hello everyone this video will cover the information in chapter two sections three and four so in section three we're going to review significant figures in calculations um eventually at first we're just going to work on rounding so for 2.24 again these are coming from the end of section example problems we have round off each of the calculator answers in problem 2.23 to two significant figures and i have included those values here so here is my public service announcement for this particular video your calculator does not give a damn about significant figures um it is up to you to determine the correct number of significant figures to report your final answer to so as i mentioned let's first focus on just rounding so um the key to rounding and we see here that we need to round to two significant figures is to identify the number of significant figures that you need to report your final answer to and then look to the digit immediately to the right of that final significant digit if it is five or more then you need to round that final significant digit up by one number if it is four or less then that final significant digit remains unchanged so let's take a look for a we have 1.854 kilograms and we know we need to report this to two significant figures so we'll underline those and then we look at the digit immediately to the right of that final significant digit and we see that it is a five which means we're going to need to round the eight up by one so if we wanted to round up or round and report this number to two significant figures we would report this as 1.9 kilograms for example b we have 88.2038 again we're going to identify those two significant figures and then we look immediately to the right we see that as a 2 so to correctly report this value to 2 significant figures would just be 88 right because this is uh less than five this final significant digit remains unchanged now for here zero point zero zero four seven three eight two six five centimeters and so let's underline the two significant figures remember these leading zeros are all not significant and so we look to the digit uh to the right of the final significant digit that is a three which means that we would write this as zero point zero zero four seven centimeters we could also write this as four point seven times ten and we say one two three times 10 to the power of negative 3 centimeters okay so if we wanted to write this in scientific notation for uh example d we have eight thousand eight hundred and seven there are our two significant figures and we see that the number directly to the right is a zero so we would report this as 8 800 or 8 800 meters which we could also write as 8.8 and 1 2 3 times 10 to the power of 3 meters here we already have this in scientific notation 1.832 times 10 to the power of 5 seconds again underlying those two significant figures and we see that the number directly to the right of the last significant digit is a 3 so this would be 1.8 times 10 to the power of 5 seconds okay there we go so now we're going to perform these calculations and the big thing is taking into consideration how many significant figures we can report our final answer to so when it comes to significant figures in multiplication and division we can only report our final answer to the same number of significant figures as whichever value in our calculation has the least number of significant figures there is a different rule when it comes to addition and subtraction uh but we will cover that later um this will also be my time to plug a calculator so in general um for my other general chemistry or organic chemistry we don't really use calculators in organic chemistry in the lab a lot but i really like um so this is the ti-30x2s if you can see that we recommend this calculator or the ti-30xa they're pretty cheap they're usually anywhere from eight to twelve dollars um you can find them most places i know target usually has them amazon has them walmart etc they are not the fancy graphing calculators and so some students kind of struggle if you have been using a graphing calculator it's okay to use the calculator that you have on hand but i know this one quite well as well as the ti-30 xa so i'll be able to help you out with those when it gets into the graphing calculators uh yeah it's it's kind of been a while so i might not be so helpful um what is important is that you know how to use your calculator um i'm sure it is frustrating you know as frustrating to students as it is for me when i see that a student has set up a problem um perfectly and uh but yet has reported um an incorrect final answer and it's usually um calculator error um so just make sure that you're comfortable with the calculator that you uh that you have um but this is the one like i said it's it's it's cheap and it has all the functionality that we need for this course so um let's take a look at a we have 400 times 185 right so if we just type in 400 times 185 into our calculator we get initially 74 000 okay but if we take a look at um the two um the two numbers in our calculation we see that the number 400 only has one significant figure these two trailing zeros are not significant and 185 all three digits are significant so we have one significant figure and three significant figures that means that we can only report our final answer to one significant figure so this is where the rounding comes in handy okay comes in handy is necessary so we're gonna underline that one significant figure and then we take a look immediately to the right and we see that that is a four so the more precise way or the correct way to report this answer in accordance with the rules for significant figures would be seventy thousand okay and now we look here and we have uh 2.40 divided by 4 times 125. so up top here we have all three digits are significant right so here we have one of those trailing zeros that occurs to the right of the decimal place so it is significant and then four would only have one significant figure and then 125 has three significant figures so yet again we are constrained to one significant figure in our final answer so we'll just go ahead and put this into our calculator 2.40 divided by 4 divided by 125 and i initially get off my calculator 0.0048 but as we just said we can only report our number to one significant figure so we underline that first significant figure there the four and then we look immediately to the right we see that is an eight so we want to report our final answer should start boxing these there you go as 0.005 okay one significant figure for example c we have 0.825 times 3.6 times 5.1 so three significant figures two significant figures and two significant figures so we can report our final answer to two significant figures so 0.825 times 3.6 times 5.1 and what i initially get off my calculator is 15.147 but we have already said we can only report this to two significant figures so and we say we underline those two significant figures we look immediately to the right that is a one so we report this as 15. all right moving on to d 3.5 times 0.261 and then we divide by 8.24 and 20.0 and oh man this is a monster so i'm not even going to write all this out but if you're following along with your calculator uh which i strongly recommend that you do we get 0.0055 0.0055 five four three zero eight three all right all right so maybe i will write it down just so we have a zero zero five five you think after the last video i would have remembered to turn off my sound on my phone okay all right so let's take a look at the initial numbers in our calculation and so 3.5 has two significant figures 0.261 has 3 8.24 has three and 20.0 also has three right trailing zero to the right of the decimal place also significant and because of where the decimal place is this zero right here is also significant so that means that we can report this to two significant figures so we'll go ahead and underline those two significant figures look immediately to the right that is a four so we would report this as 0.0055 okay now multiplication sorry um multiplication and division using scientific notation um so um you know we we can put these numbers um into our calculator in scientific notation um let's see for those of you who have a calculator like this or similar you would need to use i don't know if this is going to work but um do you see that that that you're that ee right there okay that's what you want to use and so you'd have to press the second button and then that and that's how you would be able to put that exponent term in there um if you have a graphing calculator it is a little bit easier because you can just use the the 10 to whatever function um so um yeah the other way that we can do this is we can rewrite this to just take all these digit terms okay so that would be 5 times 1.05 divided by 8.24 and then we'll handle all of the exponents so we had 10 to the power of minus 5. now when we multiply numbers in scientific notation we add those exponents together so that would look like 10 to the power of negative 5 plus 4 but when we divide by a number in scientific notation we have to subtract that exponent so that would look like um minus negative eight okay so again when we are multiplying numbers in scientific notation we are adding the exponents but when we are dividing we need to subtract the exponent that is in the denominator so again we have grouped together those digit terms and we can just do that math it makes it a little bit simpler to put into a calculator if you feel a little uneasy directly putting these numbers in scientific notation into your calculator so we can do it both ways we will get the right answer so if we take this and we do 5 times 1.05 divided by 8.24 i initially get zero point six three seven and i get i get a bunch of numbers one and i'm gonna call it there because if i look here i got one significant figure three significant figures and three significant figures so i'm only going to be able to report this final answer to one significant figure um so i'll just write those but remember your calculator doesn't give a damn about sig figs there's a whole string of numbers there and then we'll take a look at the exponent term so if we have negative 5 plus 4 so that is minus 1 and then minus a negative number uh means plus so minus 1 plus 8 and so this would be times 10 to the power of seven and now we need to consider significant figures in our digit term and so we've already gone through that and we see that it can only be one significant figure so a couple things that are wrong with this number as written um one we have too many sig figs right we're working on that to identify the proper number of sig figs to report our final answer but also even if we dropped that 3 7 1 we would have a value that is less than 1. and as i mentioned before to properly write a number in scientific notation that digit term should be a value in between 1 and 10. so let's first write this so if we take this decimal place and we move it one space [Music] right right there okay then we have already moved that one uh position to the right okay and so now this becomes six times 10 to the power of 6. okay so again a couple things there right one how to take a a calculation with multiple values in scientific notation and kind of break it up to make it perhaps a little bit simpler to work with as you maybe are noticing this does require a couple of additional steps so although the numbers may be more manageable um it it it does potentially increase the the time because we have more steps and as i said um we can go ahead and just put this right into our calculator with the scientific notation so if i did five and then i pressed the the second button and then that e e and i put negative 5 and then times 1.05 second e e 4 and then divided by 8.24 second e e minus eight and okay well it does not give it to me in any kind of nice um significant uh digit term but i do get let's see one two three four five six uh i do get uh well i get six three seven one three five nine point two two three um but again if we underline that first digit because we can only report to one significant figure we get six times ten to the power of six um so right uh definitely [Music] time wise a lot easier to just work with scientific notation in your calculator but do make sure um that you are comfortable doing so okay um and then our final question here i'm just going to go ahead and i'm going to put this into my calculator i'm using the scientific notation so 4.25 second e e to the power of 2 times 2.56 second e minus 3 divided by 2.245 second e e minus 3. and then 56.5 okay and i get eight point five seven seven five six there's a whole bunch there's a whole bunch of numbers six six six six seven um but we are already well past the plausible number of significant figures to report our final answer to um so uh let's take a look so 4.25 so this first has three significant figures three significant figures four significant figures and three significant figures so we can report this value to three significant figures and then we look to the digit directly to the right of the final significant figure it is a seven so that means we want to round that final significant digit up by one eight point five eight um so i'm sure that we'll have um ample opportunity to uh cover uh this um throughout the semester but i do just wanna caution against rounding um too early when we start getting into calculations that have multiple steps it's it's really important to not round off your answers if you do go step by step not round your answers off prematurely if you start rounding after each step of a multi-step calculation it will lead to errors in your final answer so a good rule of thumb if you are doing a multi-step calculation step-by-step is to carry one additional digit beyond what the significant figure rules would designate and that will generally help you avoid rounding errors so um you know if if this were this is a bunch of steps but i usually tell students you know if you underline what your final significant digit should be and then carry one additional digit in the subsequent calculations you should be able to avoid any rounding errors so um i thought that this was going to cover um well let's see i think there's not much here okay so yeah this this will cover 2.4 i was going to call it because it's already getting long but um we'll just go ahead okay sorry for the abrupt end to the last one i thought i got a message from the daycare uh anyway so um in section 2.4 we are going to explore prefixes and equalities so um really we're just kind of getting used to the metric system and using these multiplicative prefixes um and going from name to symbol etc and so it's a good idea for you just to to know at least the the ones that we use most commonly in chemistry so you know converting between milligrams and grams or grams and kilograms i would say um the prefixes kilo and milli are used quite frequently in chemistry nano is also used but to a lesser degree and some of these other multiplicative prefixes you might find in the medical field particularly helpful um but we just don't use them quite so much uh in in you know general chemistry mostly because we are not often working with very very very small amounts or very very very large amounts and so the whole idea behind utilizing these prefixes goes back to what i've said a couple times before already is that scientists we like nice neat compact numbers we don't like a lot of zeros and the multiplicative prefixes allow us to report numbers in a compact way while kind of disguising all of the zeros in these multiplicative prefixes so um again we're just we're just in this section getting familiar uh with what those prefixes are and and what they are telling us um in terms of of amounts so for 2.38 use a prefix to write the name for each of the following so if we have 10 to the ninth meters another way that we could write that is that we have one giga meter so giga being the multiplicative prefix and so the way that we could write this is we could say that one in a little trouble with my writing pad right now one excuse me that one so giga is capital g one gigameter is equal to uh 10 to the power of nine meters we could also say that ten the power of negative nine gigameters is equal to one meter um so this is where i find that some students get maybe a little bit confused with these multiplicative prefixes and writing these equalities um we can write them either way and they are both correct it's it's kind of more of a matter of preference or how it is easiest for you to remember this but the way to keep it straight is to at least try to recognize which is the the smaller increment okay that smaller unit so in in this case the meter being the smaller increment than the giga meter so it would take a lot more meters uh to um to make a giga meter then vice versa so again both of these qualities are saying the same exact thing it's just you know where we want to write the the exponent so again one giga meter is equal to 10 to the power of 9 meters or 10 to the power of minus 9 gigameters is equal to 1 meter now for b we have 10 to the power of 6 meters so again we have um a it's a pretty big number of meters so which multiplicative prefix could we use to to write this as one something and that is a mega meter so we could write this as one mega meter and we could also write these equalities one mega meter so capital m for mega one mega meter is equal to 10 to the power of six 10 to the sixth meters or alternately 10 to the power of negative six mega meters is equal to one meter right the mega meter being uh the larger unit okay so it would take um 10 to the sixth meters the smaller used unit to make up a mega meter now for c and d now we're moving into the the smaller um uh the smaller units so if we have 0.001 meters we could also write this as 1 milli meter and over here so millie being our multiplicative prefix and we could say that one millimeter is equal to 10 to the power of minus 3 meters right 1 2 3. and we can also say that there are 10 to the power of 3 or 1000 millimeters is equal to 1 meter and while i'm thinking about it um i do just want to say that these multiplicative prefixes you work with any um unit of measurement right so whether it's meters or grams or liters um so that liter gram even seconds um that's that's that base unit and then we utilize those multiplicative uh prefixes with those um and so we we can use the multiplicative prefixes with with any of those um units within the the metric system um or then the uh the si so we should be talking about that um maybe that's already covered uh but the si unit of um or the system international for units and so basically this is an international agreed upon units for different types of of measurements and we can use the multiplicative prefixes for for any of those so here for d we have 10 to the power of minus 12 meters so again very very small very very small and so this would be one picometer and we could write that one picometer is equal to 10 to the power of minus 12 meters or alternately we could write this as 10 to the power of 12 pico meters is equal to one meter um so again equally correct um just you know however is easiest for you to [Music] to remember all right and then um moving on let's complete each of the following metric relationships so if we have one mega gram how many grams is that well we have conveniently written that up here so we were using um the unit of meters but as i said these multiplicative prefixes can be used for any so if we have one mega gram and we see up here that one mega meter was equal to 10 to the sixth meters and so um it would be uh in the case of the mega meter that one excuse me mega grams that one megagram would be equal to 10 to the power of six grams now for b we're going between milliliters and micro liters uh so the answer to this is there are 10 to the power of 3 microliters in one milliliter or 1000 microliters in a milliliter but until you get more comfortable with these multiplicative prefixes if you needed to um you know work work this out what we could say is so our known beginning and we're going to talk about solving problems uh using conversion factors in the future and i'll talk about the known beginning the desired end and then the conversion factors in between in more detail but here we have that our known beginning is the milliliter and we need to get to microliters so maybe you can't quite remember exactly how many microliters there are in a milliliter but what you do know is that there are 10 to the power of 3 milliliters in 1 liter and you also know that in 1 liter there are 10 to the power of 6 micro liters so if you have ever worked out math uh in the past um and you know you know if you have the same thing in the numerator and in the denominator that it cancels out right it is the same way with units so if we have milliliters up here in our numerator and milliliters in the denominator we can cancel those out as such and the same here we have liters in the numerator and liters in the denominator and they will cancel out and we will be left with the units of micro liter and so if we just work this out that's 10 to 6 and then we have i just pointed at my screen like you can see that uh 10 to the power of 3 in the denominator which means we would need to subtract that exponent and so our final answer is 10 to the power of 3 microliters okay and you could also if you look at the list of multiplicative prefixes you would see that yeah there are 10 to the power of 3 milliliters 10 to the power of 6 microliters so maybe you just recognize that um the difference is 10 to the power of 3. but anyway this is how we could set it up mathematically to work that out for c we have one gram is equal to how many kilograms um so there are um 10 to the power of 3 grams per kilogram alternately there are 10 to the power of minus 3 kilograms in 1 gram so the gram being the smaller unit right a thousand grams and a kilogram so if we only have one gram that would be equal to 10 to the power of minus three grams okay so there's that um and then one gram to milligrams and so we saw up here going from milliliters to liters that there are a thousand milliliters in a liter and there are also 1 000 milligrams in one gram okay um so you know as i had mentioned um a lot of this yeah you're just gonna kind of have to spend some time with these multiplicative prefixes i've given you a little bit of a of a sneak peak for um how to work with these multiplicative free prefixes to convert between them um not a bad idea maybe i probably have this that i can post for you to just you know you know have a list with you um when you are working on um you know example problems and homework or on any exams or quizzes i tend to expect my general chemistry students to memorize these and i will say that if you can memorize them it's just going to cut down on time for you but i you know i don't want you all to go as far as you know making um flash cards or or anything like that or spending um a tremendous amount of time on it the important thing is that um you know you you are able to to to work with these multiplicative prefixes and convert between them so okay so i'm going to call it um and yeah that'll wrap up um section 2.3 and section 2.4 you