DPI

The one idea everything follows from

An image file contains a number of pixels. It does not have a physical size. A 3000 x 2000 photograph is three thousand pixels wide — not eight inches, not ten. It acquires a physical size only when something decides how many of those pixels go in each inch.

DPI is that decision. It is not an attribute the file has; it is a conversion factor between pixels and space. Physical size equals pixels divided by DPI, and pixels needed equals physical size times DPI. That is the entire mathematics, and every confusion in this area comes from treating DPI as a property rather than a ratio.

So the same 3000 x 2000 file prints at 10 x 6.7 inches at 300 DPI, 20 x 13.3 inches at 150, and 41.7 x 27.8 inches at 72. Same file, same bytes, three different physical objects.

What the DPI tag in a file does

Most formats can store a DPI value in metadata. It changes no pixels. It is a note saying what physical size the author intended, and software may or may not read it.

Which means changing the tag from 72 to 300 without resampling changes nothing about the image. A print shop's software computes the same physical size it always would have. The tag matters only where the receiving system reads it — some print workflows, some layout applications, and upload forms that validate it, which is why a passport photo specification saying 300 DPI means it.

"Save for web at 72 DPI" is cargo cult. Browsers do not read the tag at all.

How much you actually need

300 DPI is roughly where the eye stops resolving individual dots at reading distance, which is why it is standard for anything held in the hand. The requirement falls as viewing distance grows, because what the eye resolves is an angle: about 150 for a wall poster, 100 for an exhibition banner, and 10-20 for a billboard, which really is printed that coarsely.

Supplying 300 DPI for everything is a mistake. A 2 x 1 metre banner at 300 DPI is roughly 800 megapixels and indistinguishable, at its viewing distance, from the same banner at 100.