Here's a topic I'm barely qualified to write about, but want to talk through as part of the process of understanding it better myself. I'm talking about F-numbers/F-stops/F-ratios. They all mean the same thing. And for many photographers, they mean a spec that you associate with the "speed" of a lens, or a setting on a camera to adjust your depth-of-field. If someone asked me "what's the f-number mean?", my explanation would be in terms of what changing it does as opposed to what the number stands for. Lower the F-stop to get more bokeh, raise it for landscape photos, etc. That's how I've been thinking about it for the past year, meaning that I had a functional understanding of what the F-number meant for my photos, but not a technical understanding of why that number has its intended effect. Let's work together to change that.
If you Google for explanations of the F-number, there are plenty of great guides and videos that do the job well. I don't want to rehash their words; it's more useful and interesting to talk through the salient points that I took away from reading those primers and the practical effect they've had on my photography. The best guide I read was this one by Matthew Cole. It's thorough and walks you through the technical derivation of the F-number, concluding with some tips for metering light using a combination of F-number and shutter speed. That last part is a little more advanced than what I want to get into today, but the following are my three biggest takeaways from reading about F-numbers to understand how it's calculated.
Exposure as a Water Bucket Analogy
A good analogy for understanding how the exposure of a photo is determined is using the idea of a bucket of water. Think of a bucket as representing the amount of exposure (or brightness) you want a photo to have, and that to achieve that target exposure, you need to fill it completely full of water. Water, in this, case, represents light. If you have a smaller target exposure (eg. want an image to be darker), you'll use a smaller bucket, which will require less water/light to fill. Conversely, if you want to overexpose your photo, you need more water/light to fill that bigger bucket. Pretty simple so far.
In photography, three technical attributes work together to determine the exposure of a photo: aperture, shutter speed, and ISO. In the water bucket analogy, each of these attributes has a corresponding element. In photography, the aperture refers to the opening in the lens that allows light through to the camera sensor or film. So in the analogy, think of this as the opening of a funnel through which you pour water into the bucket. The larger the funnel opening, the more water you can pour during a specific amount of time. That time, then, is analogous to the shutter speed--how long you keep that aperture/funnel open. The faster the shutter, the less light can get through.
And then there's ISO, which in my mind is thought of as a compensating factor for the aperture and shutter speed. In the water bucket analogy, it's like pre-filling the bucket with sand. More sand in the bucket (higher ISO) means you need less water (light) to fill it for your target exposure, but it also means that your image will be less clear--literally grainy. Trading off these three attributes is a fine balancing act, which is why it's useful to set your camera to different priority modes to automate one or more of those settings.
So how does this relate to the F-number? Well, the F-number corresponds with the lens aperture; the effect of changing it is what opens or closes the aperture to allow more or less light into your bucket. But if you think of that opening as a circle, the F-number is not a standard geometric variable. It's not the radius or diameter of that aperture opening, nor is it the circumference. It's not even the physical area of the aperture opening, which would seem to make the most sense in the water bucket analogy. And that leads to my second takeaway point:
The F-Number is Lens Specific
The F-number is not only dependent on the size of the aperture opening, but also the focal length of the lens.
This was the first breakthrough that really helped me understand why the series of F-numbers are designated in a seemingly arbitrary sequence (1, 1.4, 2, 2.8. 4, etc). Turns out, it's all because of math. *Shakes fist*. The F-number can't be a fixed description of the size of the aperture circle because the amount of light that passes through that circle is not only dependent on the size of the opening, but also the focal length of the lens. For example, the physical aperture opening on a short wide-angle lens will actually allow more light in than the same aperture opening area on a long telephoto lens. Put another way, the longer the focal length of a lens, the wider the physical aperture opening is needed to allow the same amount of light in as a shorter focal length lens. In short: the target size of the aperture is beholden to the focal length of the lens. Physical aperture size and lens focal length are bound together, and you can't express the F-number without referring to both.
Hence, the F-number is actually a ratio of those two traits: the diameter of the aperture and the focal length of the lens (in mm).
So while f/2 on a 35mm lens and an 85mm lens designate that the same amount of light can get through onto the sensor, the aperture opening on the 85mm lens is actually larger. This explains why it's difficult to manufacture a zoom lens (lens with a range of focal lengths) with a constant aperture--the opening needs to be wider when you're zoomed in than when you're zoomed out, to let the same amount of light in. Instead, it's easier to make a zoom lenses whose smallest F-number changes as you shift focal length, as in kit lenses like Sony's 18-55mm f/3.5-5.6.
The Square Root of 2, or 1.41
Understanding that the F-number is a ratio between two aspects of a lens doesn't really explain its sequence, though. When talking about how much light is let onto the film or camera sensor, we don't speak in absolute terms (eg. lumens of lux). Instead, we describe light in terms of a relational sequence of "stops." In that sequence, the interval between each progressive stop represents the doubling of light. Going up one stop means you're letting twice as much light in. Going down one stop means you let half as much light in.
And because the physical aperture is a circle, that's where more math and geometry comes in. The area of a circle is calculated by the equation Pi times radius-squared. Given that a diameter is twice the radius, in order to double the area of a circle, you have to multiply its diameter by the square root of 2, or approximately 1.41. That factor of 1.41 is what determines the sequence of F-numbers. This should look familiar:
1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22
That sequence of F-numbers represents a full stop interval between apertures. f/1.4 allows half the amount of light as f/1. f/2.8 allows half as much light as f/2. And so on. Each number in that sequence is roughly 1.41 times the number previous. It's a sequence that's not difficult to memorize once you know the 1.41 rule.
But if you change F-numbers on your camera, you'll notice that you're not changing it in the same order as that F-stop sequence. You'll find F-numbers between 2.8 and 4, and numbers between 5.6 and 8. That's where it's easy to get thrown off. F-numbers can be adjusted not only along full stops of light, but fractions of a stop. In modern cameras, 1/3 stop adjustments is normal. f/3.5 isn't along the F-number sequence, but it's a common aperture setting that represents 2/3rds of a stop between f/2.8 and f/4. That was my big breakthrough in understanding the meaning of F-numbers. And now, I just memorize the F-stop sequence and keep in mind the multiplier of 1.4 for figuring out how to balance light stops with shutter speed.
This is why an f/2.8 lens is so much more effective than an f/4 lens in low light. f/2.8 is one stop wider than f/4, meaning it allows twice as much light in. That means the shutter can close twice as fast to get the same exposure. If a scene demanded that the shutter be set to 1/15sec at f/4, it can be set to 1/30sec at f/2.8 for the same exposure. That doubling in shutter speed can make all the difference between a blurry and sharp photo.
Update: Looking at the sequence, I now realize there's a simpler way to memorize it instead of multiplying by 1.41. Because every interval is multiplied by a factor of root-2, every other interval is differentiated by a factor of 2. Every other number is a doubling. So it may be easier to visualize it as a 2-stop sequence: 1, 2, 4, 8, 16, etc. interspersed with another 2-stop sequence: 1.4, 2.8, 5.6, 11, 22, etc.
1, 2, 4, 8, 16
.7, 1.4, 2.8, 5.6, 11, 22
Phiew--make a little more sense now? Or have I completely lost you?
If that's not helpful, or I've made too many misconceptions, here's the best video I founding explaining the F-number: