okay welcome back so well gonna start now is section number one which is entitled introduction and mathematical concepts okay now physics is a quantitative science meaning we're going to be using a lot of numbers in this course so it's gonna really help us to make sure that we understand what we mean when we use numbers to describe something for example in physics when we say to what do we mean by that well oddly enough we need to describe what we're actually talking about usually in physics when we say a number like 2 what we really mean is something that is about 2 or approximately 2 now sometimes we actually will mean exactly to some quantity is exactly 2 but usually and we'll talk about when we mean exactly - in a little while but usually when we say something is 2 we really mean it's about 2 or approximately - if we're gonna measure something for example maybe the length of a table or something like that and we say that oh the length is 2 meters what we mean is the length is about 2 meters or approximately 2 meters in other words what we mean is that the length of this thing is closer to being two meters than it is to 3 meters hopefully if the length actually were closer to 3 meters we would have said 3 meters rather than 2 meters so we mean that the length is closer to 2 than it is to 3 if it's a precise measurement hopefully hopefully we're precise as possible so let's say it's closer to two vanities to three likewise the measure is closer to to that it is to one likewise it was closer to one metre long we would have said the table is one metre long so very approximately will say something like the measurement is say closer to 2 then it is to 1 closer to 2 then it is to 3 in other words about say 2 plus or minus 0.5 roughly speaking if it was more than 0.5 larger than 2 hopefully that's well that's closer to 3 hopefully we would have said oh well that's 3 meters if it's more than 0.5 less than 2 that would have been closer to 1 so hopefully we would have said oh that's 1 meter long so roughly speaking when we say two we mean about 2 and we're talking about something that is approximately somewhere in between 2 plus 0.5 or 2 minus point 5 in other words 2 plus or minus point 5 so let's imagine we're talking about a distance maybe the distance say from where we are now to some other place and rather than 2 let's say 2000 let's imagine I say that the distance from here to some place is 2000 meters now what's the difference between saying the distance is 2000 meters or 2100 meters or 2130 meters or 2132 meters well by saying well as I say so here say if we said it was 2,000 meters remember what we're talking about that is something that's closer to 2000 than it is to 3000 or 1000 so we mean really something like 2000 say maybe plus or minus 500 meters well here if we say 2100 we mean something that's closer to 2100 than it is to 2200 or just 2000 so we mean a number that's maybe within only fifty meters if we say 2130 we're talking about something that's closer to 2130 then it is 220 140 or 2120 so we're saying maybe something within say five meters and like we were talking about before 2130 - that would be say closer to that than 2133 or 2131 so we're saying maybe plus or minus 0.5 meters so we will describe these different numbers in terms of how precise they are so this is more precise than this number this one is more precise than this one this is more precise than this one in other words the error so to speak associated with V values is smaller as we move down down the list now we'll talk about that very simply in terms of the number of significant figures how many significant figures numbers have and we will use that to kind of describe the precision that values have so how do we determine how many significant figures a number has well if the number does not have any zeros it's pretty easy so for example if we have a number like 4367 all we have to do is count up the digits and that's how many significant figures there are so four three six seven that number has four significant figures or I'll say SiC makes four short sometimes I'll just write this as SF sig figs significant figures so that number has four significant figures as long as there are no zeros all we have to do is count them up and it doesn't matter if there's a decimal point for example twenty-three point eight six four seven eight and I'm writing these numbers specifically these numbers because they are the ones in the lecture notes if you had access to the lecture notes you can see them there so in this case how many sig figs does this number have well we just count them up one two three four five six seven so that has seven six now if there is a zero in the number that makes it a little more a little bit more complicated to determine how many sig figs there are but there are three rules for counting zeroes as whether they are significant figures or not so let's take a look at those one at a time okay rules for counting zeros rule number one zeros to the left of all nonzero digits are never significant so let's look at some examples of that zero point zero zero two three these are the nonzero digits these are the zeros to the left of the nonzero digits they are never significant so how many sig figs does this number have not significant one two two significant figures zero point zero zero zero zero zero zero zero zero three six seven all of these zeros are to the left of the nonzero digits so none of these are significant one two three three sig figs okay zero point zero zero zero zero two zero four none of them still only one sig fig there we go so keep in mind zeros to the left of all nonzero digits are never significant rule number two zeros between nonzero digits are always significant these are sometimes referred to as sandwiched zeros so let's look at some examples of this 100 for now that zero is in between the nonzero digits and so it is significant so one zero four that has three significant figures and it doesn't matter if there is a decimal point two point three zero zero four six we've got nonzero digits nonzero digits these zeros are in between the nonzero digits and so they are significant so we have one two three four five six sig figs five zero five zero point zero zero three this zero is in between nonzero digits these zeros are in between nonzero digits and so they are all significant one two three four five six seven okay let's look at module number three and rule number three zeros to the right of all nonzero digits are significant only if there is a decimal point present so to the right of nonzero digits only significant if there is a decimal point so let's look at some examples of this okay some examples of rule number three 1,300 so this is kind of similar to the number we were talking about before here are zeros to the right of the nonzero digits but there is no decimal point present no decimal point written in the number so those are not significant so we have two significant figures if we would write this number as one thousand three hundred but include the decimal point then that makes those significant and we have one two three four significant figures if we were to include even more zeros after the decimal point then those zeros are still to the right of the nonzero digits the decimal point makes them all significant one two three four five six-six sigfigs okay the decimal point makes the zeros to the right of the non six but nonzero say non-zero Desa nonzero digits significant and so one two three four five five six six thirty nine point zero zero again the decimal point makes the zero is significant one two three four no decimal point so the zero is not significant that only has one however 60.000 the decimal point makes the zero significant one two three four five okay there we go now in general obviously a number might you you need you might need to use different rules to determine how many how many significant figures a number has so for example one point four five well there are no zeros so that's pretty easy one two three one point zero zero five well those are sandwiched zeros so one two three four zero point zero zero zero four five zero those zeros are to the left of the nonzero digits and are not significant this zero is to the right there is a decimal point so it is significant one two three four zero five zero point zero six zero well those zeros are in between nonzero digits that zero is to the right there is a decimal point so it is significant one two three four five six so sometimes you've got to make sure to keep well all the time you should make sure to keep all those rules in in my because you might need any of them in determining the total number of significant figures now in physics we are also going to be dealing with some very large and very small numbers and so there is a way of writing very very large numbers and very very small numbers that makes it a little bit easier for us to deal with them sometimes and that is called scientific notation for example let's imagine we've got some large number like one six seven zero zero zero zero so that would be like 1 million six six hundred seventy thousand sometimes if you're writing this number it might be very easy to miss count the number of zeros and you might put an extra zero or leave off a zero or something like that so there's a way of writing it to make sure you know exactly the size without having to worry about counting the zeros all the time now to write this in scientific notation we write the first non-zero digit on the Left which in this case will be a one we follow that with a decimal point and then after the decimal point all of the other significant digits now in this case we have six seven now one point six seven is obviously not the same as 1 million 670,000 so we need to make this bigger to make it the same size as this to do that we multiply by a power of 10 which has the effect of moving the decimal point some number of places and the number of places we make the power of 10 so if we were to multiply by just 10 that would move in one place if we multiplied by 10 to the 2 in other words 10 to the 2nd power 10 squared that would move it 2 places 10 to the 3 3 places well we need to move it from one point six seven we need to move it one two three four five six places so we say that this is point six seven times ten to the sixth and then that number is the same size as that number and this has been what we say in scientific notation form that number is written in scientific notation for example we could also write very small numbers in scientific notation for example let's say zero point zero zero zero five three again we write the first non-zero digit on the left which in this case would be five follow it with a decimal point and then all of the non all of the significant figures afterwards to the right which would be in this case three and then we multiply again by a power of ten because obviously five point three is not the same as point zero zero five three but in this case we need to make this smaller to be the same size as point zero zero zero five three so we need to move the decimal point to the left and in this case we need to move it how many places well from here five point three we need to move it one two three four places to get it back to where it should be in the original number so we have to multiply by ten to the negative four one two three four yeah negative four so five point three times 10 to the negative 4 is the same as point zero zero zero five three it's a very very general rule of thumb try to keep in mind that positive exponents of 10 are big numbers negative exponents of 10 are small numbers so positive exponents big negative exponents small just try to keep that in mind we'll come back to that a lot okay now how do we use these ideas of oh look sorry one more point the one thing that's very helpful about scientific notation it allows us to write numbers with the correct number of significant figures even if it would be difficult to write in this form so for example let's we want to write this number 1 million in 670,000 but for some reason we happen to know that it actually has five significant figures in other words these zeros here let's imagine we really do know that those are zeros they are significant how can we write this with five significant figures well if we write it in scientific notation we could write one point six seven zero zero times ten to the sixth by putting these extra zeros over here then we know one point six seven zero zero that has five significant figures while if we write it in this form this only has three significant figures so by writing a number in scientific notation we can write as many significant figures as we want for the appropriate number for the original number for the appropriate number of significant figures for the original number so perhaps this number came about as the result of a calculation of something like that and you really want to make sure that that has five significant figures if we write it in scientific notation we can always be sure to emphasize how many sig figs it has there's some some notation kind of underlying significant figures or something like that I think this is just a neater way of doing that so we'll do it this way we'll write it in scientific notation if ever necessary to emphasize how many sig figs it has okay now how do we do how do we use these things why do we do we worry about these kinds of things