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Remember taking your SATs? They had questions like this:
What is the next number in this sequence: 1, 1.4, 2, 2.8, 4, 5.6…?
At this point, you’d fill in the circle for “8” (or should have). Each number doubles by leapfrog. And so the next number after 8 should be 11.2. But let’s simplify things and round it off to 11.
This is the f-number sequence. Math-wise, it’s a geometric progression on the square root of 2 (1.414…).
OK, why such weird numbers?
First, what the f-number is: the focal length of a lens divided by the diameter of its diaphragm opening. A 100mm lens with an opening of 50mm is f/2. With a 25mm opening, the same lens is f/4. This explains 1) why the f-number gets larger when the opening gets smaller; 2) why long lenses with small f-numbers are so damn fat; and 3) why teeny-tiny lenses can have f-numbers like f/1.4 (tiny lenses—like those on digital compacts—are often very short focal lengths).
When you turn a dial to go up by one f-number step (say f/2.8 to f/4, called a full stop), you are cutting the light coming through the lens by half. If you go down the f-number sequence (from f/4 to f/2.8), you are doubling the amount of light coming through the lens.
Then why doesn’t the f-number itself double or halve for each single step?
Consider: to double the amount of light coming through an aperture (or double the volume of water through a pipe), you have to double the area of the opening. If you double the diameter of a circle, the area increases four times, not two. (Remember the math: area of a circle is π times the circle’s radius squared.)
So, to double the amount of light, you don’t increase the diameter of the opening by 2X; you increase it by the square root of two. Hence those f-numbers that bump up by √2 (about 1.4X) each step.
Each successive stop doubles or halves the amount of light over again. So if you skip from f/8 to f/4, you increase the light by four times (double, and double again). From f/4 to f/8, you’re quartering the light (half of half). Same for wacky in-between f-numbers. What’s two stops less than f/7.1? Easy: f/14.2, which your camera will show as f/14.
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