Determine dpi-values from pdf- and image-files

I just received an e-mail inquiry from a customer with the question how to determine the dpi values from PDF and image files ...

The dpi value can be calculated in an easy way.
1 dpi means 1 pixel per inch and 1 inch has 2.54 cm.
An image file with a width of 1024 pixels and 10 cm width has a dpi of (1024 x 2.54) / 10 ... so in this case 260 dpi.

An A4 PDF has the standard dimensions of 595 x 842 pixels.
With 21 cm width and a pixel width of 595 this results in:
(595 x 2.54) / 21 ... so you get the common 72 dpi.

This means if an A4 PDF page is rendered directly to an image file, the quality won't be great, because the quality has a fix limit of 72 dpi.