Celsius to Fahrenheit

°F = °C × 9⁄5 + 32. Handy anchors: 0 °C = 32 °F (freezing), 37 °C = 98.6 °F (body temp), −40° is the same in both.

Four in every nine Fahrenheit degrees can never come out of a Celsius reading

One Celsius degree is 1.8 Fahrenheit degrees, so whole Celsius readings land 1.8 apart and skip as they go. Between freezing and a mild day:

Nothing produces 33, 35, 38, 40, 42, 44, 47 or 49. Between 32°F and 50°F that is 11 reachable values and 8 that are not. The pattern repeats every 9°F exactly, because 5 Celsius degrees are 9 Fahrenheit degrees — five reachable, four not, forever. Over a wide sweep it comes to 44.3% unreachable, which is four ninths. So a Fahrenheit forecast translated from a Celsius one uses a little over half the scale, and the gaps are not random — they are the same eight numbers in every block of nine.

Which is why one direction survives a round trip and the other does not

Take a whole Fahrenheit temperature, convert to whole Celsius, convert back. Of the 213 values from 0°F to 212°F, 94 come back different — 44.1%. Do it the other way and every single Celsius value returns intact, over a range far wider than any thermometer you own.

The values that fail are exactly the values that were unreachable in the panel above. Not a similar count — the same set, every one. The cause is that the Celsius step is the coarser of the two, so an average of 1.79 Fahrenheit degrees land on each Celsius degree — one or two of them, never more — and only one can come back. When it fails it is always off by exactly one degree, never more. Which makes Fahrenheit the finer scale to report a whole number in, and Celsius the safer one to store it in. If a figure has been through both, it may not be the figure that was measured.

"Double it and add 30" — exactly how wrong, and where

The mental rule everyone uses is not vaguely approximate. Its error is a straight line, 0.2 × °C − 2:

It is exact at 10°C and nowhere else, and drifts one degree for every five you move away. Across 0°C to 20°C it stays within 2°F, which covers most weather and is why the rule has survived. Take it near an oven and it collapses: 200°C comes out 430°F against a true 392°F, 38 degrees high. The fix costs one step and is not an approximation at all: double it, take off a tenth, add 32. Doubling and removing a tenth is multiplying by 1.8, which is the entire conversion.

One curiosity that falls out of sweeping the whole two-digit range rather than picking favourites: there are exactly two temperatures whose Fahrenheit equivalent is the same two digits backwards — 16°C ≈ 61°F (truly 60.8) and 28°C ≈ 82°F (truly 82.4). Nothing else in the range comes within half a degree, and nothing above 28°C can, because the arithmetic runs out of digits.