in section 1. 5 we're going to Define accuracy and precision distinguish between exact and uncertain numbers correctly represent uncertainty in quantities using significant figures and apply proper rounding rules to compute quantities so being absolutely certain of a value is actually a very rare thing and the only way to actually do this is through counting so you can say that there's one egg two eggs three eggs but second you start talking about how much mass say of eggs you have like I have or how much weight of eggs you know like I have three pounds of eggs or something like that you don't have exactly three pounds you may have 2. 99 pounds and if you measured it even more carefully you might find you have 2.
993 pounds of eggs so you can't absolutely be certain unless you're counting a discrete unit the result of counting measurement is an example of an exact number the numbers for defined quantities are also exact for instance one foot is exactly 12 inches because we said one foot is exactly 12 inches and we can say that one inch is exactly 2. 54 centimeters just because we say that this is true so we can Define things to to be exact or we can count them other than that all numbers have some level of uncertainty with them so uncertainty is a common thing quantities derived from measurements other than counting are uncertain to varying extents these numbers are not exact there are always practical limitations of the measurements Process Measurement processes that we use a measured number must be reported in a way to indicate its uncertainty in general when recording a measurement you are allowed to estimate one uncertain digit so let's see what that means so here for instance uh we have a graduated cylinder it has 20 here has 25 here it has five markings in between so each one of these is one uh milliliter so here we don't have 20 this is 21 this top line here would be 22. we get to estimate between the two so I could say maybe that's a little closer to 22 so I can say maybe it's like 21.
7 or something like that um so I get one estimate between the graduations and that is the best that I can do using this instrument uh on the previous slide if one recorded the volume in the graduated cylinder to be 21. 6 remember we estimated this last value so you might estimate a six where I estimated a seven uh two and one are certain digits so it's not debatable that it is larger than 21 and less than 22 but 6 is the estimate someone else might perceive the volume be 21. 5 or 21.
7 milliliters all the digits in a measurement including the uncertain last digit are called significant figures or significant digits frequently we need to know the number of significant figures in a measurement reported by somebody else and I would say pretty much always we need to know that um so there are some cases where we're going to write numbers that are not significant okay um non-zero digits are always but there are some cases we're going to write numbers that aren't significant but these ones are always going to be significant any non-zero digits any captive zeros so those are zeros that are in between two non-zero digits any trailing zeros so the zeros we write at the ends of numbers um when they are to the right of the decimal place and when in scientific notation these numbers are always not significant uh leading zeros so zeros at the beginning of a number and trailing zeros when they're to the left of the decimal place so we're going to look at some examples here so here is our captive zero it's got two non-zero digits on either side so that is a significant digit this is a trailing zero and there's no decimal place so it is to the left of the decimal place if we had written it this is not a significant digit here we have our leading zeros here here and here doesn't matter if they're to the left or to the whoops does it matter if they're to the left or right of the decimal place they are not going to be significant here we have our capped of zero and again just like when it was to the left of decimal in place it is significant and now we have our trailing zero and unlike this one because it's on the left side of the decimal place because this one's on the right side it is a significant digit okay and one way I like to think about that is if I'm going to write out the number 3090 or I have to have this zero here or else it's just not 3090 right there's no way for me to write that uh uh and not have that zero there okay so that means that it's not necessarily significant because it's just a way it's just the artifact or the way that we write numbers over here though I had to go out of my way to add another zero I could have just stopped writing numbers right here . 00802 I went out of my way to write this and I went out of my way to write that number because it was significant so let's talk about how we work with sig figs results calculated from measured numbers are at least as uncertain as a measurement itself so we're indicating some something about the measurement technique that we used how accurate was this when we're writing our significant our our measured numbers with significant figures when performing mathematical operations a set of rules must be followed because we can't wind up with numbers that indicate a higher level of precision than we actually had when we made our measurement and you can definitely do that again just as an artifact the way that we do math numbers must be rounded to ensure that the result of a mathematical operation doesn't imply greater Precision than it actually has so let's go through our rules here when we add or subtract numbers we should round the result to the same number of decimal places as the number with the least number of decimal places the least precise value in terms of addition and subtraction okay so here we have three places past the decimal place here we have two past the decimal place we only get to write our number 2 past the decimal place okay we got to go to the least one and make sure that we stick to that so we only get the two here so we would have added these together uh we would have gotten a seven as the last digit but we round that up to get 0. 78 grams when we multiply or divide numbers we should round the result to the same number of digits as the number of the least number of significant figures the least precise value in terms of multiplication and division okay so before we were only concerned about how many numbers we had past the decimal place but when we're doing multiplication we're going to look at the whole number and the number as a whole so here we have three sig figs here we have two sig figs when we write the number out we only get two sig figs okay so this one's significant the 3 is significant but these two trailing zeros to the left of the decimal place are not significant all right so we had to round it to that number because we were limited to two sig figs from the 42 centimeters here we have three significant figures here we only have two so we have to round our number to only have two significant figures even though we had four here we only get to use this two okay remember that these leading zeros were not significant because leading zeros are never significant if the digit to be dropped the one immediately to the right of the digit is to be retained uh is less than five we round down and leave the retained digit unchanged if it is more than 5 we round up and increase the retained digit by one if the drop digit is 5 we round up or down whichever yields an even value for the retained digit so this is a little caveat here so we're used to if it's less than five we round down if it's more than five we round up here they kind of make this distinction usually often in America if it's a five we just round up but this does lead to some error over time so uh one way to get around that is they say well if it's a five and it's right in the middle we're going to round down or up depending so that we get an even value why do we want to even value there's no real reason why we get an even value it's just that in the long run you're going to wind up rounding up just as much as you round down because half of the numbers are even and half of them are odd right so it's just to give it like a 50 50 round up and down on a five uh so it's not a bad way of going about doing it um and it is the standard in like Baccalaureate programs and stuff internationally to do this uh but you know it is a little different than what you may have been taught where in America it would just round fives up uh the following examples illustrate the application of this Rule and rounding and a few different numbers to three significant digits so if we're going to round this one to three significant digits so we're looking to round to this number and we're going to base it off of this number right here which is a 7.
so 7 is greater than five so we round up and we get 2. 67 we rounded this 6 up to a 7. um here we're going to round to three significant digits so we go 18.
3 so we're going to round this number we're going to base it off of this one this is a three Which is less than five so we round down and we'll get 18. 3 here uh we're going to have 6. 87 we get that 5 right so we're going to decide whether or not we're going to round up or we're going to round down if we round down we'll get 6.
7 which is an odd number because it ends at 7 which is odd um if we round it up we'd get 6. 88 which is an even number because 8 is even so we do that and we round it up okay uh here we got 92. 85 if I round that up it'd be 92.
9 which would be odd so I'm going to round that down to give 92.
