Why do lower f-stop numbers mean larger apertures? [duplicate]

I thought this would have been asked before but it seems not.

I don't understand how f-stops are numbered. From what I read, the lower the f-stop number, the larger the aperture. That is, f/2 is larger than f/22.

This seems odd to me. The maximum aperture width is (at least theoretically) infinite, while the minimum is 0 (i.e. the lens is completely closed and not letting in any light).

The way the numbering system is set up, it allows for infinitely tiny apertures but a limit on how large they can be (i.e. I guess the largest is f/0)

Can someone clarify why the aperture is numbered in this way?

• – Tetsujin Nov 22 '18 at 20:16
• There may be limits on how large an aperture can be, but practical considerations make lenses faster than f/1 very, very, very rare. The fastest I have ever heard of is f/0.7. – Jim MacKenzie Nov 22 '18 at 21:08
• There are many measuring systems where a larger numerical value means a smaller physical size. – whatsisname Nov 23 '18 at 2:44
• f/2 is larger than f/32 in the same way that a 1/2 gallon (64 oz.) of milk is larger than 1/32 gallon of milk (4 oz.). – Michael C Nov 24 '18 at 5:38

/ is division.

A number divided by 22 (f/22) is smaller than the same number divided by 2 (f/2). As simple as that.

Now, WHY?

f is the focal length. Describing the aperture as a fraction of the focal length has one advantage: it immediately reveals the image brightness.

Example:

20mm aperture can be...

• Telephoto lens, 200mm f/10 - pretty dark
• Wide angle lens, 30mm f/1.5 - pretty bright
• Aha didn't realize that we were dividing `f`. So what is `f` ? What does `f/1` equal? – CodyBugstein Nov 22 '18 at 19:56
• Aha So aperture of `f/6` means something very different on a 30mm lens and a 200mm lens? – CodyBugstein Nov 22 '18 at 20:18
• f/1 would be aperture equal to the focal length (which means quite big aperture and bright lens but otherwise not a special case of any kind. it is always possible to push a little harder and make lens a bit bigger and more expensive) – szulat Nov 22 '18 at 20:18
• f/6 means the physical diameter would be different but they produce the same brightness. – szulat Nov 22 '18 at 20:19
• "Aha So aperture of f/6 means something very different on a 30mm lens and a 200mm lens?" It means the same thing in terms of exposure - because exposure is measure in terms of light per unit area. It means very different in terms of the physical width of the entrance pupil (the size of the aperture as observed looking through the front of the lens). – Michael C Nov 23 '18 at 19:31

The f stop is a comparison between the physical size of the focal length and the aperture.

If the aperture is 25mm and the focal length is 100mm, then you'd have f/4 because the aperture value is 1/4 the focal length.

So, the f stop is the fractional representation of this comparison. As with fractions, 1/2 is bigger than 1/22.

It's no accident that f-stops are written with a "division" slash.

You say, "f/2 is larger than f/22". Here, "f" stands for the focal length, and the result of the expression gives the physical size of the aperture opening. So, let's take a 50mm lens... At f/2, the aperture opening measures 25mm. At f/22, the aperture opening measures ~2.3mm. Obviously an opening measuring 25mm is going to let in more light than one that measures ~2.3mm.*

You say, "I guess the largest is f/0". No. You can't divide by zero. Or in other words, there is no lens such that the ratio of the focal length to the size of the aperture is f:âˆž.

* on the same lens

• I always say... photography is the art for mathematicians :-P – osullic Nov 22 '18 at 23:52
• I have literally been told, you're about the science and I'm more about the art, when trying to teach darkroom principals. Le sigh. – OnBreak. Nov 23 '18 at 0:46
• What does the aperture have to do with the focal length though? This means f/1 is a different amount of opening on different size lenses – CodyBugstein Nov 27 '18 at 3:02
• @CodyBugstein exactly! f/1 is indeed a different physical opening size on different lenses. Because of the way lenses project light onto the sensor (or film), a 50mm lens with a 50mm aperture opening lets in the same amount of light as does an 85mm lens with an 85mm aperture opening. That's exactly why we use f-stops - to easily compare across different focal lengths. Each lens is f/1, so the amount of light is the same. Ditto if each lens was f/2.8, or if each lens was f/4...or whatever. – osullic Nov 27 '18 at 9:43

Because the lens operates much like a funnel in that it gathers light, the greater the working diameter of the lens, the brighter the projected image. How bright this image will be is dependent on the brightness of the scene and the magnification realized by the lens. The longer the focal length, the more the lens magnifies. The deed of magnifying to produce an image, takes its toll on image brightness. In other words, the longer the focal length, the more the lens magnifies. This higher magnification result is a larger but dimmer image of objects.

Another way to say this, image brightness intertwines the working diameter and the focal length. Because these two factors are so interwoven, gauging image brightness is demanding. We are forced to fall back on a mathematical ratio that will take the chaos out of figuring out image brightness. This is true because a ratio is dimensionless. If I tell you the ratio of boys to girls in a 6th grade class is 3 boys for every 4 girls, I have given you a ratio that works independent of the number of students. For example if the class consist of 28 kids, then 12 boys to 16 girls is the breakdown (ratio is dimensionless).

For the camera lens: if the working diameter is 4 inches and the focal length is 4 inches, then the focal ratio (f-number) = 4 Ã· 4 = 1 (written as f/1 (f/1 is produces a very bright image). If the working diameter is 2 inches and the focal length is 4 inches, then the f-number is 4 Ã· 2 = 2 (written as f/2.).

The splendor of using a ratio is, any lens operating at the same f-number as another lens, yields the same image brightness regardless of the dimensions (diameter or focal length), for an identical scene. Itâ€™s complicated; but the f-number system actually takes away the chaos.

The "why" is that f/stop number is focal length / aperture diameter.

So a 100 mm lens at f/4 has an effective aperture of 25 mm. This is not exactly the physical diameter opening, but is the effective diameter, specifically the front entrance pupil, as seen by the magnification of the front lens elements.

Because of this definition, as aperture diameter becomes larger (and exposure increases), f/stop number becomes smaller. f/2.8 is wide, and f/22 is narrow.

It may seem backwards at first, but we get over it easily, and the huge advantage of the f/stop number system is that it applies for any lens, of any size and focal length and aperture, so then we all know what we mean when we say f/8. f/8 is f/8 exposure on any lens. This is a plus. It wasn't that way in the early beginning. And of course, light meters meter with the same meanings.