Minor Math Monday

Those pesky “f-numbers”…  my new f/1.2 lens prompts me to write about them.

As you know, f-numbers determine the depth of field (download the sample chapter of my DSLR book here, if you need a refresher). But what does the number mean? A little math for you today.

What is the f-number? The f-number is actually an “f divided by” number. An expression where “f” is the focal length of the lens (say, 50mm for a 50mm lens, or 70mm for a 24-70 lens that is zoomed in) that has as its result the diameter of the opening. Or in formula form:

d = f/n

..where d = diameter, f = effective focal length, and n = “the f-number”. So a 100mm lens set to f/4, for instance, would have an opening with a diameter of 100/4 = 25mm.

Why use this measure? Why not just say the diameter? Because the f-number describes the light-gathering ability of a lens. A 10mm f/2 lens and a 100mm f/2 lens can gather the same amount of light. How so? Surely the opening of the 100mm lens is much larger? Ah yes, but it is also much farther away! The amount of light is the same.

Why the funny numbers? Why f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22 and so on? That’s because:

  • Each number was chosen to be a stop apart. A stop just means a doubling or halving of the light; i.e. each subsequent number lets in half the light of the previous number.
  • And the light that enters the camera is proportional with the area of the diaphragm circle.
  • Hence, to go to the next main f-number, we want half the area.
  • Which means reducing the diameter not by 2 (that would give 1/4 the light gathering ability, since the area = pi x r squared). Instead. we need to divide the diameter by the square root of two (√2) to get half the light gathering ability.

So the series is: 20x0.5, 21×0.5, 22×0.5, 23×0.5, 24×0.5, 25×0.5

For half stops, the series is: 20x0.5, 20.5×0.5, 21×0.5, 21.5×0.5, 22×0.5, 22.5×0.5

For third stops, 20x0.5, 21/3×0.5, 22/3×0.5, 23/3×0.5, 24/3, 25/3×0.5

So now you know a little of the math behind aperture numbers. And here’s what they do, again,.. here’s f/1.2:

Talking of f/1.2: I am selling my pristine, boxed Canon 50mm f/1.2L lens (I have the 85 and the 50, and one is enough). Interested? To my readers here I am extending a $125 discount. It is listed for $1,500 on Kijiji (slightly cheaper than what this lens has been selling for on eBay) , but it is $1,375 for you. If interested, let me know quickly!




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