Why Shutter Speed and Aperture Numbers are Upside Down
There's a lot of halving and doubling of numbers in photography. Exposure calculation just works that way. But what's confusing for some is that when you double the aperture from, perhaps f/8 to f/16, the image ends up darker. Both aperture and shutter speed numbers can seem upside down. But there is a method to the madness once you know what the numbers actually mean.
Let's start with shutter speed since it's the easier of the two. When you press the shutter release, the camera goes click. But what sounds like a single click is really the sound of two events — the shutter opening, and the shutter closing. The typical shutter speed is fast enough that those two sounds blur into what we perceive as a single click.
Indeed, shutter speeds are so fast that, as you probably already know, they're almost always less than a second. So when your camera shows numbers like 15, 30, 60, 125 and so on, those are really 1/30, 1/60 and 1/125 second. Constantly putting that "1/" fraction stuff on every number must have seemed like too much bother to somebody long ago and now we're all stuck with it.
When shooting long exposures it is possible to go above one second and things can get even more interesting. Those same 15, 30, 60 and so on numbers now have to have two tick marks (essentially a quotation mark) after them to clarify that now we're talking about seconds rather than fractions of a second. A single tick mark is a short hand for minutes, and two tick marks stands for seconds. So rather than putting the one-over fraction prefix in front of all the short number we get to put the seconds double-quote after then long ones.
It does make for a kind of short hand I guess, but it can confuse new photographers. It really doesn't take that long to understand any of this, but it can take a while to get used to it. It can be all too easy to instinctively increase shutter speed when you mean to decrease it because of these upside down numbers.
Aperture numbers can similarly seem upside down and for a somewhat similar reason. When we say that a shot was made at f/4, that number does look somewhat like fraction, doesn't it? But in this case, the top number isn't a number at all, it's the letter "f." Aperture values are indeed fractions. The numbers are the ratio of the focal length (that's where the "f" comes from) to the effective diameter of the lens opening.
This helps explain why shorter focal length lenses often have a faster maximum aperture (widest open aperture) than do telephotos. It's not unusual to have f/4 wide angle lenses. Standard 50mm lenses with f/1.8 or even f/1.4 are common. But if you own a telephoto that can go to that wide it weighs a bit and costs a lot. To achieve an f/4 aperture on a 40mm lens requires a lens opening of only 10mm. That same f/4 aperture on a 200mm lens means you'll need a 50mm opening.
It's worth pointing out that the opening at issue here isn't the front lens element, the one you can immediately see when you take the lens cap off. It's the effective opening of the lens which lies deep in the middle of the lens body and is dictated by design of the lens optics. In rough terms though, you can think about the front of the lens when making aperture comparisons, assuming the lenses have similar construction.
This fraction definition also helps explain why the standard series of aperture numbers don't double or halve as you go from one to the next. Instead, they change by multiples of 1.4, or the square root of two. They amount of light that makes it through a lens relates to the area of the opening while the aperture fractions are in terms of the lens opening diameter. The area increases as to the square of the diameter, so if we multiply or divide the area by two, the diameter changes by the square root of two.
And as with shutter speed numbers, intellectually understanding that they're fractions is only half the battle to really understanding them. To really become comfortable with these number scales requires using them. So go take some pictures.