all right significant figures i think significant figures get a little bit of a bad reputation and that's probably rightfully so however they're super important so what we will do is talk about significant figures now we will look at how to apply them in calculations and you will apply them in calculations near the beginning of this class and then you will use them in second semester lab if you take the second semester lab so significant figures it's the same number of digits as if you were reading the value off of an instrument and all men measurements contain some amount of uncertainty so an example of what that looks like if you're reading the volume of a liquid in this piece of glassware which is called a graduated cylinder the volume might not fall perfectly on one of the lines that's given so we have markings that tell us a certain number of decimal places and then we always estimate the last digit right if it's exactly on the line the last digit estimate might be a zero if it's in between the lines then you will estimate based on how far it is from one of the other lines and significant figures reflect that uncertainty in the measurement now this can be kind of discussed in two different parts so one way to look at it is first just how many sig figs are there or is a digit significant and then the second thing that we'll do is when you have multiple measurements with different numbers of significant digits how does that propagate when you do a calculation so the first question of is it significant or how many sig figs we have it's kind of broken down like a set of rules i'm going to go through this with some examples so first all non-zero digits are significant so in 28.03 the two the eight and the three definitely significant in point zero five four zero the five and the four definitely significant the second thing to consider are zeros that are interior so a zero that falls between two non-zero digits so 408 based on the first rule we know that four and eight are definitely significant and then the zero is kind of trapped in between them that zero is significant and in the number 7.0301 we know 7 3 and 1 are definitely significant and then the two zeros in there there's one trapped between the seven and three and one trapped between the zero and the one those are definitely significant now let's talk about zeros in different places leading zeros are not significant now the two examples that i have here are both with numbers that are less than one because you're really only gonna see a leading zero if you have a really small number we tend like they you could technically write it in front of a bigger number but we tend not to by convention so in the number .0032 there are no zeros that are trapped in between the three and the two we only have leading zeros and none of those are significant in the number .0006 all of those zeros are leading they only exist there to be a placeholder so you can't communicate the value of this really tiny number without putting a zero there but it is not significant it's just a placeholder that helps you tell the value of this number now trailing zeros have two different cases uh i'm gonna say two cases plus one unclear case that we're going to do our best to rectify so if you have a trailing zero after a decimal place it is significant so 45.000 those zeros don't need to be there to tell you the value of the number we could have just written 45 and you would understand the magnitude of that number because we explicitly included the zeros after the decimal place they are significant so this number would have five significant figures the number 3.560 is the same scenario you don't need that zero to communicate the magnitude but we did include it so it is significant now there's one more case if you have something a zero before a decimal and after a non-zero digit so it is there to show magnitude but we have put a decimal place after it to give an indication that it is place holding but also significant so 140.00 is five significant figures 250.6 now this also follows that trapped between two non-zero digits rule this would have four significant figures now that's this rule that we just did is before a decimal but what about before an implied decimal so what about 1 200 with no decimal place those zeros are there as placeholders there's an implied decimal place after this because that last zero is in the ones place it's just it's unclear because we didn't explicitly write the decimal place it's super unclear whether or not those zeros are supposed to be significant so we will avoid this at all costs the best way to avoid this is by reporting your answers in scientific notation so for scientific notation everything that i have written here is the same value and it's also 1200 but when you write something in scientific notation the number that you have before the times 10 to whatever power you need to use is all of those digits are significant so if i write 1.2 times 10 to the third the 1 and the 2 are significant this number has two significant figures if i write 1.20 times 10 to the third the 1 2 and 0 are all significant so 3 significant figures and if i wanted to write it with all four digits significant i would write 1.200 times 10 to the third we will also sometimes encounter exact numbers so these are numbers that have no uncertainty and because there's no uncertainty they don't affect significant figures and calculations this will usually come up when you are counting discrete objects um so if i say that i have three apples right this and especially something like an apple this macroscopic thing you can tell whether or not that apple is whole it is there's no uncertainty about whether i have slightly more slightly less than three apples if they all appear to be whole um three atoms now it's a little bit harder to discreetly count three atoms since they're so tiny but three atoms it's not gonna you can't go any smaller and still be an atom so this is sometimes talked about as having an infinite number of sig figs so you know my example here i've written 3.000000 it has a lot of significant figures because it is that certain we don't have the uncertainty in that measurement the other thing that will be an exact number is a conversion and a definition so the conversion between centimeters and meters there's 100 centimeters in a meter there are exactly 100 centimeters in one meter because that's the definition of the prefix centi it means exactly 10 to the minus 2. you will also not count sig figs for numbers that are appearing in an equation so right there's an equation to convert from the radius of a circle to the diameter of a circle where you multiply by two that two is an exact number it's not going to count against your sig figs