 I think that we should add a unit of measurement for emotional mass, based on that depressing feeling that you get when you see yourself in a really bad selfie. I give you the Instagram. Randleman Rowe of XKCD writes an intermittent blog called What If, where he answers some interesting questions sent in by his readers, like whether it's more efficient to melt the snow on your driveway with a flamethrower or a microwave. If you don't work with numbers for a living, his answers may seem a bit like magic. Of course, Google can tell you things like the weight of a car or the average radius of the earth, but how is someone supposed to translate those facts into things like swimming pools full of saliva? Does he just have a really massive physics book somewhere with an incredibly comprehensive index? Well, no. Randle uses a method that's intimately familiar to people who use mathematics on a regular basis. So much so that many of us don't even think about how it was freaking magic the first time that we learned about it, called dimensional analysis. It's super easy, which probably explains why we don't think about it very much. Just treat the unit attached to every number that you're working with, as though they were variables multiplied by that number. 100 miles is the number 100 times the unit miles. Nine gallons per tank is nine times gallons divided by tanks. 30 miles per gallon is 30 times miles over gallons. Then when you need to figure something out using those numbers, like how much of your gas tank you'd burn driving 100 miles, you don't need to look up or remember any equations. You just look at the units themselves as variables and think about how you could multiply or divide them by each other to get the units that you're looking for. In that example, we just want tanks left over when we're finished. We want everything else to cancel on either side of the fraction bar. Some of these things look upside down, so we just divide by those numbers instead of multiplying. Cancel, cancel, cancel, and there's the answer. Just knowing how to do basic fractions lets you calculate almost anything. It also provides a guiding principle for working backwards to discover equations from a series of measurements, which is very handy when you're a researcher trying to develop a formula. Again, for many scientists and engineers, thinking this way is second nature whenever they're working with numbers, so much so that they don't even realize they're doing it. But surprisingly, it's a relatively modern technique. Before the popularization of the method in the early 19th century by Lord Raleigh, scientists had to depend mostly on their intuition to discover new equations. But when it blew up, it prompted a flood of very important scientific discoveries in many different fields. For example, one of its earliest advocates, James Maxwell, was playing around with the units in his equations for electromagnetism until he got these ones, meters per second, without any other variables attached to them. You know how the speed of light in a vacuum is constant? This is how we discovered that. When I first learned about dimensional analysis, it felt a little bit like I had received the keys to the castle. Like I could figure out anything. I mean, yeah, E equals MC squared is a brilliant idea and all, but it has to be MC squared. You need units of energy. Duh. But it turns out you still sort of need to know what you're doing if you're trying to discover new equations. It's not obvious in many cases, but almost every unit that we use for measurement actually contains a whole bunch of hidden variables that we often don't list, any one of which can affect the usefulness or truthfulness of our final answer. It takes a keen intuition to know when those variables can be safely ignored and when they can't. This is sort of hidden behind the scenes in Randall's calculations, a cautious hand steering the mathematics in the right direction. Rather than just plugging and chugging, it's clear that he's done a fair amount of research and reflection on the broader context for these questions. For example, when someone asks him how powerful a radar gun would have to be to physically stop a car, he checks to make sure that that amount of power output wouldn't just vaporize it. Although dumping numbers into the equation for conservation and momentum would give him some sort of answer, it turns out that that actually makes a difference. It takes a decent amount of insight to anticipate these kinds of factors that could meaningfully affect the final result, and unfortunately it seems that human brains aren't wired particularly well to do that. In 1999, some neuroscience researchers from Orsay and MIT published a study in which they analyzed the activity of human brains while processing different sorts of questions about numbers. In one scenario, test subjects were asked to report the sum of two two-digit numbers. Just add these together in your head. In another scenario, they were asked to estimate which of a set of numbers was closest to the sum. If you were to add these together, would it be about 100 or 200? The results solidified an idea which had been hinted at by several previous studies. We used totally different areas of our brain for approximation and calculation. When we're trying to figure out the exact right answer, regions responsible for verbal memory are most active, like what you'd use to memorize a list of names. There's a lot of research that shows that our ability to process large numbers with precision depends on language, and this seems to support that. On the other hand, if we're trying to approximate numeric answers, we use areas normally reserved for spatial reasoning and visualization, like figuring out what an object would look like if it were upside down, areas which have very little to do with language. In fact, case studies of people with brain damage show just how distinct these processes are. You can get a lesion in this area of the brain that totally destroys your ability to process numbers precisely, but for some reason, your ability to estimate answers is totally unharmed. All of this is at least suggestive that there are two distinct systems at play when we're working with math problems. There's the exact number crunching, symbol pushing, language-based system, which we use to discover the precise answer, and the rough estimation spatial system, which we use to get a sense of about what shape we can expect the answer to be. Although they can hand information off to each other, they don't really work in tandem. Like if I ask you to solve this equation, your brain does something drastically different than it does if I ask you about how many football fields are in a mile. You might shift gears from imagining football fields laid end to end on a stretch of highway to this equation, or vice versa, but they don't usually happen simultaneously. If you've ever thought that word problems felt a little different than other math problems, this is probably why. Our brains have to switch gears from pushing around numbers to thinking about what those numbers actually mean. And that's kind of a problem. The insight necessary to think of which important factors might not be included in an equation depend on a sort of real-world grasp of what these units are meant to represent, the sort of insight that you gain from stepping back for a moment and thinking, wait a second. Are we including the end zones? So how thick is that border around the outside? But while forgetting to check for radar gun car oblation or field border width while you're humming along in your dimensional analysis probably isn't a huge concern for you. Getting stuck in that pure calculation mode without engaging the other system to think of these things in context has many important practical ramifications. Market collapses, engineering screw-ups, incorrect scientific conclusions. It's relatively rare that any of these problems are caused by people doing math incorrectly. The majority of the time, they're caused by a failure to account for something important in the equations before the calculator comes out. Anyways, even with powerful tools like dimensional analysis making raw calculation easier, it's still important to occasionally engage that system of approximation that lets us interpret those results meaningfully and imagine what other sorts of factors might be worth considering. Your hobby is abusing dimensional analysis. In which case, go crazy. Do you still remember the first thing that you used dimensional analysis on? Please leave a comment below and let me know what you think. Thank you very much for watching. Don't forget to subscribe, blah, share, and don't stop thunking.