The full-stop aperture scale explained
Why f-numbers climb by the square root of two, how half and third stops fit between them, and how aperture stops trade against shutter and ISO.
Why the numbers climb by root two
The classic aperture scale, f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, looks arbitrary until you see the pattern. Each step multiplies the f-number by about 1.414, the square root of two. That factor is chosen because the light an aperture gathers depends on the area of the opening, and area grows with the square of the diameter. Multiplying the f-number by root two shrinks the area to exactly half, so every full stop halves the light. That is why the numbers repeat in a doubling rhythm: 1, 2, 4, 8, 16 appear, and so do the root-two neighbors 1.4, 2.8, 5.6, 11, 22.
Full, half and third stops
Modern cameras rarely jump a full stop at a time. Most divide each full stop into thirds, and some offer halves, so the aperture dial clicks through finer steps. Between f/2.8 and f/4, the third-stop marks are f/2.8, f/3.2, f/3.5 and f/4, and between f/2 and f/2.8 they are f/2, f/2.2, f/2.5 and f/2.8. These printed values are rounded, so a single third-stop step like f/2.8 to f/3.2 is not exactly one third of a stop. This calculator uses the real f-numbers you enter, which is why that step comes out as 0.3853 stops rather than a clean 0.3333.
Reading the sequence on your camera
The f-numbers engraved on lenses and shown in viewfinders are convenient roundings of the true root-two values. The exact sequence would read f/1.414 and f/2.828, but cameras display f/1.4 and f/2.8 to keep the scale readable. That rounding is why some pairs that look a clean stop apart, such as f/2.8 and f/5.6, are only approximately two stops when measured precisely. When you need the exact stop difference, for flash ratios or exposure compensation, enter the printed numbers and let the math sort out the small discrepancies.
Aperture is only one of three stops
Aperture is one leg of the exposure triangle, alongside shutter speed and ISO, and all three are measured in the same stops. Because a stop is a stop, you can trade one for another to keep exposure constant. Closing the aperture two stops, from f/2 to f/4, cuts the light to a quarter, and you can recover it by slowing the shutter two stops or raising the ISO two stops. Understanding the aperture scale in stops therefore makes the whole exposure system easier, since the same doubling and halving logic runs through every setting.