 About apertures and f-stops...

When I was first getting started in photography, I often wondered what those aperture settings in my camera meant: f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16… And how does one memorize these numbers? Once I realized what it was and how it worked it made sense to me and, dare I say it, I even found it interesting. So this is why I decided to write this article: to share my understanding with you and explain an easy way to remember apertures.

## apertures

In photography, “aperture” is simply a measure of how much you open the diaphragm on your lens.  Open the diaphragm and more light comes through (wide aperture). Close the diaphragm and less light comes through (small aperture). This is a pretty simple and intuitive concept so far. What is not so intuitive is how these aperture values are traditionally specified:

f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 …

What does this mean? Well, we must first explain that “f” in this context stands for the lens focal-length. This is the length expressed in mm that is typically found on the front of a lens. For example, my Canon Digital Rebel shipped with a 18-to-55 mm zoom lens, meaning that I can set “f” to anywhere from 18 up to 55 mm. So as an example, when we say that the aperture is f/8 (for a given focal length) all this means is that the diameter of that aperture diaphragm is set to 1/8th of the focal length.  If we use f/16, the diagram shrinks accordingly to cover only 1/16th of the focal length and so on and so forth…

The important thing to remember here is that, the higher the denominator “n” in the “f/n” fraction, the smaller the aperture and hence less light that is allowed to pass through the lens. In other words, the smaller the fraction of focal length you use, the smaller is the aperture.

Someone was intent on reverse logic on this one because now we have to get accustomed to the fact that larger numbers mean smaller apertures… but such is life. Call it part of the mystique of photography :)

The question still remains though - where did these numbers come from? And how can one memorize them? Enter my friend, the realm of square roots:)

## square roots

Some people just find it easier to memorize the sequence above and leave it at that. If that’s your case, don’t bother reading anymore. I always found easier to remember this type of thing when I actually understand how it works. So how does this work?

f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 …

Look for a moment just at the denominators: 1.4, 2, 2.8, 4, 5.6, 8, 11… The more trained mathematical eye will recognize that each term in this series equals the preceding term times the square root of two (√2). Examples:

√2 = 1.414 (approximately 1.4 which is our first term in the series)

2 = 1.4 x 1.4

2.8 = 2 * 1.4

4 = 2.8 * 1.4

22 = 16 * 1.4

Another interesting thing to note is that, every number in this sequence doubles from the one before last. For example: 16 = 2x 8, 5.6 = 2x 2.8 and 8 = 2x 4. This makes total sense since, if we represent the series as s(n), then

s(n) = √2 x s(n-1) (ie each term equals previous term times √2)

then it is easy to see that:

s(n) = √2 x s(n-1) = 2 x [√2 x s(n-2)] = 2 * s(n-2)

In other words, the current (n) term in the series equals twice the term two positions before (n-2).

What a concept! Now it is really easy to memorize these apertures. You just need to memorize the first two terms: f/1.4 and f/2. All the other terms are obtained by doubling these two terms sequentially! This is how it works:

then: 1.4, 2, 2.8                                                 (doubled 1.4 to get 2.8)

then: 1.4, 2, 2.8, 4                                             (doubled 2 to get 4)

then: 1.4, 2, 2.8, 4, 5.6                                       (doubled 2.8 to get 5.6)

then: 1.4, 2, 2.8, 4, 5.6, 8                                   (doubled 4 to get 8)

then: 1.4, 2, 2.8, 4, 5.6, 8, 11                              (doubled 5.6 to get approximately 11)

then: 1.4, 2, 2.8, 4, 5.6, 8, 11,16                         (doubled 8 to get 16)

then: 1.4, 2, 2.8, 4, 5.6, 8, 11,16, 22                    (doubled 11 to get 22)

I find this very easy to memorize, and provided I remember some of the apertures, I can easily get the others. Sometimes it is not even that difficult to do the “1.4 times” calculations in your head if you want to move one stop in one direction or the other.

## so why is that?

Why the series of apertures is defined this way? Well, it is common knowledge that for each of the steps in the sequence above, you decrease the amount of light that reaches your sensor (or film) by half. This is also referred to in photography terminology as decreasing exposure by “one stop”.  The reason why each step in the sequence reduces light by half can be easily explained by going back to the definition of aperture. Remember that “aperture” is the diameter of the diaphragm. The diameter of the diaphragm is directly related to the surface area through which light comes through.  That is to say that “half of the light” really means “half of the surface” through which light comes through. Because the surface in a circle is proportional to the square of the aperture ( S = Π/4 x D2 ) one can intuitively see that the aperture (D) only needs to be cut by √2 to yield half of the light (half of the surface).

Comments, questions, suggestions? You can reach me at: contact (at sign) paulorenato (dot) com 