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):