on the $5000 trip, you should know what f-stop to use for maximum sharpness, and you should practice practice practice adjusting the f-stops and shutter speeds until you don’t really have to think about it. All you need is the camera and one lens to practice, and you can do it watching TV, having a quick lunch, etc. The $5000 trip is not the place to finally take your camera manual out to read either.
Good shooting. Charlie
ok guys.. you have me so confused here.. you say that hand held one should not go below 1/500
are you nuts? sorry.. I have tried what you say.. and it does not work for me..
I shoot manual mode.. hand held.. and shoot at 1/15 when in the house.. sometimes even 1/8.. I have a 200 telephoto lens.. and the ap depending if at night or during the day.. (still in the house ) I use 5.6 or 4.5..Iso usually set to 400 and they come out just fine…. I really am trying to wrap my brain around all your saying.. I really don’t know that much.. am self taught and have a long way to go.. but really don’t understand your hand held thingy..or your settings..
I tried your setting of the shutter speed at 1/500 and the ap.. at 5.6 the iso at 800.. focused on the TV no less and .. and all I got was a pretty dark photo so what am I doing wrong ?
I have a nikon d40..