in this video we will be looking at significant figures in calculations and measurements first thing we have to think about in significant figures is you know we're dealing with numbers these numbers are supposed to reflect what we know about a physical object or something in our physical world the first thing we really want to be aware of is that all measurements contain some level of uncertainty it doesn't matter if you're measuring something with a ruler whether you are measuring something a simple balance a complicated balance any measurement you take whether it's done by an instrument or by hand has some degree of uncertainty in it and depending upon what instrument or what measuring device you use in order to make that measurement that's going to affect how uncertain that measurement is or how much certainty there is versus uncertainty a lot of times when we do a measurement there's an estimated digit a digit best estimate is called uncertain we're not really sure that based upon what that instrument can provide for us that's where the certainty ms and an estimation begins and that's usually the last recorded digit the two balances in the picture below do a good job of kind of representing how we take measurements and what significant figures really tell us so these two balances have different levels of certainty and uncertainty the one on the left is a relatively simple balance it can estimate down to the hundreds place its certainty runs out in the tenths place meanwhile the balance on the right we see that we have certainty all the way down to the thousands place but now we're uncertain at the ten thousands place so there's a different degree of certainty in the measurements if we were to weigh something on each one of these bounces and that's what we want to think about imagine putting the exact same object on each one of these balances now regardless of which balance you put it on the object didn't change but the information we're getting back is going to be a little bit different based upon what device we are trying to measure that solid object on so that's really important when we start thinking about going beyond numbers because sometimes we can get lost in the rules for significant figures and we lose sight of what's actually physically happening we're trying to tell a story about what that measurement is what that measurement is actually doing so significant figures tell us the level of uncertainty in a measurement it can be used to determine what instrument was used to conduct that measurement if we look back at the previous slide and you think about those two balances well based upon you would report that mass to somebody else they would know kind of what level of scale or what level of balance you use to make that measurement anytime we're taking a measurement or allowed one place one digit of estimation so our rules for counting significant figures remember all these kind of play back to this idea of we're getting some physical knowledge about that measurement that we took so the first rule is pretty straightforward that's anytime you have a non-zero digit so not indeed zero digits or those one through nine those are always going to be significant there's no exceptions to that those are always going to be significant so for an example if we look at the number 213 point five so are all nonzero digits so this would have four significant figures in it all of those would count as significant in the number one point five four all of these are nonzero digits so we would have three significant figures in that non it gets a little bit more complicated and potentially confusing when you start getting into zeros and that's because there's different types of zeros you have to identify what type you're dealing with they all kind of have little slang catchy names that go along with them that help you realize what type of 0 you have the first we call leading zeros so these leading zeros come just what the name is implying are zeros before the first non-zero number these leading zeros are not significant sometimes they go by the name placeholder 0 so here we have two examples of looking at some leading zeros in the first one zero point zero zero two three we look at these zeros all of these zeros appear before the first non-zero digit that means those zeros are not significant so the zeros aren't significant but the two and the three are that's where we're going to get our two significant figures from in the number zero point zero zero zero zero zero zero six 75 all of those zeros are leading zeros those are not going to be significant so our only significant figures in this would be the six to seven and the five and we'd wind up with three significant figures can be almost the exact opposite of the leading zeros are the trailing zeros these are zeros that appear to the right of the last non-zero digit and and it's a very important and to the right of the decimal place so both things have to be true they have to be to the right of the last non-zero digit and to the right of the decimal place these end up being significant and we can kind of think that they're going to be significant because they're telling a different story so let's look at some examples of some trailing zeros so first nineteen point three four zero zero well we know the one 9 the 3 and the 4 are definitely significant because they're non zero but we look at these zeros these are trailing zeros there to the right of the last non-zero digit and to the right of the decimal place those are going to be significant as well for six significant figures now you may be looking at this and say well what are those zeros even matter because they don't change the numerical value of the number well you'd be right they don't for what they do do is they tell you what level the measurement was at where your uncertainty happened if you didn't have these zeros well then your estimation was here at the hundreds place with the four here you're showing that your measurement was much more certain than that your measurement told you that I could estimate all the way down here to the ten thousands place and I already had certainty at the thousands place so you know those trailing zeros don't change the numerical value of the number any they're very important because they're telling you something about that measurement what level did you know that AK if we look at another example zero point zero zero zero zero zero zero three two zero zero here I have a mixture of some leading zeros all the front zeros are leading zeros those are not going to be significant we know the three and the two will be significant these two zeros are trailing zeros so I'm going to have a total of four significant figures in that digit another type of zero is a captured zero a captured zero are ones that are between two other significant figures so these captured zeros kind of lie in between they're captured by other significant figures if you have a significant figure to the left and to the right well I'm probably pretty safe to assume that these are going to be significant as well so here let's look at some examples of some captured zeros so 2300 for point one well we know the 2 the 3 the 4 and the 1 are all significant this 0 is captured between the 3 and the 4 so being it's captured between the 3 and the 4 that's going to be significant all of these are going to be significant figures doesn't matter where in the decimal place you know to the left to right these captured zeros appear our next example thirteen point six zero zero nine well the one the three and six are definitely significant they're nonzero same with the nine here I can have more than one capture zero here I have two captured zeros well they're between the six and nine those must be significant all of these are significant we wind up with six significant figures the 1,900 points zero zero zero well we know the one in the nine are significant we actually don't know anything about these zeros these zeros that are kind of above the green here those first two zeros we don't know anything about yet but what we do know is that the zeros to the right of the decimal place are going to be trailing zeros why these three zeros to the right of decimal place are to the right of the decimal place and to the right of the last nonzero number those are gonna be trailing zeros those three at the end or significant now that those are significant these two zeros these first two zeros that we found that we saw in the number these are now going to be captured zeros and are going to be significant as well there's another type of zero and this is of zero we really want to avoid because it can be very confusing how to deal with this and these are ambiguous zeros and just like the name applies they're very ambiguous it's tough to tell what is meant by that zero these are zeros that sometimes appear in a large whole number difficult to tell if these are going to be significant or not example is 2400 are these zero significant or not or maybe are some of these may be significant while the other ones are not it's very very tough to tell so being it's so tough to tell we need to have a way of avoiding this while still being able to represent these large whole numbers that may contain zeros so to avoid writing these ambiguous zeros when you have this large whole number where it's confusing it's best to rewrite the number using scientific notation to be clear what the zeros are let's kind of look at that 24,000 well how can I write that such that it was really clear that I don't want the zeros to be significant or I want the first one to be or the second one or the third one it's really tough but there is a way way to do that is it change this using scientific notation if I change this a scientific notation I can write it such a way that these ambiguous zeros I can clearly put as a different type of zero and let's see that if I don't want those zeros to be significant I can rewrite this in scientific notation to be 2.4 times 10 to the 4 I'm chopping those zeros out not showing very clearly two significant figures now not those zeros weren't significant if I need the first zero to be significant but not the second two if I rewrite that in scientific notation is 2.4 zero times 10 to the 4 I'm showing that first zero as significant I get my three significant figures there if I need to show four significant figures or that the first two zeros are significant but the last one is not I can rewrite it in a way that shows that two point four zero zero times 10 to the fourth I can clearly see those four significant figures what I'm doing with these ambiguous zeros going through in scientific notation is if I don't want them to be sick leave them out of the scientific notation if I need them to be significant when I rewrite the number in scientific notation those ambiguous zeros are now becoming trailing zeros which are perfectly clear to the reader that hey these are meant to be significant if I need all of them to be significant I can rewrite that way 2.40 0 times 10 to the 4th I'd get the five significant figures out of here now here's another case where when you look at this the numerical value of all of these numbers is identical mathematically in a pure mathematical sense those are all exactly the same all of these numbers it's still 24,000 a pure abstract mathematical sense but how I choose to represent that number can display some physical meaning to what that number really represents and that's why scientific note herbs aren't significant figures are important so we can use scientific notation to display the correct number of significant figures when we're dealing with some of these ambiguous zeros there's some other type of numbers these are called exact numbers an exact number does not come from a measurement and just kind of what their name is applying they are known exactly if you know something exactly there's zero uncertainty since there is no uncertainty these numbers would have an infinite number of significant figures regardless of how you write them so you have to understand when you're working with numbers where that number is coming from there's two types of exact numbers the first type of exact numbers are counting numbers these are things you can physically count you know you may look at and say physically counting is a form of a measurement it's generally assumed that if you physically count things there's no uncertainty to it so when we say pencils on a table or a desk in front of us that six came from a counting type number it was a counting number we physically were able to count those six pencils we're not looking at one significant figure here for the sixth it really has an infinite we don't write 6.000 because that gets very cumbersome instead we recognize hey that was a counting number that's going to have an infinite number of significant figures same thing with 14 sheets of paper you have 14 sheets of paper on your desk in front of you well that's not two significant figures from the one on the floor you physically counted out those 14 sheets of paper you counting out those 14 sheets of paper there's no uncertainty to that this would really have a infinite number of significant figures to it your second type of exact number our definitions and if you have a definition you're going to define that relationship to be exactly true so that definition is an exact definition and we use these all the time twelve inches equals one foot well the 12 inches isn't two significant figures and the one foot isn't one significant figure this is exactly 12 inches equals exactly one foot that's the definition we understand that that's exact I didn't come from a measurement that came from an exact definition those that have an infinite number of significant figures other ones we could talk about a thousand grams equals a kilogram again that's a definition infinite number of significant figures seven days in a week not one significant figure from the seven days or one significant figure from the week an infinite number of significant figures for cups on the court again exact number because it's coming from a definition a lot of times we're not just measuring something and then it goes away we're doing something with that measurement so there's rules for dealing with significant figures especially in terms of the rounding basically what you're doing is at the end of a calculation you're dropping in significant digits things that don't have any physical meaning this is only at the end of calculations and this really involves kind of the weakest link principle and you may have heard of this principle and reference to a chain may be heard the old saying a chain is only as strong as its weakest link what that means is no matter how strong the other links in the chain are the overall strength of that chain is based upon whatever the weakest link in there is that's where it's going to break even if the other links can hold a lot more it just doesn't matter because it's gonna break at the weakest link that same idea that same principle applies to our calculations and our significant figures the number of significant figures in the final result cannot be greater than the weakest link used in the calculation and that kind of makes sense your end result can't be any better than the numbers that went into it it just can't happen because you already have a lot of uncertainty in a initial number you can't all of a sudden get less uncertainty out of your final answer so the actual rule depends upon the exact mathematical operation let's have a look at a couple examples of that multiplication division is the biggest one you end up following this rule all the time number of significant figures in the result is the same as the number in the least precise measurement used in the calculation so here we're looking at six point three eight times two point zero well the mathematical answer to that is twelve point seven six but when we're dealing with physical objects we have to represent that certainty in that measurement well here six point three it had three significant figures 2.0 only had two significant figures so that means we need to round our final answer to those two places so I need to round here at the ones place so that's why I'm rounding up to 13 to represent the two significant figures works same way with division sixteen point eight four divided by two point five four well the sixteen point eight four that's four significant figures the two point five four three significant figures when you just start lending to your calculator you get the six point six two nine nine that's the mathematical ones but that doesn't have a physical reality the physical reality here is I only know up to three significant figures so I need to round to three places one two three that would be in the hundreds place here so rounding to the hundreds place I'd go six point six three my three significant figures would match the least precise measurement I use in that calculation if you're doing a whole string of five or six multiplications or divisions in a row you just look for your least precise measurement that was used in that and then round your final value to that addition subtraction works with the same general idea a little bit different mathematical rule here this weakest link idea is that the number of significant figures in the result depends upon the number of decimal places in the least accurate measurement so here six point eight plus eleven point nine three four what that means is six point eight I'm and at the tenth spot eleven point nine three format the thousands place the tenths place is less precise less accurate therefore my final answer I lino down to the tenths place that's where I have to round things at same thing for subtraction thirty seven point six five seven minus six point two one well the thirty five point five five seven that is the mathematical answer but that doesn't represent the physical reality very well so here I'm looking at my decimal places again because that's how we work with addition subtraction thirty seven point six five seven is at the thousands place the two point one is at the tenths place that means I need to round at the tenths place so my final answer I need to round at that tenths place and that's going to give me my final value so multiplication and division we're looking at least number of significant figures addition and subtraction least number of decimal places