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When I type the following command in the Terminal:

xdpyinfo  | grep dimensions

I get the following output:

dimensions:    1366x768 pixels (361x203 millimeters)

As far as I know these are the number of pixels that my monitor has along the horizontal and vertical axis respectively.

Here is my doubt: When I am on Youtube and I am watching a video with a 720p resolution I understand the fact that my monitor can handle it since 720p<768p.

However if I play the video at a higher resolution, for example 1080p, I can't understand how it is possible to actually appreciate the better quality of the video if my monitor has less pixels than the resolution I have selected (768p<1080p).

Hopefully the question makes sense.

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It's called scaling.

There is both hardware scaling (which is done by the video card or the monitor) and software scaling.

That is an example of software scaling.

It's much like watching a big picture, at some point if the image can't be rendered directly the image is passed to an algorithm which tries to resize it in a reasonable way (notice that resizing itself implies quality loss); the same goes for videos.

More on picture scaling

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You can't appreciate the extra quality. Your computer down-sizes the video to fit within the pixel dimensions of the screen.

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