Intro to significant figures

let's see if we can learn a thing or two about significant significant figures sometimes called significant digits and the idea behind the significant figures is to just make sure that when you do a big computation you have a bunch of digits there that you're not over representing the amount of precision that you have that the result isn't more precise than the things that you actually measure that you usually use to get that result but 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 seven zero zero so over here you have three three significant significant figures and it might make you a little uncomfortable that we're not including these zeros that are after the decimal point and before this seven that we're not including those because just like you know that helps that does help define the number and that is true but there's not it's not telling us how precise our measurement isn't try to understand this a little bit better imagine if this right over here was a measurement of kilometers so if we measure zero point zero zero seven zero zero kilometers that same measurement we could have this would be the exact same thing as seven point zero zero meters maybe in fact we just used a meter stick and we said it's exactly seven point zero zero meter so we measured to the nearest centimeter and we just 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 zeros are just taht kind of telling you are just are just shifting it based on maybe what the units of measurement that you're using but the numbers that are really giving you the precision are the seven the zero and the zero and the reason why we're counting these trailing zeros is that you didn't though 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 zeros off and they would have just told you seven meters 7.00 let's do the next one so based on the same idea we have the five and the two the nonzero digits are going to be significant figures you don't include this leading zero by the same logic that if this was point zero five two kilometers this would be the same thing as fifty two kilometers which clearly only has two significant two significant figures so you don't want to count leading zeros that are at leading zeroes before before this before the first non zero before the nut first non-zero digit I guess we could say you don't want to include those you just want to include all the nonzero digits and everything in between and and trailing zeros trailing zeros if a decimal point is involved I'll make those 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 take 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 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 zero here which means we got to the nearest tenth so in this situation once again we have three significant figures over here we you know the seven is in the hundreds but we got all the way down the measurement went all the way down to the thousandths place and even though there are zeroes in between those zeros are part of our measurement because they are in between non-zero digits so in this situation every digit 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've got an exact number you got exactly 37,000 or maybe you only measured to the nearest thousand so it depends on what you know there's there's a little bit of ambiguity here if you've just see something written exactly like this you would probably say if you had to guess you or not guess if there wasn't any more information you would say that there's just two significant digits 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 there was 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