While traveling to Australia earlier this year, a cab driver taught me a quick Celsius-to-Fahrenheit conversion trick:

“Double the temperature and add 20 degrees.”

Because I’m good with the math, that means converting from Fahrenheit to Celsius would be:

“Subtract 20 degrees and divide by 2.”

A quick inspection of the actual conversion formulas—be they derived or simply looked up—shows why this trick works pretty well:

\(F^\circ = \frac{9}{5}C^\circ+32\)

and

\(C^\circ = \frac{5}{9}(F^\circ-32)\)

The \(\frac{9}{5}\) is a bit less than 2, which we correct for by adding only 20 (instead of 32) and, conversely, \(\frac{5}{9}\) is a bit more than 0.5, which we compensate for by subtracting only 20 (instead of 32).

For no good reason whatsoever, I wanted to see two things:

  1. At what point—if any—does this shortcut actually work perfectly?
  2. How far away from that perfect point can we go before that estimating technique really breaks down?

This could be an algebra problem…but why not make it a brute force visualization problem instead?

We’re going to stick with something of a reasonable air temperature range (we’re not boiling water or burning paper) for this exploration. As a matter of fact, we’ll just estimate the coldest and warmest temperatures I’ve personally experienced.

Converting Fahrenheit to Celsius

First, we’ll convert Fahrenheit to Celsius using the actual conversion formula and the shortcut formula:

The lines never cross…but they would if we went up to 140 F\(^\circ\) (which would be 60 C\(^\circ\) if you’re playing along at home).

Perhaps a better way to visualize this would be as the difference between the shortcut-calculated temperature and the actual temperature:

So, basically, this tells us that, in the summer months, if an American is converting his Fahrenheit temperature to Celsius with this shortcut method, he’ll almost certainly be overstating the temperature, but just by 1-4 degrees. In the winter, that overstatement will be more like 4 to 8 degrees.

Converting Celsius to Fahrenheit

Now, let’s convert Celsius to Fahrenheit and do the same comparison:

The lines never cross…but they would if we went up to 60 C\(^\circ\) (140 F\(^\circ\)).

Let’s now look at this as the difference between the shortcut-calculated temperature and the actual temperature:

So, basically, this tells us that, in the summer months, if a non-American is converting his Celsius temperature to Fahrenheit with the shortcut method, he’ll almost certainly be understating the temperature, but just by 3-7 degrees. In the winter, that overstatement will be more like 8-15 degrees.

The Conclusion

The shortcut formula works, but, to be a bit more accurate (without having to do math that involves odd numbers in fractions), the shortcut conversion should be qualified:

And I’ve already forgotten which is which.

The R code used to build this silly little analysis is available at https://github.com/gilliganondata/miscelleneous-explorations.