hey everybody back in chapter one we talked about quantification in chapter two we're going to carry that further we're going to talk about measurement problem solving uh get into conversions and equations and really dig more into uncertainty and units so let's get started first let's focus on uncertainty when we report values in the scientific community we want to report really only what we're sure of we don't want to wing it when we're sharing results so uncertainty is presenting our values and the last digit that we report is the first one that we're uncertain of first one where we have some wiggle room we're not entirely sure so let's begin by looking at this example involving average global temperature now the data says that the average global temperature is 0.6 degrees celsius higher in the last century now because it says there was an increase in temperature we're going to express this as a positive because the temperature went up so this is how much on average the temperature went up in the last hundred years now we said that the last reported digit is going to be the first digit where we have some uncertainty so here the last reported digit was the six so really this means 0.6 degrees celsius plus or minus 0.1 degrees celsius so what that means is if we take 0.6 degrees minus that 0.1 that would be an increase of 0.5 degrees celsius or it could be an increase all the way to 0.6 plus that 0.1 degrees celsius that would bring us up to 0.7 degrees celsius so that means that the average global temperature wasn't right at 0.6 degrees increase over the last 100 years what that means is somewhere between an increase of 0.5 degrees celsius to 0.7 degrees celsius that's what this plus or minus means at the end here so why does this matter because it's values like these that we make political decisions off of these political and policy decisions make changes and impact on our lives so we could be looking at investing in a spaceship investing in the next line of electric vehicles values matter certainty matters and only reporting what we're certain of which is why we're going to spend the chapter talking about how to handle questions like these and interpretations like these correctly so first one of the tools that we're going to use is scientific notation there are two parts to scientific notation we're going to have a decimal part between 1 and 10 so let's say that we've got 1.0 that's what i mean a value between 1 and 10 where we start we're going to have an exponent part so in a scientific notation we're going to have the decimal and we're going to have the exponential part which we would say times 10 to the power so this is going to be a whole number it could be negative it could be positive let's say it's 1 times 10 to the 3. we don't have to write positive 3 that positive is going to be assumed we could also have 9.6 again the decimal portion times 10 to the negative 62. so these values can be positive they can be negative but they are whole numbers now what you won't see is something like this 0.8 times 10 to the minus 1. now our exponent this is fine but here our decimal point is 0.8 that should start out with a value between 1 and 10. so it's too low we do need a value between 1 and 10 in a properly written scientific okay so let's dig into what these actually mean so let's say i have 10 to the three what that really means is i'm going to have 1 times 10 three times that's where that 3 comes in so if i see something times 10 to the third really it's times ten times ten times ten or one thousand what if i have ten to the fourth that's one and then multiplied by 4 tenths or that would be 10 000. so using a positive exponent like this is giving me a way to express very big numbers in an easy way so when i say big number i mean greater than one so associate positive exponents with big numbers when i come over to negative exponents we have to change our interpretation a little bit let's say instead that we have 10 to the minus 3 if i have 10 to the minus 3 we're going to take 10 but put it on the bottom so i'm going to put 10 times 10 times 10. 310 still because i have this 3 here but the negative makes it a fraction so this is going to be one over one thousand or one thousandth so we're going to associate these negative exponents with small values in a similar way if i had 10 to the minus 4 i would have added an additional 10 on the bottom making it 1 10 000 just like we added an extra 10 over here when we were working with positive exponents so in short positive exponents big values greater than one negative exponents are giving us small values less than one okay so let's practice this this is in standard or uh decimal notation we've got 5983 it doesn't show us right off the bat where the decimal point is but this is where it sits it sits at the end of the value so to get into scientific notation i'm going to take where the decimal place starts and i'm going to move it 1 2 3 times until i get it to a place where i have a number between 1 and 10 for my decimal portion of the exponent or of the scientific notation value so decimal portion has to be between 1 and 10 so this is going to give me 5.983 now if this is the decimal portion we've done it right so far because i have a value that i'm starting with that's between 1 and 10. now i'm going to add the exponent portion that's going to be times 10 to some power and look at the number of times that we had to shift it to get the decimal place in the right position we had to shift it once twice three times so this is going to be 5.983 times 10 to the third oops so 5.983 times 10 to the third and why does that make sense because this 10 to the third really means 10 times 10 times 10 which was really a thousand so this is 5.983 thousands or 5983 so quick recap we take the decimal place where it starts at the end of the value we move it until i get to a position where i have a number between 1 and 10 and then for our exponent we take times 10 to a whole number and that whole number is the number of times that we shifted that decimal place now notice i didn't talk about shifting it left or shifting it right when i look at this this is a big value this is larger than one so based on what we learned at the start when we were talking of scientific notation big numbers positive exponents that's why i left this as 5.983 times 10 to the positive third so let's try our next example our next example shows the decimal place but to be able to get this decimal place where i need it to where i can have a number between 1 and 10 i got to move it in the other direction so i'm going to move it 1 two three four times that leaves the decimal portion of my scientific notation as 3.4 and for the exponent portion i'm going to write times 10 and then the number of times that i shifted the decimal place one two three four that's going to go up top now in deciding if this is a negative or a positive exponent this number right here is small it's less than one so i'm going to be looking at a negative exponent so this is 3.4 times 10 to the minus fourth now when we report scientific numbers remember we said that the last number that shows up is going to be the first number that we have uncertainty in so this 45.872 this 45.87 these are digits that i'm certain of because they come earlier in the value but because this 2 is the last number that i've written down that's the first number that i'm uncertain of and the first number that i'm estimating i've got a little bit of wiggle room there so there's more precision in the 45.87 less precision more uncertainty in that too so let's look at a skill that you're going to need once you get more into the sciences especially the lab sciences and that's reading off of scales now we said that we're going to be reporting what we're sure of and the last digit is where we have to start to guess so if i were to blow the scale up a bit then it looks to me that i have 0 this 0 here and then i've got 1 2 3 4 5 and it doesn't mark it but we'll make a mark for ourselves 6 7 8 9 10. so every line that's on here on the gram scale is going to represent a single gram it's important to consider what unit we've got also because the unit helps tell the story now the line that we're trying to measure comes right about here i'll draw it as a dashed line now we can tell that this dotted line that we're trying to measure is after the one but before the two so it comes a little closer to the one than the two it's not quite halfway in between so the number i'm certain of is that i have one gram i feel confident that this dashed line falls between the one and the two so i'm going to record one but then i have to guesstimate this digit because i know that i'm going to have to read between the lines here so i'm going to say it's not quite halfway it's about 1.4 now you may have said 1.3 or 1.2 or 1.5 and that's okay and that's because this is the last digit that we're going to record here we're done because here we're uncertain and we're starting to guesstimate now to finish this off we do need to say that this is 1.4 grams we can abbreviate that as a g so to be able to read the scale properly i record what i'm certain of and we were certain it went past the one but not to the two that's why i put this one here and then this four came in because it wasn't quite halfway between the one and the two so i decided on four but again we guesstimated so there will be some variation here but that's where we stop reading i wouldn't try to say 1.427 because i can't possibly guesstimate accurately here we're reporting what we're certain of and cutting off on the first digit where we start to guesstimate okay so let's try another one of these we're going to estimate the hundredths of a gram so when i look at this scale here if we were to blow this up then i've got 2 and i've got this so instead of having each of these lines representing a whole value i'm looking at each of these little lines here representing tenths so one two three four five lines in this wouldn't be five this would be halfway between one and two it would be 1.5 now the number that we're trying to measure it goes right here see if i can trace this so it falls one two three it looks like it falls between the two and the three so it falls right about here so this dotted line is what we're trying to measure on our scale this means the number that i'm sure of is going to be 1 because i start with a 1 here point 1 2 because it's after the 2 and before the 3. so this is going to be 1.2 and these numbers i can read off of the scale and i'm certain of now for my last digit here i can read the whole unit the one i can read that it's past the two so it's definitely a 2 here but i'm going to have to read between the lines to decide where it falls between 0.2 and 0.3 now it looks like if you look closely it's about halfway between these two lines so i'm going to call this 1.25 this last number i was uncertain of i had to guesstimate so that's where i'm going to cut it off i'm just going to leave it as 1.25 and not try to go any further now the unit on this again is gram now notice here we recorded to two decimal places before we just recorded to one decimal place the difference was in the scale in this example the scale was marked to the tenth of a unit so we could record all the way to the hundredth the second decimal place in the last example it was only marked to the single unit so the furthest we could go would be the tenths so we record exactly what we can read in this case the 1.2 and then the last digit is the first one that we have to guesstimate all right so when we talk about precision when we talk about certainty we're going to start talking about sig figs now significant figures hold the most value when we talk about the significant figures and reported values in science we're talking about the values that carry weight and meaning in the number so we've got four rules we're going to have to be able to look at a value and determine what digits are going to be significant or not we're going to hit the first two rules here the first is non-zero digits are significant that means that non-zero or the numbers one through nine are always going to be significant so if i have the value 13 then the one is significant and the three is significant because they're non-zero that means that i'm going to have a total of two sig figs if i have the number uh point two seven the two is significant and the seven that gives me two significant figures so spot a digit that's non-zero it's definitely going to be a significant value okay coming down to rule number two we're going to have three rules for zeros we're going to have trapped trailing and proceeding now trapped zeros are always going to be significant what do i mean by trapped if we take two non-zero numbers and we trap a zero between it like 101 well from rule number one we know these non-zero numbers are significant and for this zero here this is trapped between the two ones and trapped are always significant so this would be also a total of three sig figs what if we had .207 again non-zero numbers we've got a zero trapped between the two non-zero numbers so we'll have a total of three sig figs trailing zeros are going to come at the end of the value now these are probably the most complicated trailing zeros are only going to be significant if most important word in the statement there if a decimal place is shown explicitly in the number so let's say that we've got 1.00 versus 100 now in the top example i've got a 1 here that's non-zero that's definitely going to be significant these zeros are trailing zeros because they're going to come at the end of the number and it says if a decimal place is shown explicitly that's when trailing zeros are significant so explicitly meaning it's shown i've got the decimal point so that means that these trailing zeros are significant what about this bottom one the top one had a total of three sig figs these look pretty similar the difference is this doesn't show an explicit decimal place the non-zero value at the front the one is definitely significant rule number one these two trailing zeros though because there is no decimal place shown over here explicitly we know it's there but it's not explicitly shown these are not significant digits really it's ambiguous maybe they are maybe they aren't but we're going to play at the conservative route and say that this is only one significant figure so pause the video and try this let's do 1.207 zero i want you to see if you can count the number just using rules one through three count the number of sig figs present in this value give you a moment to pause and then we're going to come back and try this together okay so let's get into this we know that non-zero numbers rule number one are significant so one two and seven definitely significant digits this zero here is trapped trapped zeros always significant we know that this value is a trailing zero and trailing zeros are significant if we have a decimal place or a decimal point shown explicit and the number and i've got it right here it's shown explicitly so i'm going to say this is significant for a total of five sig figs now our last rule we've already had trapped zeros we've had proceedings where we've had trailing zeros now we're going to have preceding zeros if something precedes it comes before so zeros to the left of the first non-zero number so zeros that come at the start of the number let's look at zero point zero let's break this down so all three of these are preceding zeros now here's a common misconception this is not the only preceding zero to proceed to come first doesn't mean it comes before the decimal place it has nothing to do with where the decimal place is these preceding zeros precede this one they come before the first non-zero number because these are all preceding zeros they are not significant now are they still there yes did they tell us how big or small the number is absolutely so they're still important they give us the scale but as far as meaning for the value they're not significant so when i look at the non-zero digits here the ones these are both significant and pause the video and practice what type of zero is this is it trailing preceding or trapped it's caught between the two ones because it's caught between these two ones it's trapped and it's significant so i do have a total of only three sig figs in this value so um when these preceding zeros don't count we said they're still there we said they still give us a feel for how big or small the number is if i were to write this in scientific notation i would have to move the decimal place three times to get 1.01 times 10 and this is a small value looking at a negative exponent negative third so there are three sig figs in the value once have expressed it in scientific notation that's tied to the idea that i also have three sig figs when i express it in standard or decimal notation now we do have something called exact numbers exact numbers have infinite number of sig figs so if i said that there was one chair at my desk which there is it's the one that i'm sitting in right now i know that infinitely well i know there's not 1.2 chairs at my desk and that's because it's a discrete counted number just like you could count the number of pickles on a chick-fil-a sandwich you can count the chairs at a desk that's an exact value that we don't have to worry about for sig figs and we know it infinitely well constants that are a part of an equation so if i give you a constant like 6.022 times 10 to the 24th that's going to be avogadro's number we don't worry about the number of sig figs in a provided constant we assume that we know that to an unlimited number of sig figs and finally definitions if i think about a carton of eggs there are 12 eggs and a dozen because there are 12 eggs in any dozen of eggs across the planet it's a definition i could have a dozen shoes that's going to be 12 eggs so one dozen is 12 we know infinitely well now it's a little hard to chew on but this last idea of one inch equals 2.54 centimeters it feels a little weird to say we know this infinitely well because this seems like an odd number to know infinitely well it's not a nice whole number but this is a conversion factor it's a definition it relates to different units so we are going to take these relationships between units these conversion factors really a definition to be exact so let's put this into practice if you'd like you can pause the video work through the work through these yourself and then i'll work through them with you so feel free to take a moment pause all right so this is a counting sig figs exercise we want to know how many sig figs are in each number so in this first one we've got zeros that come at the start of the number that means these are preceding preceding zeros were never significant so i'm going to cross these out those are not significant the 3 and the 5 non-zero numbers definitely significant so for .0035 i've got two sig figs 1.080 let's go ahead and get what we know right off the bat non-zeros the one and the eight are significant this zero here is a trapped zero and trapped zeros are significant this last zero is going to be a trailing zero and we said trailing zeros were significant if i had a decimal place explicitly shown which we do so this is significant total 4 sig figs here 23.71 this might be the more straightforward of all four of the examples because we don't have to worry about zeros these are all nonzero all significant four sig figs our last example 2.9 times 10 to the fifth we don't worry about the exponent when we're determining sig figs here we worry about the decimal portion of the scientific notation these are both non-zero numbers both significant this would be two sig figs all right let's work through a couple more examples now feel free to pause practice yourself and we'll get going in just a moment okay so let's dig into these oops sorry jump forward a bit let's go ahead and get into these got my pen pulled up we've got a relationship here that one doesn't equal 12 this could be shoes it could be grains or rice it could be eggs but what we have here is a definition and we said that definitions like this are going to be an infinite number of sig figs because this is an exact value so our answer here would be infinite for 100.00 the 1 is nonzero definitely significant all these here are going to be trailing and we said that for trailing zeros if we have a decimal place explicitly shown we got it right there then these are significant total of five sig figs for our last example again we've got trailing zeros here but i don't have a decimal point shown anywhere in the number so none of these trailing zeros are going to be significant the one being non-zero is so this works out to just one sig fig in the way this is expressed on that last problem so our next skill that we're going to work on is rounding so when we do calculations we're going to wait until the very end to round if we have a five-step problem and we round it every step by the time we get to the final answer we are not going to have a very sound very precise answer so you don't want to have rounding error that's what happens when you're rounding snowballs through the whole problem you round on step one step two step three step four and step five you end up with a very different answer at the end so what we want to do is learn how to round properly so let's do a little bit of practice here we want to round all the values below to just two sig figs now this banks on our ability to count sig figs and express sig figs properly now if i want two sig figs i'm going to start at the left of the number over here and i'm going to read just like i would a book left to right and i'm going to take this 2 and i'm going to take this 4 but then i'm going to going to put a little line i'm going to start to cut things off because these are going to be my sig figs the 2 and the 4 but i can't just write 24 because 242.4 is not going to round to 24. i look at this 2 the first number and only the first number that i'm cutting off it's important that you don't look further just look at the first digit that you're cutting off if it's 5 or higher we would round this to a 5. if it's 5 or if it's lower than 5 we would leave it as is so i'm going to leave this as 2 4 not 2 5 i'm not going to round up and then i'm going to put a 0 to wrap things up now why am i just putting a zero because we said we couldn't round 242.4 to 24 what we can do is take 242.4 to 240. when we look at this value that we rounded to the 2 is significant the 4 is significant i'm not showing a decimal place so this trailing zero at the end is not significant that's why i've only got two sig figs let's try this next one 1892 we're going to do it the same way we're going to go left to right i'm going to pass the 1 i'm going to pass the 8 and i'm going to cut here so that means that my number is going to start with 1 8 but this number this first digit that i'm cutting off is 5 or higher so instead of 1 8 that's going to cause this to round up so this is going to be 1 9 and i'm not just going to put 1 0 i'm going to put two zeros one for each of the whole numbers that i'm cutting off here so i'm going to say that 1892 rounds to 1900 that's a reasonable rounding we wouldn't want to round 1892 to 190 so make sure your rounding makes sense when you're done now just to make sure we're keeping two sig figs the one and the nine are significant these zeros are trailing i chose not to write an explicit decimal place that means no decimal point shown it's trailing it's not significant that's two sig figs and that's what i was aiming for the last example works very similarly left to right again only these zeros are all preceding so they're not going to be significant so i'm going to keep going past the zeros i'm going to move to the 7 and the 2 and then cut so my answer is going to be 0.0072 but because that value that i'm cutting off is a 9 this is 5 or higher that's going to cause this to round to 0.0073 and that's going to provide us with all we need here to express that value i've got these preceding zeros as not being significant the seven and the three are significant we had to round that two up and now i've still only got two sig figs all right so let's cover the multiplication and division rule with sig figs so for multiplying and dividing the rule is we're going to keep i'll write this out for you the fewest sig figs i'm going to abbreviate that sf so keep the fewest sig figs multiplied or divided now what do i mean by that let's say that we have point two times four oops go back here lost by then so two point two times 4 that's going to give me a value in my calculator of 8.8 but in this case you were smarter than your calculator and your calculator i guarantee you doesn't know the multiplication and division sig fig rules so with your calculator answer we've got to look at what we started with 2.2 has two sig figs these are non-zeros so both are significant this 4 is non-zero so i'm multiplying something with two sig figs by something with one sig fig and my job is to keep the fewest so in my answer i should only be keeping one sig fig so let's do what we did before go left to right that means after this 8 i'm going to cut it off this first number that we're cutting off is going to be 5 or higher so we're going to have to round up so 8.8 is going to round to nine i'm only recording this to one sig fig that's my one sig fig it's my final answer let's do one more do 4.4 times 3. now that would give us a calculator answer of 13.2 but again we've got to be smarter than our calculator we've got two sig figs here we've got one sig fig here so i need to keep the fewest in my answer we read left to right so i'm going to cut things off right after that 1. now what i can't do is round 13.2 to 1 but what i can do this 3 isn't high enough to make that round up to a 2. so i'm just going to leave that one as is and we could say that 13 would round down to 10. that way i'm only reporting one sig fig this trailing zero i didn't list any uh decimal points explicitly so it's not significant and that's my one sig fig so make sure when you're done that your rounding is reasonable so it stays roughly around the same scale there okay so if multiplication was few of sig figs addition and subtraction is slightly different we're going to keep the fewest decimal places subtly different that are added or subtracted so let's try one let's do 25 plus 0.1 now your calculator gives you a value of 25.1 can't be right because your calculator is not as smart as you are we've got to keep the fewest decimal places i don't show any decimal places here there are no values after the decimal place or after the decimal point there's not even one shown so this is zero decimal places this here shows one decimal place so which one am i going to go with i'm going to go with the so in my answer i should only keep zero decimal places that means that my answer i'm not going to keep this one i'm just going to keep my answer as 25. mine's just exploded i just added something to 25 i added 0.1 to 25 but my answer didn't change but that's because the number i added was so small that when we look at significant figures that number didn't significantly change so it may be counter-intuitive at times but fall back on those rules get comfortable with those rules let's try 8.3 plus 0.25 our calculator gives us a value of 8.55 if i look at the decimal places that i started with i had 1 here i had 2 here remember we're talking about the digits that come after the decimal point i'm going to keep the fewest so i'm going to keep one decimal place in my answer so that means i'm going to come over to this 8.55 and i'm going to cut off after that first decimal place this 5 causes it to round up we look at the first number that we cut off and that's going to round this to a final answer of 8.6 so the take home here make sure you're careful when you're adding and subtracting or multiplying and dividing that you're applying the appropriate rule for addition and subtraction we're worried about decimal places for multiplication and division we're looking at significant figures subtly different but they both help you carry out significant figures properly in calculations