Digression #1
A note about the digressions: The digressions are intended as
"extra" material that is not really important to understand. They're
here in case you're interested and are willing to slog through some technical
talk on the way to grasping an esoteric concept. If you don't really care, or
you just don't want to spend the time grappling with the math or whatever, then
just skip the digressions. You won't be missing anything important.
Sometimes you'll hear photographers say "fstop" instead of aperture.
Then they'll talk about an exposure being "off by a full stop." Then
they'll mutter something about needing a "wider" lens. What does all this
gook mean, and why on earth is f/4 not twice as much light as f/8? All that and more
you will learn in this digression.
What exactly is an aperture, and why is it written f/8 ?
Figure 1: the Aperture Iris

Towards the back of the lens there is an iris that can open and close. The iris
is typically made of 7 blades that can slide over each other. Twisting the aperture
ring on the lens one way moves the blades closer together, closing the iris.
Twisting it the other way moves the blade farther apart, opening the iris. As you
open it, it allows more of the light gathered by the lens to reach the film. As you
close it, it cuts off some of the light. Figure 1 shows a schematic diagram of an
aperture iris. There isn't really a gap between the blades; the white lines are there just
to show you where the iris blades are located.
The aperture number you see represents the diameter of the lens opening as the physical
diameter of the opening (in cm, mm, inches, whatever; it doesn't matter) divided into the
focal length of the lens (in the same units, of course). In other words, the f/stop is the
diameter of the opening represented as a fraction of the lens length. That's why the
aperture is written as f / some number. f is the focal
length of the lens.
For example, let's say you have a 80 mm lens. If the aperture is set to f/8, the
diameter of the lens opening is 80mm/8, or 10mm. At f/4, the lens opening is set to 20 mm.
This is why increasing the aperture number (or "fstop") decreases
the amount of light; you're dividing a fixed number (the focal length of the lens) by an
increasing one, so the fraction gets smaller. When you go from f/4 to f/8, you've
decreased the lens opening from 1/4 the lens length to 1/8 the lens lengththus admitting
less light (not half as much, but one quarter as muchkeep reading for an explanation of
why).
By the way, you'll frequently see people write the aperture setting as "f8"
or "f 8". Both of these are technically incorrect.
The proper way to write the aperture setting is "f / 8". So when you read
a photography magazine in the future and the reporter writes "1/125 at f16,"
you're permitted a small knowing chuckle. :)
Apertures and Buying a Lens
It's worth mentioning that when you purchase a lens, the lens will be described as
something like this: 200mm f/4. That aperture is the maximum opening the
lens can do, and is functionally limited by the diameter of the front element of the lens.
In other words, a 200mm f/4 lens has a front element 200mm/4 or 50mm (around two inches)
in diameter. It's never quite that easy, of course. If you look at the front
element of a 200mm f/4 lens, it won't be exactly 2 inches; it will usually be a
bit larger than that. The reason is that there are always some inefficiencies in
the geometric designs of lenses, which result in having to design a slightly
larger lens than you would expect.
When it comes to apertures, the smaller the number, the bigger
the diameter of the lens. That's why a 200mm f/2.8 lens costs a lot more
than a 200mm f/4 lensthat 200mm f/2.8 lens is 71mm (2.8 inches) in diameter! That
sounds like the front lens is only 1.4 times the diameter (it is)but that means it's
twice the area! Since the amount of glass used in the lens is proportional to the
area, the front element of that f/2.8 lens costs twice as much in raw materials as the
same element on the f/4 lensand it's harder to manufacture, too.
That's also why you see zoom lenses represented as 2880mm f/3.55.6, as some lens of
yours is probably marked (just a guess). The diameter of the lens doesn't change as you
zoom, but the focal length does. By dividing the same lens diameter by
a larger focal length, you get a smaller maximum aperture (i.e. a larger
aperture number). At 28 mm, the max. aperture of the lens is
f/3.5 When you zoom it out to 80mm, it has the same diameter, but it's almost 3 times as
long...so it becomes an f/5.6 lens. Zoom lenses are strange in their optics, so
the straight computation of element size / aperture opening doesn't apply, but
you get the idea.
The terminology gets worse, believe it or not. "Faster" and
"slower" are terms applied to lenses. An f/2.8 lens is "faster"
than an f/4 lens. Why? because the f/2.8 lens will have an aperture setting of
f/2.8, which admits twice as much light as the max. f/4 that the other lens can do.
That means you can set the shutter speed to be twice as fast with the fast lens as you
could with the slow lens.
Why does twice the aperture number mean one fourth the light?
Think about this: the amount of light admitted by the lens is proportional to the surface
area of the lens. So if you double the diameter, you admit 4 times the light right?.
Remember: Area = p r^{2} (A = pi * r^2 if
you have a symbolchallenged browser). So what does this have to do with the
aperture? Recall that the aperture number represents the diameter of the lens
openingthat is, the radius times two. If you double the radius, you get 4 times
the area. Similarly, if you halve the radius, you get one fourth the area.
Thus, doubling the aperture number, which cuts the iris radius in half, gives you one
quarter the area, and one quarter the light.
Figure 2 shows this graphically. In each succeeding step, the diameter has
been reduced by half. The thing to look at is the size of the white portion in
the center of the lens; this is the portion of the lens that admits light. It
should be obvious to you that when you go from f/2.8 to f/5.6, the central
portion does not halve in size. It's hard to estimate that it quarters,
but I assure you that's the case.
Figure 2: Apertures and the Iris Diameter

So let's look at some common aperture stops on cameras: f/2.8, 4, 5.6, 8, 11, 16, 22, 32
Here's a quick quiz for you:
What is... 

4 divided by 2.8 ? 
1.43 
5.6 divided by 4 ? 
1.40 
8 divided by 5.6 ? 
1.43 
11 divided by 8 ? 
1.35 
16 divided by 11 ? 
1.45 
Get the idea? Basically they're all 1.4, plus or minus a little. Why?
I'll give you a hint: what is the square root of 2?
Ahhh...Did you figure it out? Every time you go up "a full stop" (as it's
called), you decrease the diameter by a factor of 1.4, which means you decrease the
surface area by a factor of 1.4 * 1.4 = 2. So every "full stop" either doubles
the light or halves it. Also notice that the numbers basically come in pairs:
multiples of 4 (4,8,16,32,...), interspersed evenly with multiples of 2.8
(2.8,5.6,11,22,...) This is because the square root of two is roughly 1.4.
Multiply 2.8 by 1.4 and you get 4. Multiply that by 1.4 again, and you get 5.6.
Multiply that by 1.4 and you get 8,and so on. There's no mystery there; every
time you do two multiplications by 1.4, you actually multiply by 2.
So every two stops gets you
to double the number you started from (i.e. 2.8 > 4 > 5.6 ), which
corresponds to half the
aperture, which in turn yields one quarter the light.
What the hell is a "stop"?
Sometimes you'll hear photographers talk about exposures and they'll say something like
"that was off by a full stop." The "stop" specifically means
"aperture stop," because even the oldstyle "photographer stands under a
hood and opens the shutter with a cable release" cameras that had no shutter speed
controls had aperture settings on the lenses. And those aperture settings were
usually marked with little bumps that would cause the aperture ring to stop
at places known to be "full" aperture settings (like f/2.8, f/4, etc.).
Each "stop" would get you twice as much (or half as much) light as the one
before it.
In general what the photographer really meant was that he had twice too much (or half
as much) light as needed. But the phrase "fstop" gradually got shortened
to "stop," and now it's stuck that way. So if you're a "full stop
underexposed," you need to open your aperture "one stop" to get the right
exposure, or go for twice the shutter time (i.e. half the shutter speed).
Stop Down, Open up...what the hell are you talking about?
Basically, there are dozens of terms which all mean the same thing. "Open
Up," "wider," etc. all mean "go to a smaller aperture number," or
"open the iris more." "Stop down," "clamp down," etc.
all mean "go to a higher aperture number," or "close the iris more."
