One centimetre is 0.3937007874 inches, because one inch is exactly 2.54 centimetres. That 2.54 is not an approximation or a measurement — it is a definition agreed internationally in 1959, and it is one of the cleanest numbers in the whole metric-imperial boundary.
About the units
The centimetre is one hundredth of a metre, and it occupies an odd position in SI: the system officially prefers multiples of a thousand, which would mean millimetres and metres with nothing in between. The centimetre survived anyway because it matches the scale of the human hand, and it dominates everyday measurement across Europe, Latin America and Asia. The inch has a much older and stranger history. Its name comes from the Latin uncia, a twelfth part, and its length was at various times defined as three barleycorns laid end to end, or as the width of a man's thumb — the same origin visible in the French pouce and the Swedish tum, both of which literally mean thumb. Those definitions varied by region until the twentieth century.
The exact factor
The exact relationship is 1 in = 2.54 cm, which makes centimetres to inches a division: 1 ÷ 2.54 = 0.39370078740157480..., repeating. The reverse direction terminates cleanly. A rough mental method that works surprisingly well is to divide centimetres by two and subtract a fifth of the result: 40 cm halves to 20, minus 4, giving 16 inches against the true 15.75. Before 1959 the inch varied slightly between countries — the British inch and the American inch differed by about two parts per million — and the 1959 agreement chose 2.54 precisely because it was round enough to end the argument.
Where you meet this conversion
Screen sizes are the most visible case: televisions, monitors and phones are sold by diagonal inches almost everywhere in the world, including in countries that use nothing but centimetres for everything else. Clothing and body measurements switch too, with waist and chest sizes in inches on garments sold in centimetres. Photography uses inches for sensor formats and print sizes. And anyone assembling furniture or working on a bicycle meets both on the same object, since fasteners and frame sizes often follow different conventions than the packaging.