Dear Uncle Colin,

Inspired by a recent XKCD cartoon, I want to start measuring temperatures in radians celsius. How can I quickly convert between the two?

Made Up Nonsense? Réaumur’s Octogesimal

First up, MUNRO, that’s a really bad idea. I’ve said elsewhere that I don’t like degrees for measuring angles, as has Matrix the dragon; that doesn’t mean they’re bad for measuring other things. (Degrees Fahrenheit are obviously ridiculous, though. No argument there.)

But – with a heavy heart, and noting that water now boils at 59π radians celsius, here’s a quick and easy way to get at least close to converting between the two.

Noting that π227, and that 180º is the same thing as π radians, we can say 180º×7221 radian.

That simplifies down to 63011, but it turns out that 4017 is both easier to work with and closer. Incidentally, that corresponds to an estimate of π as 1260401, which I doubt will catch on.

This method boils down to “multiply or divide by 4007 and you’ll be in the right ballpark. Adjust by a quarter of a percent and you’ll be closer still.”

Let’s take the current temperature in Weymouth, which is 6ºC. The radians temperature will be smaller, so I need to multiply by 7400, giving me 42400 or 0.105 radians. Adjusting downwards by a quarter percent makes that 0.1047 or so – which it is indeed. (In fact, it’s π30, if you want to be picky about this.)

The temperature in the comic is 0.173 radians (although it’s not specified whether it’s Celsius or Fahrenheit). This, you multiply by 4007 to get 69.279.89º; or slightly less; adding a quarter percent (call it 0.025) makes that 9.91º. A slightly strange temperature to pick, but who am I to second-guess Randall?

-- Uncle Colin