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Question: If I make a 1024 x 768 photo on a 2 megapixel camera and another one on a 4 megapixel camera, are there twice as many pixels in the 4 meg image? I assume the latter is true, as the image sensor is the same size in both cameras, isn't it?

No, you're doing one of two things.

You're either using a 'digital zoom', cropping out a center part of the image and blowing it up. If you do this then the 4 meg camera will give you more of a 'close up'.

Or you're taking a 2 meg or 4 meg full frame image and down-sampling the camera capture to arrive at a smaller pixel array. In this event the 4 meg camera might give you a slightly better image as you are probably going to 'average out' any digital noise.

The sensor may be the same size in the two cameras but the individual photosensitive sites is most likely much smaller in the the 4 meg camera.

Think of it like when you set a pixel size for display, you get a picture 1024x768, 800x600, etc. If you change the monitor resolution, the picture gets larger or smaller, it doesn't add or drop pixels.