How Accurate Is That Number? ➶

From the transcript of Charles Bailyn’s ASTR 160: Frontiers and Controversies in Astrophysics Lecture 2: Planetary Orbits (emphasis mine):

Student: How accurate is that number?

Professor Charles Bailyn: How accurate is that number? It’s accurate to about one digit, which is why I only wrote down one digit of accuracy out front. This is a pro–excellent question thank you very much. This is appropriate because remember I’m using 7 times 10 minus 11 for G, where it’s actually 6.6. So, I’m already about 10% off from there. I did my little calculation to come up with one year equals 3 times 107 seconds. That’s about accurate to one digit or so. And so the whole thing is done to one digit accuracy. If you’re dealing with one digit accuracy, it is true that 7 divided by 4 is 2. It really is true, because if that wasn’t true, then you would have to have more digits on your unit for G or something like that. In particular, let me make an official rule for this course. Three equals π, equals the square root of 10, all right. That will solve an enormous amount of arithmetic problems and it will not get you into any serious trouble. So, we don’t have to worry about the .14159 and however many more digits you all memorized it to. And when you multiply it together you get ten. Yes?

Student: Are you expecting this kind of calculation for problem sets?

Professor Charles Bailyn: Yes. The question was, “Am I expecting this kind of calculation for problem sets?” The answer is “yes.” Here’s the rule about calculators. Let me put it this way: You can only use calculators if I can’t tell that you’ve done it. So, that means you can check your work to make sure you’ve it right or something. But if you start coming up with numbers like 7.1516397, that’s eight digits of accuracy and I’m pretty sure you haven’t worked it out yourself. So important, no calculators on the tests, okay? So, get some practice doing this kind of thing. And this will–this I promise you will be useful to you in everyday life because this is how you catch the politicians doing screwy things with big numbers. You do it in your head in scientific notation and you figure out whether the answer is meaningful or not.

This whole business of significant digits, I think, is badly distorted; by the way, it’s taught in high school. In high school you, and also I should say in laboratory courses sometimes at the college level, you often get situations where people say–give you a whole sheet of rules on how to figure out how many significant digits you have. This is nonsense. All you have to do is behave like a human being. We say to each other, I’ll meet you in the dining hall in ten minutes. That doesn’t mean–that means something different from I’ll meet you in the dining hall in eleven minutes and twenty-six seconds. Even if the person happens to show up in the dining hall in exactly eleven minutes and twenty-six seconds. Ten minutes means I’ll meet you there in ten minutes, we all know what that means. I’ll meet you there in eleven minutes and twenty-six seconds means you’re a character in a bad spy novel who’s just synchronized his watch. So, this shows up in science fiction too.

I don’t know how many of you are Star Trek fans, I certainly am [laughter]. And in all the different Star Trek movies [inaudible comment]–thank you. In all the different–a friend [referring to person who made comment]. In all the different Star Trek movies there’s always a second in command who isn’t a human being, right? A Vulcan or an android or some damn thing or another. And to emphasize the non-humanness of these characters, what they do is they make them use too many significant digits. And so that makes them inhuman and so the captain will say, “When are we landing on omicron M?” The second in command will say, “Well, we should assume standard orbit in 2.6395 minutes,” emphasizing somehow superior brain power or something. But it’s nonsense because it takes the guy ten seconds to say that sentence, so what is this time calculated to a 100th of a second? Does it start from when he begins the sentence? From when he ends the sentence? What’s the other end of that time interval? Can you say you assume standard orbit to the 100th of a second? What does that even mean? When you start beaming down? When you end beaming down? Also, keep in mind it takes more than a 100th of second for the sound to travel from his lips to the captain’s ears, so the whole thing is just nonsense. And so, you don’t need any special rules, just behave like a human being; don’t behave like an android. So, no androids. And that’s the only rule I’m going to give you [laughter]. These two are the only rules I’m going to give you about significant digits, just do the right thing, okay.

Two rules:

1. $3= π= 10$
2. No androids!