When I first learned about significant figures, back in high school, they seemed totally arbitrary and annoying to me. Now, though, I know they’re incredibly important. The rules seem silly at first, but actually they make sense. So, if they seem dumb, please, please stay with me.
Ok, here we go. Let’s say I give you a ruler that only has inches marked on it. (So, none of the little tick marks in between, just 1 inch, 2 inches, etc.) Now, I ask you to measure the height of a cup I have. What answer might you give?
Really, you can only say for sure that it’s close to 4 than it is to 5 or 3. You might be able to guess that it’s halfway between 4 and 5, so you could say 4.5 pretty reasonably, but you definitely couldn’t say 4.56712. Because how would you know that?
In science, we’re measuring stuff all the time, and not only do we need to know what we’ve measured, but we also need to know how precisely we’ve measured it. Like, is the acceleration due to gravity really 9.8, or were we only able to measure it to one decimal place? (Actually, it’s different in different places, and even points in slightly different directions if you’re near a big mountain or a pile of gold.)
For example, I was out in the woods one day looking for crime scene evidence, and I was pacing out distances. If you’ve never paced out a distance, when most people take two steps they go a pretty constant distance, maybe 4 feet, so you can use pacing to measure how far you’ve gone. Well, I was out in the woods with the other members of Explorer Search and Rescue and we were pacing along looking for evidence, and I found a piece of trash. I reported it to my teach lead and he asked me how far I was from the place we started. I have a 4.5 ft pace, and had gone 5 and a half paces, so I did the math in my head and then, without thinking, said something like 24.75 ft. The team lead was laughed and was like “oh yeah? wow.” Because, here I am being incredibly imprecise, wading through bushes, and I’m telling him that the piece of trash I’ve found is exactly 24 ft and 9 inches away. But, that’s what my math told me.
So, the problem is, when we do math with imprecise numbers, we sometimes get things that look more precise than they are.
Like, if I take 1 and divide it by 3. I get 0.333333333333333333333333333333333333333…
The first two numbers (1 and 3) are integers, I have no idea if they’re 1.4, 1.6, 3.1, or what, but when I divide them, I get something that looks like I’ve measured it out to the infinite decimal place. Silly.
So, we have to watch out for situations where math spits out numbers that don’t mean anything.
After that first 3, the other 3’s don’t mean anything at all. 1 divided by 3 might be 0.32 or 0.34447. Because, I don’t know whether the 1 was really a 1 or whether it was actually 1.1 but I didn’t have good enough tools to measure that carefully.
Does this make sense? The key here is that 4 is different from 4.000000000 because if I just say 4, then that might actually be 4.1 or 3.9 or 4.12321111.
Because really, in the real world we don’t often have things that are precise, we only have measurements (unless I say something like “I have 2 dogs.” That’s probably exact. But, I’m 5′ 4″ tall, but I have no idea how many nanometers that is.
I hope that wasn’t boring. I realized partway through that maybe no one cares about sig figs, except that you’re made to calculate them. If you just wanted to learn how to calculate them, you can look through this short powerpoint I made that summarizes all the rules and shows some examples. Feel free to steal it and use it however you like (I mean, within reason):
In physics, math suddenly becomes something useful. It’s now a tool that will tell you how quickly something gets somewhere, or how high it will go if you shoot it out of a cannon. This is awesome, but it can also be hard, because if your math isn’t solid then when you’re in the middle of a tough physics problem you might get distracted as you try to figure out how to solve for the root of a quadratic. By the time you figure it out you’ve forgotten why you were solving for it. (Also, this happens to everyone, so it’s not bad, we just want to minimize it as much as possible.) Hopefully, the math has been practiced enough that it’s somewhat second nature, so you can focus on the reasoning behind the math you’re doing.
Luckily, there isn’t too much math you need to know, it’s not too fancy. Here’s a quick quiz you can take to see if you’re ready:
Work through the problems, and if there are some you can’t get that’s no problem, it just means there is some stuff you might want to review before you start your class. Here is the answer key so you can check your answers:
Next, for each question you missed, here’s a homework assignment for you:
Over the next week I’ll be writing some posts about good things to practice and know before you start taking a physics class. This includes making sure your math is solid, and getting a head start on the basics before school starts. Each day I’ll post a short practice assignment. You can post answers here in the comments, or email them to me with questions.
Here’s what I’ll be covering:
1. Introduction (ie. this post 🙂 )
2. Are your math skills ready for physics?
3. What are significant digits and why are they significant?
4. Why does everybody think physics is hard? (An introduction to problem solving.)
5. SI Units and the Metric System
6. When should you start studying for the AP test, and what can you do at the beginning of the year to make that easier?
7. A head start on 1-dimensional kinematics (usually the first thing you learn).
In each of these, I’ll be giving links to resources (using them will be part of the homework) so that when school starts you’ll already have a ton of places to go to get extra information (plus you can always ask me questions here or on my facebook page.)
Every year I have a couple of students who are taking calculus-based physics at the same time as calculus. Traditionally, you take calculus first, and then you take physics with calculus. The calculus used in these introductory physics classes is usually simple enough that it’s not impossible to take them both at the same time, but it’s definitely a challenge. I wouldn’t recommend it if you can avoid it, but if you can’t there are some things you can do over the summer that will make next year go much more smoothly:
- Recognize that you’re at a disadvantage. The worst thing possible outcome is that you end up thinking you’re bad a physics. Even if there isn’t much calculus in the course, a lot of the ideas in physics are covered in calculus. There is a lot of conceptual overlap. So, just keep in mind that everyone else who took calc last year has had a whole year to get comfortable with these ideas and to practice them. Also, you’re having to learn two challenging things at once. Not only that, but you’re having to put them together. It’s as if you just learned how to juggle and you also just learned how to ride a bike, but you’re forced to practice them both at the same time.
- Get a head start on calc. I recommend Calculus Made Easy, which is actually the book I used to help me learn calc when I was in high school. Start at the beginning and work up through chapter 8. Take your time. Focus more on the ideas than on the math.
- Work through my book Rates of Change. I feel like recommending this to you is a bit of a conflict of interest on my part, and I agonized over whether to include it in this post, but I really do think it helps make the fundamental ideas much clearer.
Lastly, once school starts, don’t hesitate to ask for help, either from your teacher, a tutor, or me.