One millimetre is 0.0393700787 inches, because one inch is exactly 25.4 millimetres. This is the conversion that engineering actually runs on, and 25.4 is worth memorising outright — it appears in machine shops, drawings and tool catalogues constantly.
About the units
The millimetre is a thousandth of a metre and is the working unit of mechanical engineering across most of the world: technical drawings in metric countries carry dimensions in millimetres by default, with no unit marked, because everyone in the trade assumes them. The inch is the corresponding default in American and British engineering, but it is usually subdivided differently. Machinists work in thousandths of an inch, called thou or mil — one thou is 0.0254 mm — while carpenters and general trades use binary fractions: halves, quarters, eighths, sixteenths. Those two subdivisions do not mix well, which is why an inch-based drawing can carry both 0.375 and 3/8 for the same dimension.
The exact factor
The exact relationship 1 in = 25.4 mm comes directly from the 1959 agreement. Millimetres to inches is therefore 1 ÷ 25.4 = 0.03937007874015748..., repeating. A useful reference point for anyone working across systems: one thou, the machinist's basic increment, is 0.0254 mm, so a hundredth of a millimetre is a little under half a thou. Metric machining tolerances of ±0.01 mm and imperial tolerances of ±0.0005 in describe almost the same precision.
Where you meet this conversion
Drill bits, taps and fasteners are the daily case, and the two systems produce sizes that are close enough to be dangerous: an M6 bolt and a 1/4-inch bolt differ by about 0.35 mm, enough to strip a thread but not enough to notice by eye. Camera lenses are named in millimetres worldwide, while sensor sizes are quoted in inches using a legacy convention that does not even match the actual dimensions. Rainfall is measured in millimetres in metric countries and hundredths of an inch in the United States.