Chapter 2. ƒ Number Arithmetic

The “ƒ Numbers” tab lets you perform arithmetic on so-called ƒ numbers, also known as aperture values. This tab understands two different calculations:

  1. ƒ2 as ƒ1 plus stops: This calculcation lets you add an arbitrary number of ƒ stops to an arbitrary aperture value. Most SLR users know that ƒ/2.8 plus two stops is ƒ/5.6, but what is ƒ/2.8 plus three and a half stops? (Answer: About ƒ/9.4.)

  2. stops as ƒ2 minus ƒ1: This calculation lets you find out how far apart two aperture values are, in stops. This is especially useful when comparing variable-aperture zoom lenses, since these often do not use “even” aperture values. For example, Canon makes a 28-105 mm ƒ/3.5-4.5 lens. Clearly, ƒ/3.5 is slower than ƒ/2.8, but by how much? (Answer: Roughly ⅔ of a stop.)

In the second calculation, you can also choose to have the answer rounded off to the nearest ½ or ⅓ stop. You can change this behavior in the program’s settings.

About ƒ Numbers

ƒ numbers, or aperture values, are a measurement of the size of the hole that the light passes through in the rear of a lens, relative to the focal length of the lens. The smaller the ƒ number, the more light gets through the lens. Each additional ƒ stop means that half as much light gets through the lens. So, at ƒ/2, twice as much light gets through the lens as when you set that same lens at ƒ/2.8.

The ƒ number notation is a clue to the nature of ƒ stops: it’s a slash separating an “ƒ” from a constant. We thus have the optical physicists’ symbol for the lens’ focal length, ƒ, divided by some number. So, if the focal length is 50 mm, the aperture diameter at ƒ/2.8 is about 18 mm. This is also why you sometimes see ƒ numbers written as a ratio, like 1:2.8; it’s the ratio of the aperture diameter to the focal length.

The term “stops” comes from the early days of photography, when photographers would place thin wooden panels with holes cut in them between the lens and the film (photographic plates, actually), to “stop” a certain amount of light from reaching the photographic plates.

Formulas Used

ƒ stops are powers of the square root of 2. The first ƒ stop is the square root of 2 to the zero'th power, or ƒ/1. Next is the square root of 2 to the first power, or ƒ/1.4, then the square root of 2 squared for ƒ/2, etc. As you can see, common values like ƒ/2.8 are actually approximations of the unwieldy “true” values. When you pick ƒ numbers from ƒ/Calc’s aperture value pickers, it shows you the approximation, but calculates with the accurate value internally. (More information on how ƒ/Calc handles aperture values is elsewhere in this manual.)

To calculate a new aperture value ƒ2 from an initial aperture ƒ1 and a given number of stops, we first convert ƒ1 to a number of ƒ stops from ƒ/1.0 with:

Equation 2.1. convert aperture value to number of stops from ƒ/1.0

log base sqrt(2) of f

Then, we add the result of that to the given stops value and convert back to an aperture value with:

Equation 2.2. convert number of stops from ƒ/1.0 to aperture value

sqrt(2) to the n'th power

To calculate the number of ƒ stops between ƒ1 and ƒ2, use the two formulas above to first convert both values to a count of ƒ stops from ƒ/1.0 as above, subtract the results, and convert the difference back to an aperture value.