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While this topic uses no pictures, it's one of the most important concepts in photography.  Don't let all the numbers prevent you from reading and understanding it.

The aperture is the hole that allows light to come through when you release the shutter, to expose the film or to allow the digital imaging chip to process and record the image.

The relative size of the aperture opening is indicated by the f-stop number.   If the f-stop were to have a value of "4", it would be shown as f/4.   The "slash" in f/4 is there intentionally because this value is fractional in nature.

f-number defined

The f-number is defined as the focal length of the lens divided by the diameter of the aperture.  This value has no dimensions, or "units of measure."  For example, at a setting of f/11, the focal length is 11 times greater than the effective aperture diameter (called the entrance pupil).  The larger the f-number, the less light gets through.  The size of the entrance pupil is controlled by a diaphragm, which allows the photographer to adjust the f-stop in order to control the amount of light that strikes the film or imaging sensor.  This further allows the photographer to control (within the limits of the lens' capabilities) the exposure time, depth of field, etc.

It's all relative

A 400mm lens set at f/4 has a pupil diameter of 100mm.
A 100mm lens, also set at f/4, has a pupil diameter of 25mm.

Even though the entrance pupil of the 400mm lens in the above example is physically larger than the pupil of the 100mm lens at an f/4 setting, they will both pass the same amount of light through, to expose the film or imaging sensor.  It's also why the higher quality super-telephoto lenses are so big and heavy - they need to be!

For any given focal length, the larger the f-stop number, the smaller the opening.  An aperture of f/4 is larger than an aperture of f/8, just as a value of 1/4 (0.250) is larger than 1/8 (0.125).   However, because we are talking about properties of light, the relationship between these values is not quite this simple (it's not linear), but it's important to understand exactly what these numbers represent.

The following information may seem difficult to comprehend at first, but I believe that understanding this material will go a long way to helping you take better pictures.   I have not found the following information presented in any photography books that I have read.  This doesn't mean that it isn't out there, it just means that I haven't come across it yet, so I'm thinking that maybe you haven't either.

Where do these f-stop numbers come from?

As light travels away from its source, the brightness of the light does not decrease in a linear fashion.  In other words, if a person is standing 50 feet away from a bright light, and another person is standing 100 feet away from the same light, you might assume that the light seems twice as bright to the person who is only 50 feet away (or half as bright for the person 100 feet away.)  This assumption is completely wrong!  See the last example at the bottom of this page for a detailed explanation why.  [The answer is 4 times as bright for the person standing 50 feet away.]

The scientific reason why:  Light intensity obeys an inverse square law.  In other words, as you move away from a light source, the intensity of the light decreases in a manner that is inversely proportional to the square of the distance.

Why is this important to me, as a photographer?

As you change the f-stop numbers on your camera, you are telling the camera how much light you want to allow through the aperture when the shutter is released.  Changing the aperture by one full f-stop will allow exactly twice as much light to come through (or exactly half as much, depending on which way you go with it).  An aperture of f/2.8 will allow twice the amount of light as f/4, while an aperture of f/5.6 will allow half the amount of light as f/4.  This also means that an aperture of f/2.8 will allow 4 times the amount of light as f/5.6, just as f/5.6 allows 4 times the amount of light as f/11.  But where did these numbers actually derive from?

The intensity of a light will double (or halve) as you move closer (or farther) from it, based on the square root of 2.  Therefore, common aperture values are also based on the square root of 2.

The following table will demonstrate where these f-stop numbers come from, and what they mean for calculating exposure values.

The following values will
continually be multiplied by
this number

( f / ? )

Fast Lens vs. Slow Lens

When you hear the terms "fast lens" or "slow lens", this is actually a reference to the lens' largest possible aperture (smallest possible f-stop number).  Because a lens with a larger aperture will allow greater light transmission, it will allow for a faster shutter speed in order to achieve any given exposure.  An f/2.8 lens is faster than an f/4 lens.  An f/2.8 lens with a shutter speed of 1/500 will give the same exposure as an f/4 lens with a shutter speed of 1/250, only the f/2.8 lens can do it in half the time.  Faster lenses can be stopped down to achieve slower shutter speeds when needed (to capture the motion of flowing water, for example).  Either of these two lenses above could be stopped down to f/5.6, and provide an equivalent exposure with a shutter speed of 1/125.  An f/2.8 lens might have a full range of f/2.8 to f/32.

Slower shutter speeds can result in unwanted motion-induced blur, and is why a tripod, or some other stabilizing device, is needed when using slower shutter speeds.  In general, faster lenses have greater capability, and allow you to stop action with faster shutter speeds, corresponding to their larger maximum aperture.  Sports photographers, for example, will use a fast telephoto lens to stop the action at a soccer game or race track.  With everything else being equal, a faster lens will, of course, be more expensive than a slower lens.

Using Filters

Let's say that you're using a neutral density filter that changes your exposure by two f-stops.  This means that it allows only 1/4 of the light to pass through.  So, a 2-stop filter reduces light transmission by a factor of 4, not 2.  A 3-stop filter allows 1/8 of the light to pass through, etc.  The table below will show you various common designations that you may see, in reference to neutral density filters.

Changing light intensity based on distance

Suppose that you're using an external light source to illuminate a subject.  The picture is overexposed, so you want to put half as much light on your subject.  How far from the subject do you move the light?  Hint: It's not twice as far away.  Remember, light intensity doubles or halves based on the square root of 2.

If your light source starts out exactly 100 inches away from your subject, you need to move it the square root of 2 (1.4142) times away.  Conversely, you need to move it 1/1.4142 times (0.7071) closer to put twice as much light on the subject.

With a base point of 100 inches between your light and the subject, the subject will be twice as bright if the light is moved to a point 70.71 inches away, and it will be half as bright if the light is moved to a point 141.42 inches away.  If you moved the light to a point 50 inches from the subject, it would be 4 times as bright as it was from 100 inches away.  If the light were moved to 200 inches away, it would be only 1/4 as bright as from 100 inches away.