Understanding f Numbers

In photography the size of the aperture is measured in f numbers, a lot of people just learning photography find f numbers confusing, this starts with them being backwards (smaller aperture = larger number) and continues with the fact that to double the amount of light you multiply the number by 1.4.

The aperture has a huge impart on the photo you make, so being confused about it is not a good thing.

Basically it’s rather simple, an aperture of f/2 (for example) means the width of the hole in the lens is the focal length divided by 2 (and f/8 is divided by 8, f/2.8 is divided by 2.8, etc.) so if you have a 50mm lens at f/2 the width of the hole in the lens is 25mm – it’s even written as a math equation.

Now the reason for the numbers being backwards is obvious, dividing by larger numbers give smaller sizes.

The reason for multiplying by 1.4 now also becomes simple – you just take the formula for the area of a circle and you get that to make the area twice as large you have to increase the diameter by the square root of 2 (or approximately 1.414) – I’ll skip the math, you can easily check this for yourself if you want.

Now the only question left is – why do we specify the aperture size in relative terms? and to answer that I took a few pictures, both pictures were taken with the Canon EFS 18-135 f/3.5-5.6 IS using the same camera settings (ISO 100 f/7.1 1/160sec) from the same location – the only thing that changed between pictures was the focal length (I zoomed in and out) – the first picture was taken at 18mm and the second at 135mm:

And for you convenience I’ve cropped the 18mm photo to show the same area as the 135mm photo:

Apart from the effect of the focal length on depth of field (that’s the topic of a different future post) both images look the same, if you look closely at the middle flower you will see the same colors and the same exposure in both pictures.

But the diameter of the hole in the lens is 18/7.1 = 2.5mm for the first picture and 135/7.1 = 19mm for the second – a 7x difference! and yet they both let the same amount of light into the camera – and that’s why we measure aperture in f numbers – if we used the size of the aperture the exposure would have changed radically when zooming in or out.

And, as a side note, it’s now also easy to see why cheaper lenses have variable max aperture – the same size hole has different f numbers at different focal lengths, and making a lens that can open up at the same ratio at both ends for a large zoom range requires a very big lens (or a tiny aperture at the wide end).

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