Full frame is derived directly from 35mm films. The sensor size is 35mm width as in films.
The advantage of having a full frame is
a) You have got around 2.5 times more surface area on the sensor than in an APS-C sensor; it means more photo sites in the sensor which directly leads to capturing more details.
b) By the very nature of the sensor you can shoot up in high ISO without any significant noise.
c) The lens focal length will shoot what exactly it shows when it is mounted on a full frame sensor. (details below)
d) You want to take a shot at f 1.8 in a full frame; you get 1.8, but in APS-C you get 2.88 and so on. (details below)
For instance you shoot in 200 mm zoom on a full frame camera you get exactly what it is in 200 mm, but in a APS-C (1.6 crop factor) actually you get 320mm zoom (1.6*200), it could be an advantage of using APS-C for some zoom shots. When you want to take a wide angle shot you actually zoom in a bit. In that case you definitely need a full frame camera and that too, too many details in a wide angle shot, without a doubt you need it.
The shot you take on f 22 in a full frame is not the same when you take the very shot in f 22 in APS-C. To get the shot equivalent to f 22 in FF you have to change the f no. to 14 in APS-C.
Advantages of APS-C sensors
a) You zoom in according to the extra crop factor, say if the crop factor is 1.6 multiply your focal length by 1.6 (as mentioned above), without any extra zoom lens.
b) The cost of the APS-C sensor/camera is far less than FF. (Details below)
c) A very good one for starters.
Why a FF sensor does cost much than an APS-C?
Sensor is made of Silicone wafer with other technical parts in it. The silicon panel comes in a size where APS-C sensors can be cut without much wastage. Whereas lots of wastages in cutting in to FF sensors. So the manufacturers want to make the lost money in wasting the wafers. (I strongly believe they will be recycling the waste portions)
a) If you are a starter you do not have to buy a FF camera, APS-C is far better for learning.
b) If you are a wildlife photographer, you buy an APS-C high end camera (say canon 7D) you get 1.6 converter attached to it by its very nature.
c) If you are high end fashion photographer and your photographs are going to print often, better you own a FF apart from having some L lenses.
d) If you are a night bird; you have to take a FF camera. APS-C cameras are very bad for lowlight, night photography.
e) This above list is not exhaustive.
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