Intro to significant figures
Let's see if we can learn a thing or two about significant figures, sometimes called significant digits. And the idea behind significant figures is just to make sure that when you do a big computation and you have a bunch of digits there, that you're not over-representing the amount of precision that you had, that the result isn't more precise than the things that you actually measured, that you used to get that result. Before we go into the depths of it and how you use it with computation, let's just do a bunch of examples of identifying significant figures. Then we'll try to come up with some rules of thumb. But the general way to think about it is, which digits are really giving me information about how precise my measurement is? So on this first thing right over here, the significant figures are this 7, 0, 0. So over here, you have three significant figures. And it might make you a little uncomfortable that we're not including these 0's that are after the decimal point and before this 7, that we're not including those. Because you're just like, that does help define the number. And that is true, but it's not telling us how precise our measurement is. And to try to understand this a little bit better, imagine if this right over here was a measurement of kilometers, so if we measured 0.00700 kilometers. This would be the exact same thing as 7.00 meters. Maybe, in fact, we just used a meter stick. And we said it's exactly 7.00 meters. So we measured to the nearest centimeter. And we just felt like writing it in kilometers. These two numbers are the exact same thing. They're just different units. But I think when you look over here, it makes a lot more sense why you only have three significant figures. These 0's are just shifting it based on the units of measurement that you're using. But the numbers that are really giving you the precision are the 7, the 0, and the 0. And the reason why we're counting these trailing 0's is that whoever wrote this number didn't have to write them down. They wrote them down to explicitly say, look, I measured this far. If they didn't measure this far, they would have just left these 0's off. And they would have just told you 7 meters, not 7.00. Let's do the next one. So based on the same idea, we have the 5 and the 2. The non-zero digits are going to be significant figures. You don't include this leading 0, by the same logic that if this was 0.052 kilometers, this would be the same thing as 52 meters, which clearly only has two significant figures. So you don't want to count leading 0's before the first non-zero digit, I guess we could say. You don't want to include those. You just want to include all the non-zero digits and everything in between, and trailing 0's if a decimal point is involved. I'll make those ideas a little bit more formal. So over here, the person did 370. And then they wrote the decimal point. If they didn't write the decimal point, it would be a little unclear on how precise this was. But because they wrote the decimal point, it means that they measured it exactly to be 370. They didn't get 372 and then round down. Or they didn't have kind of a roughness only to the nearest tens place. This decimal tells you that all three of these are significant. So this is three significant figures over here. Then on this next one, once again, this decimal tells us that not only did we get to the nearest one, but then we put another trailing 0 here, which means we got to the nearest tenth. So in this situation, once again, we have three significant figures. Over here, the 7 is in the hundreds. But the measurement went all the way down to the thousandths place. And even though there are 0's in between, those 0's are part of our measurement, because they are in between non-zero digits. So in this situation, every digit here, the way it's written, is a significant digit. So you have six significant digits. Now, this last one is ambiguous. The 37,000-- it's not clear whether you measured exactly 37,000. Maybe you measured to the nearest one, and you got an exact number. You got exactly 37,000. Or maybe you only measured to the nearest thousand. So there's a little bit of ambiguity here. If you just see something written exactly like this, you would probably say, if you had to guess-- or not guess. If there wasn't any more information, you would say that there's just two significant figures or significant digits. For this person to be less ambiguous, they would want to put a decimal point right over there. And that lets you know that this is actually five digits of precision, that we actually go to five significant figures. So if you don't see that decimal point, I would go with two.