## Sunday, July 29, 2012

### Quick and Dirty Unit Conversions

Part of getting used to my new city is acclimating to new units of measure. Which have rather annoying conversions to perform mentally on the fly. In time, I won't need conversions, because my internal sense for these things will calibrate, so that I know what 10C feels like, and how painful it is to pay 15 Argentine pesos for a good cup of coffee (not that painful!), and about how long it takes to walk 3K. In the meantime, I have noticed myself devising some "good enough'" estimation shortcuts. Of course, I've also analyzed the reliability of these shortcuts, because I'm like that. And when you're a Precalculus teacher, everything looks like a function transformation.

For example, I have never been able to reliably remember the formula to convert between Celsius and Fahrenheit. I know it has a 5/9 or a 9/5, and I know you add or subtract 32, but good luck remembering what goes where when you need it. And then mentally working on a number that is not a multiple of 9 or 5? No thanks. (Dear mansplainers: before you head to the comments to mansplain this to me, I can look up formulas any time I want. And yes I know I can use my phone here in any number of clever ways. The point isn't exactness - it's estimation with good-enough accuracy while simultaneously not dying in traffic.)

My first tactic was to devise a quicker formula. Mostly, I am having to hear a temperature in Celsius, and convert it to Fahrenheit, so that I know whether I will need a jacket or whatever. So I need something close to F = (9/5)C + 32. Okay so 9/5 is pretty close to 2, and 32 is pretty close to 30. "Double it and add 30" is a function I can work with, even after two glasses of Malbec. Let's see how useful this is with a graph.

The two rules give the same result, 50F, at 10C. The estimation is worse the farther you get from 10C in either direction, but how hot/cold does it really get here? Even at extreme temperatures for Buenos Aires, we are not that far off. 0C should be 32F, but with the new rule it's 30F. (And yep that's as cold as it gets here. Feel free to hate me. I did my time.) 40C is really 104F, but with the estimation we get 110F. I don't know about you, but I don't know the difference between what 104 and 110 feel like, because they both mean "ermahgerd, stay inside and have a beer or something." We can also summarize this with a difference function: 2x + 30 - (9/5)x - 32 which simplifies to (1/5)x - 2. Let's put this on our graph.

So there's a picture of what we just said in words. The x-intercept at 10C is when the difference is 0. The graph shows us how far off the estimation is for any temperature. Looking at x (celsius) values from 0 (hace frío) to 40 (hace mucho calor) we don't get far enough away from the x axis to matter for our purposes.

Converting in the other direction is not as useful to me, because I'm not generally hearing Fahrenheit temperatures and needing to know what they are in Celsius. However, we can use our knowledge of functions to realize we don't have to go through this analysis again. We can easily picture the inverse estimation function, "subtract 30 and halve the result" along with the inverse of the actual formula by reflecting both over the line y = x. Therefore the differences will have the same distance, therefore we're safe to use the estimation going the other way.

A second tactic I have found myself using is to mentally carry around a table of anchor temperatures. If I remember…
0C = 32F
10C = 50F
20C = 68F
30C = 86F
40C = 104F

…I can do a good-enough mental interpolation to know how it's going to feel outside. But that doesn't make for as fun a blog post. It's about the best I'm doing now with pesos to dollars, though. I just think of \$100ARD notes as \$20US notes (but I'm dissatisfied with that -- see below.)