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?

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 $\frac{5}{9}\pi$ radians celsius, here’s a quick and easy way to get at least close to converting between the two.

Noting that $\pi \simeq \frac{22}{7}$, and that 180º is the same thing as $\pi$ radians, we can say $180º \times \frac{7}{22} \simeq 1$ radian.

That simplifies down to $\frac{630}{11}$, but it turns out that $\frac{401}{7}$ is both easier to work with and closer. Incidentally, that corresponds to an estimate of $\pi$ as $\frac{1260}{401}$, which I doubt will catch on.

This method boils down to “multiply or divide by $\frac{400}{7}$ 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 $\frac{7}{400}$, giving me $\frac{42}{400}$ or $0.105$ radians. Adjusting downwards by a quarter percent makes that $0.1047$ or so – which it is indeed. (In fact, it’s $\frac{\pi}{30}$, 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 $\frac{400}{7}$ to get $\frac{69.2}{7} \approx 9.89$º; 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