Celsius to Fahrenheit (C to F)

Celsius to Fahrenheit is F = C x 9/5 + 32. Written the way it computes, that is 9/5 times the Celsius figure plus 32, and the two parts do different jobs: the 9/5 accounts for the two scales having different degree sizes, and the 32 accounts for them having different zero points.

That second term is what makes temperature different from every other conversion. Length, mass and volume units all share a zero (zero centimeters is zero inches), so converting one is a single multiplication. Celsius and Fahrenheit do not, so the conversion is affine rather than scalar and needs an offset as well as a ratio. It is the reason a temperature and a temperature difference convert by different rules: a rise of one degree Celsius is a rise of 1.8 degrees Fahrenheit, because an interval scales with the ratio and takes no offset at all.

Worked through at a value people look up: 21 °C x 9/5 + 32 = 69.8 °F.

Going back, Fahrenheit to Celsius is 5/9 times the Fahrenheit figure minus 17.777778. Note what does not happen here: with a scalar conversion the two factors are reciprocals and knowing one gives you the other, but an affine conversion has two constants and the reverse offset is not the negative of the forward one unless the slope is 1.

For mental arithmetic: Double it and add 30. It is exact at 10 C and two degrees off at 0 and 20 C (20 C gives 70 against a true 68). At 30 C it is four degrees high, and the error keeps growing the further you get from 10 C.

This is the pair that exists because two countries never agreed. Both scales are in daily civilian use, which no other temperature pair in this converter can say: Kelvin is a laboratory unit and Rankine is a thermodynamics one, but Celsius and Fahrenheit are both what somebody's weather app says. The practical trap is confusing a temperature with a temperature difference. A rise of 10 degrees Celsius is a rise of 18 degrees Fahrenheit, because intervals scale by 9/5 and ignore the offset. As a temperature, though, 10 C is 50 F, not 18 F. Interval or value is the first thing to settle.

On precision: the 9/5 and the 32 are both exact, so the conversion introduces no error of its own. What it cannot do is add precision the input never had. A weather reading of 21 °C is good to about a degree, and reporting 69.8 °F from it implies a thermometer nobody owns. The converter scales its decimals to the magnitude of the input for that reason.