welcome to 109 and in the beginning here uh we're going to actually do a couple of videos to help prep you uh as we start this class uh for a variety of reasons but to get ourselves on the right feet but also because we have Labs coming up they're going to be using some of this information so as we do this one of the first things is going to be significant figures um as we're going to get into measurements um and that's going to be a really important part for the first lab as well as really for the to the rest of the class so in science we do all kinds of measurements this is how we connect ourselves learn understand everything essentially about the world uh as well as all of the questions that we're going to try to answer with science uh we make measurements this is one of the things we do uh significant figures plays a huge role in this um even though we don't really think about it as we're doing it we do at the beginning have to recognize that sign ific figures and what are they so every time we measure something a tool or instrument that we're going to it's going to give us this information um it's going to give us numbers and we have to know what part of those numbers the whole number or part of them what is actually significant I'll I'll give you another word uh that really could be used there is real uh we have instruments that can have really really sensitive um amounts of information we have scales that can go to 10,000th of a gram and we have scales that can go to tenth of a gram but the difference between those is we have to be cognizant of what is real if I use a scale that can only go to 10th of the gram I can't take numbers past the tenth point if we do wind velocity we do precipitation we can measure all kinds of things when it comes to the physical world we measure things in lab like pH um there are limit to the instrumentation and we need to say what are the actual real numbers that we've measured versus other numbers that are not what we call significant in the health World we're measuring things all the time if you go into the hospital you might get a blood P blood buffer panel um with all kinds of numbers that actually have been measured as well as more maybe even a CBC a complete blood count there's tons of numbers that are um sped out of there and and looking at that we need to know what are the significant digits the significant figures the ones that are real to what was actually measured and that is why we actually have to go through this it's really important because we're always looking for the truth we're always looking for realness and in this case we have to go through significant figures it's not always the most fun but I can tell you that it's really really important now in doing this um we're essentially going to be be able to go through some of these uh these rules you may have actually um uh gone through some of these um and I will say in this case I um normally I I may not have uh the lecture notes all written out I do in this case uh simply to make the video a little bit uh shorter um now significant figure these are different um rules for this your book also has uh it laid out in a nice table but essentially just I want to go through these rules pretty quickly that is um any nonzero number is always going to be significant that's just that's always a good thing if you have a digit 1 through n it's going to be significant um and that one we'll get to but that's that's the great place to start a zero between significant figures is going to be significant as well and so I've tried to give little examples under some of these so if I see the number 701 because I know that is significant and one is significant then zero between them must be as well so that right there is going to be three significant figures so that's pretty easy now once we get to decimals it gets a little bit trickier um now a zero to the left of the first nonzero okay so this is why I tried to underline these um is not significant so it says a zero to the left of of the first non zero so if I have something like 0.01 many people start to get this wrong they start thinking oh anything right of the decimal point is significant that is not true we're looking for that first nonzero so in this case if I go left to right that first nonzero is one anything to the left of that is actually not significant so in this case 0.1 is actually only one significant figure it might seem like wow down to the hundreds but again it it is a measurement and out to the hundreds if I only have 0.001 I really only have that one significant figure now once we get to the right of it though so if I go a zero to the right of the last nonzero that is going to be uh significant so in this case right here and this is the key and why I shaded this if there is a decimal now I want to go to example three first since we just started with 0.01 01 then down here I have 0.010 so now I have a zero after right to the right and it's after a decimal point so after the decimal point I have my significant figure one and I have a zero after it now that is also significant so this this decimal point plays a key role here there are two sign significant figures here now if I go back to example one here this is 70 70 again there is no decimal so if I go to this here's my significant figure but zero after that is not unless there is a decimal point and so that decimal point being placed in there is letting me know that that zero is now significant all right so if that's the case then I'm going to actually have 70 point or 70 point that means two significant figures now I can tell you that um in the case of this if I were to even actually up this and so I could actually up this to we'll say 70.0 if I were to do that now I actually have again this is a zero to the right of a decimal point it is to the right of a significant figure this would actually be then in this case three significant figures and so we have to be careful about the the decimal point and then zeros after um when we have a decimal point there zeros after would be significant um this gets easier as we go I can just tell you there is one bit of confusion sometimes and that is when we get to exact numbers and so what this is this is an exception um because if if we were to count the the students in this room for an example 95 students in this section so as we're sitting in our classroom and we have 95 people there and I count you that is an accurate counting there is no uncertainty I can't make that 95.5 students that would be uh that' be bad if we had that if it's 95 students that is an exact number there are no limitations to significant figures at that point and so every calculation that I ever do with an exact number this is not a limitation now the the number 95 if I were to measure something measure in this case 95 gr of a substance that would be two significant figures but because it's actually an exact count it's an exact count it can't be anything other than that's what we call an exact number now we're going to use um significant figure I mean we're going to calculate things that's one of the big things in uh we're going to do in all of science after this is we measure things and then obviously there's going to be calculations I can tell you that even in the health realm when you get that blood blood buffer panel back there are calculations baked into the um actual instrument that takes and does a lot of those um those a lot of those measurements there are all kinds of calculations going on to give you bicarbonate concentration and things like that so there's always going to be calculations and when we deal with significant um figures in this case we really have to see what's the difference between addition subtraction uh with significant figures as well as multiplication division and there's really two different rules one when it comes to addition and subtraction is we actually look at this decimal point and so when we add and subtract numbers um with different amounts different numbers of significant figures right I can see one here with three four and five significant figures um but in addition subtraction I go to the one with the fewest uh places past the decimal point so you can see if I line my decimal points up this one right here 1.01 only goes to the hundreds or two past it and so I'm going to draw a line here and say that that is how many past my answer is going to be now now it doesn't mean that I disregard these numbers because we want to do the math out because it might round up and in this case it does you can see that we get 8.9 067 that is going to round up to to one here but our answer has only two pass the decimal point because of that limitation there so that's addition and subtraction that's not too bad multiplication and division this really just comes down to the number of significant figures and this is why sometimes it's it's a great way to is just to track as you go in your calculations how many significant figures in each number and then you can do the math as if you're not worrying about that yet but then when you get your number you're going to round it back to that fewest amount of significant figures so the number with the fewest 4.2 has two 2.87 3 has four I'm going to do the math out you can plug it into your calculator and you get this number now your calculator and and I'm just going to say this over and over I maybe I should have introduced it at the beginning your calculator is always going to spit an amazing number of numbers out it it sometimes will go to nine decimal points the problem with that is your calculator doesn't know what you do which is in many cases that's no way you can actually measure to that that kind of accuracy and so this is why we have to then um impart these significant figures into this so in this case we do the math out we get 11915 4 but we know that the fewest ones we had here were two so I need to round that to two significant figures and that's going to be 12 so that's that's really some of the um first things now and again we once we start talking about measurements we really need to talk about algebra um the algebra is if there's one part of math that we need to remember I would say for the rest of our life it's going to be algebra um because it's algebra in in my mind the reason I like it and the reason why I can tell you that we still use it um all the way in real life but also uh even in in science and in my research lab we use algebra all the time um and we have to remember that there is an order of operations uh when we do Algebra uh I know it sounds weird but when we look down into some of these down here and we're solving for something like X again there are lots of these that are happening in health settings measuring the pH how much bicarbonate do you have these are all actual algebraic expressions after the first measurement of pH um there are all kinds of these that happen and the way that I like to look at it I know it might seem weird um maybe makes me sound a little nerdy but X we're always searching for the unknown this is what we do in science we're always looking for the unknown whether it be curing cancer while that obviously is a lot harder than just an algebraic expression we're looking for an answer that we don't know yet um and so this is kind of why I do like uh algebra because we're always looking to solve something now in order to do that we we obviously I think most of you probably remember this but we have our order of operations where if there's ever a parentheses that's in a mathematical expression you have to do that you have to do it whatever is said in that parentheses you have to do that that's first um anything with an exponent you have to do that as well um and so these are you know you do these first because as we know in many cases you can have ma mathematical Expressions that are listed left to right and if you do it left or right you may get the inaccurate answer or wrong answer because you have to find follow this um order of operations now once you get past parentheses and exponents then it's multiplication and division there really is um this is either order but those two come next followed by the next two which is addition and subtraction and so we'll do a few of these um we're going to actually do some of these in discussions um as well as a few in class as well um but we're going to do um we're going to essentially start these and go with some easier ones right now and so this is um so whenever we're going to do mathematical Expressions here we're going to end up um calculating things one of the things I really always like to say is we're usually going to have equalities and what that means is we're these are conversions so any operation that you do on one side will have to be done to the other and you're going to see in even in conversions we're going to continue with this idea of equality but in this case anytime you do something in the left or the right side of the equal sign it's valid to do and that's a great way to what what I remember and I don't know if back in my math days um teachers saying just isolate the variable this is how we solve things isolate the variable and so if I have something as easy as x - 2al 0 you know almost really everyone can see that and go immediately well I know x equals 2 because you're already doing the math in your head you're already saying oh well 2 - 2 equals z so I know x equals 2 but sometimes it it's not so simple and so how do we do this and you know obviously it's all about getting that X by itself that isolating the variable so if I add two to that um that's going to cancel this out over here and isolate this as X and if I add to I if I did it over here then I got to do it to the right this is then how I get this equation x - 2 + 2 = 0 + 2 so x = 2 all right that will always be the case that will always be a correct answer if you do it the right way now something like this though I have something like 2x - 2 equals 0 it is a little bit different because uh you start to go oh I have to isolate ol that variable I have to remember this isolate the variable I can't in this case remember if I were to divide by two here this is not the same that would be 2x - 2 ided 2 that is not so easy what I really need to do is isolate that variable and so what I'm going to do is get rid of this one first so I add two to both sides I get 2x would equal 2 then I can now that this is isolated I can then do my division by two both sides to get xal 1 so we're going to do math like this a lot um and it's going to be relatively simple algebra but I just want you to remember these two um rules now as we do this though uhoh we're going to do the math we have to now Loop in what we just talked about which was significant figures and so we're going to do a few of these here and that is and and then we'll um uh just to to do this now what I will say is that when you do these left to right in a lot of cases these are always presented left to right you can if you wish to take this over here and or any space that you would want on a quiz or an exam or anything in that case homework and line them up if that makes it easier um than doing what I did right here which is just to say hauh this is now a plus sign I have two numbers with different significant figures but what do I care about the most is it should it be how many significant figures no because this is a plus sign and so now I'm looking for how many pass the decimal point and so what I did here was to say well that's five pass that's two pass so whatever number I get when I do this actual math I'm going to have two pass the decimal point so I did the math I added these up you can line them up if you wish or you can just make a note but I I lined that up and said okay the answer is 13599579 and but now that these yeah but they're zeros but yes they're past a decimal point past a significant figure decimal point so therefore to their to the right these are going to be significant and that's valid now I do the next one and I say okay pretty easy math I can plug that into my calculator but ooh this is a multiplication and so now it's how many significant figures that counts 0 again these are to the right of the decimal but there is no significant figure prior to that so I'm then looking at one two significant figures so then when I get my actual number I'm then going to actually round that in this case it stays at 0.61 I am going to make a point to say this right now that when we actually do anything less than zero I'm going to just say this right now please do not do this which is 61 versus 0.61 they are yes the same number but this is the one to do this is not the way to do it now I I can tell you at least as you're writing down uh things and and it's very important uh sometimes even when you do things in homework it's important to put the zero out there one reason is because that point is difficult to see if you're on an exam and you have an answer and I can't tell if it's 61 or 61 it's very difficult for me to grade that whereas if it's zero Then I then you see the point there's no reason to put a zero in front of that if there's not a decimal so you see the decimal this is the way to do it okay so ding ding ding make that a um an absolute method for you every time you have something less than one and when I say you know so I'd say Well when when should you do that starting now and for the rest of your life for the rest of your life this is just something in science that we we feel is very important now what about the last little thing is is saying oh now we're going to combine some functions and so now we have math to do okay and then I have but then I have things like well there's a plus there's a minus um and so what do I do here there's you know some things to consider first of all I'm going to think about the math before I do anything I'm going to think about the math and I'm going to go back and say everything that's in parentheses I'm supposed to do first so I have math here that I need to do and I have math here in this parentheses that I need to do and then I'm going to um add those together because that's what this equation is telling me those parentheses take um absolute priority and so I'm going to end up getting these numbers now in this case I need to remember that because I did a subtraction here that in significant figure world it's one past the decimal point that matters so I did the math and left it in two but I need to remember that one past the decimal was the lowest that I had so I need to remember that then over here when I did this math I have five significant figures three significant figures so really I need to have only three here but I also have to take a no count how many past the decimal in this case if I have two past the decimal you can see this right here okay if I have two pass the decimal um and again this was back over here then this right here is three significant figures then I'm good okay and in fact I I would AR that at this point I'm actually going to I'm actually going to come in here and say while this is too past I like the idea that this is three significant figures and so three significant figures good good to go um and then we have our number now again we do the math out you just do it out to get this full number right as you're plugging these into calculators you're more than welcome to do that and get this long answer but I got to remember that while I might have two past three significant figures this is one past right because of three significant figures I had two past now I got one Past One past the decimal is the lowest so I have to go with one past um and the best way to do this is just to kind of um jot this down near your numbers as you go okay um the last one here is really now an algebraic equation we have an an x value over here that we need to isolate and so in this case right here you can see the plus sign you can see that what I need to do is bring this over so I'm going to actually subtract both by 12.99 to get that to come to the right side of the um equal sign and in doing that again this is I'm going to do addition subtraction here so this right here is going to be two pass to pass when I do this I'm at two past the decimal now uh in then I'm going to have this number here x over 4.05 equals that number 1.29 and as we have our two pass decimal it ends up with a number that's three significant figures 4.05 is also three significant figures so that's really nice multiply by 405 4.05 on the either side and that's going to give me X on the left and I get this number I have to remember that this was a multiplication I have three and three significant figures so therefore I need to end with three significant figures and so there it is now I draw a box around all of these because as we do all of our assessments quizzes tests homework whatever you're going to have numbers in a lot of places I need to know what the final answer is and so boxing your answer is always the best way to go and so that is um one of the one of the few things now we are going to go the last little bit which is going to be scientific notation and scientific notation is just as we get into science we have things that are really really big and really really small in many cases in chemistry we're dealing with really really small in Biochemistry um I study proteins entire proteins are still nanometers it can be in length and these are really small numbers and so we need to it's very difficult when you start seeing things like that low this is a by the way the weight of a human cell that's just crazy to deal with mathematically we need to be able to do this sometimes then in something like scientific notation so when we take the largest tree in the world which is actually in this case um about a million uh kilog um in this case right here what do we do to get to scientific notation and every time we have a large number and we go to the right this is a positive every time we have a small number right in in this case past the decimal point then what we're really do is is saying if I have this number I'm going over here to get to this decimal that's 13 times then that's a negative okay so when it's lower than one you're going to see something with a negative number number and so that's what we have down here when we have this general form we're always going to have an integer so some sort of number 1 2 3 4 up to nine and then it's going to go so one number on the left side and then you're going to have numbers on the right side depending on how many significant figures you have in this case if these were all significant I'd have four significant figures but let's say the number is really large or really small well in this case if it's really large then we're multiplying by 10 in other words if N is a positive number then we're multiplying by 10 to get that very large number if we're if we're if uh that number is a negative number right here then we divide by 10 um n times whatever whatever that number is to get that number so something like 3.82 * 10 the 4th that's a very large number again you can see there's an integer and then my significant figures here but I'm going to multiply that by 10 four times and that's how I can get to 3 or 38,200 and so notice in this case I have three significant figures I have three significant figures here now when it's small 1.02 * 106 we're dividing dividing by 10 six times and so we take that 1.02 there and you can see there's my number and I'm going to go backwards right 1 2 3 4 five six times and that's how I know the decimal is really in that place so in this case again I have three significant figures I have three significant figures those both match but when we deal with a number like this due to that being negative this is a really really small number and yet this is going to be something that we're going to see over and over as we do calculations because we deal with very small things and and some large things so if you wanted to convert these and I'll do this really rapidly um mainly because you can kind of see this for yourself pretty pretty simply um and just to give you a little bit of of um uh application here that these are real um this is the number of bacterial cells in humans there's actually this this number varies but this is kind of an average number this is a crazy high number 3.8 * 10 13th and you can see how would I convert it to that I simply just start and say if I want scientific notation I can simply just start between that integer value 3 point something and just start counting how many places it gets until I get to the end of that number and it's 13 times and so that's that's that number um if this gives you a little bit of trepidation um my research actually do I I am involved in research here Grand Valley um that does work on antibiotic resistance uh I do know a little something about this as far as bacteria um just to give you uh an idea this is more right here bacteria and other organisms outnumber human cells on you so I'll just let you know that if we counted cells between humans and non-human you are less nonhuman or excuse me you are less Human Than nonhuman you have more bacteria and other things growing on and in you than you do your own cells so just before you think they're all bad they're not there's a lot that are fantastic um now um when we talk about light we actually talk about things in nanometers uh what the visible light that we see from 400 to 700 nanm that's really small wavelength of light and so if I had 6.08 * 107 what I do whenever I have this is I go oh if this is 6.8 * 107 I just start drawing zeros to the left and even if I draw too many that's fine but then I just start counting and I just you know 1 2 3 4 5 6 7 there's my decimal and what I'm going to do at that point is remember remember my Golden Rule which is I always keep the zero to the left of it so I know where that that decimal is but that is then the number I get and you can keep doing this with all of these uh whether it be the the speed of of a typhoid fever bacterium in meters per second um which is it seems obviously very slow which it it is when we start thinking about 3.0 * 10- 6 but I do want to say why did I in this case why is at 3.0 again remember that there's two significant figures there and so both of those are significant the decimal would start here how do I how did I get there I count back six times that's my -6 miles of Highway in the US um and then uh the radius of a hydrogen atom and inches obviously very small and I wanted to point out this idea that sometimes you're going to see this in calculators or even in homework work um in lab and in other notes as well when you see this e that means time 10 tothe that is not to be um confused with little e which is the anti-natal log okay so this is not the thing on your calculator that's little e so whenever you see something like this on your calculator that is not the same thing that is an anti natural log okay so that's different this e is * 10 the 9 so that's what in many cases your calculator will have I just wanted you to know that lastly the number that is famous and and it should be um we're going to talk about this later but this is avagadro's number so this is something that is huge and in this case you know the number of of particles or number of molecules in a mole um so this is this is a very famous uh very very famous number yeah that is a very very large number I can't imagine doing math having all those zeros that's you're just asking to be uh wrong essentially so this is why we use exponents so this is doing that and later in another video here we're going to go over then conversions and Factor label method and that is going to be next