significant figures and calculations okay so we've already discussed how to tell how many significant figures a number has and now we're going to apply that in calculations and so basically remember that the concept of significant figures is used to limit a measurement to the proper number of digits and we've already learned how to recognize how many sig figs a number has and now we're going to limit our mathematical results so if we multiply divide add or subtract we're going to learn how to limit our result to the proper number of significant figures so for addition and subtraction calculations basically we're going to limit that reported answer so our you know our final answer to the rightmost column that all numbers have in common okay so so for instance I like to think of that as the lowest number of decimal places so if we have one point two and four point seven one so four point seven one then this number one point two has one decimal place this one has two so we would limit it to one decimal place so what we're gonna do is we're going to add our two numbers together we're not going to round this one first we're gonna add the two numbers together as they are we're gonna get our result five point nine one and then we're gonna round it down to just five point nine rounding to two significant figures one decimal place okay all right so again we want to round our final answer to the tenths place and I can't stress this enough don't round the second number first and then add them together okay in this case it would actually work but there is no guarantee of that especially with more complicated calculations so don't do that so add the numbers as you have them and then round them to the proper number of significant figures or decimal places in this case at the end and the rules for rounding that we're going to use for both addition and subtraction and multiplication and division is we're going to round the final answer up if the last digit is equal to or greater than five and we're going to round down if the last digit is less than five alright so here's a little bit of practice so let's add these two numbers together and then round them to the proper number of significant figures so take a minute do that and then we'll go to the next slide and we'll talk about them all right so put that in your calculator and when you do that you're going to get 119 point nine zero two okay now 101 point two only has one decimal place eighteen point seven oh two has three decimal places so we know that our final answer has to have only one decimal place so we're going to round our final answer to the to one decimal place so let's go up here one hundred nineteen point nine the next digit is zero so we're just going to round it to one hundred nineteen point nine so we rounded down so for the next one we're going to subtract off one point zero one three from two hundred two point eight eight this number has two decimal places two hundred two point eight eight and this number has three decimal places and so so the calculator would give us two hundred one point eight six seven now we need two decimal places in our final answer so we're gonna go and look at the third number and we see that it's seven and it's greater than five so we're going to round the six up so we're gonna have two hundred one point eight seven as our final answer with the proper number of significant figures all right for multiplication and division calculations it's very similar but now instead of worrying about the number of decimal places we're going to basically limit the answer to the least number of sig figs that the all the numbers in the problem have so for instance if we divide 20 three by four hundred and forty eight twenty three only has two significant figures four hundred and forty-eight has three so we're going to limit our final answer to two significant figures so let's go ahead and do that division we end up with zero point zero five one three three nine two eight six we know that we need two significant figures remember this leading zero is not significant so we're not going to count it this one we're also going to ignore so one two well zero point zero five one and then the next number is three so we're going to round down so our final answer is zero point zero five one alright so the same rules for rounding are used for multiplication and division and so I've already told you so that if the last digit is equal to our greater is equal to or greater than five then we're going to round up and we're gonna round down if it's less than five all right so let's do a few practice problems here also so multiply these two two sets of numbers together so seventy six point four times one eighty point four and then do the other one and then that's a division problem and then we're gonna talk about how to limit the answer to the appropriate number of significant figures all right so the first problem so we are doing multiplication so we're gonna choose the number that has the lowest number of six six which is seventy six point six four it has three sig figs 180 point four has four sig figs so we have to limit it to three sig figs we're gonna multiply it and look at this look at what we get so we get thirteen thousand seven hundred eighty two point five six that certainly looks like all those numbers are significant but they are not so what we're gonna do is we need three significant figures one two three okay now the rest of these are gonna have to turn into zeros and actually you can just ignore the decimal place after that so we're going to need two placeholder zeros here so we're going to actually round this up to 13,800 okay and and that's going to be our final answer now I have a decimal place here that is wrong okay that is probably just a function of putting sentences on my my board but that is incorrect this should be 13,800 with no decimal place all right and we're going to have the same problem with this one just as so now the first number has four significant figures here okay the second number has three significant figures so we're going to limit our final answer to three significant figures and so once we do the division we're gonna end up with two hundred five thousand four hundred and seventy two point five two seventy-five we want three sig figs so that's the two hundred and five thousand so remember the sandwich zero is significant and then we're gonna have three placeholder zeros and we can ignore the entire decimal part so again there should be no decimal place here okay this should be two hundred five thousand just without that decimal place okay and so anyway so excellent so we know how to round down we know how to round up and we know how to use placeholder zeros now okay now sometimes zeros can be very tricky to figure out whether they're significant or not okay now if we want to really determine or communicate that a zero is significant we're gonna write it in scientific notation so scientific notation always includes zeros in the coefficient of the number only if they are significant so look at these two numbers here so eight point six six six times ten to the sixth that has four sig figs so scientific notation makes it really easy to see how many sig figs there are if we knew that this like let's say this was our estimated digit in a measurement so known and then our estimated all of those are significant it just happened to be a zero and so that is going to give us five significant figures and then of course we have the same power 10 so if a zero is not significant it will not be written in the coefficient so if you see a zero in the coefficient like that you can safely assume that it is significant and so yeah it is good practice that if you have a situation like this and you want to communicate that that zero is significant make sure you write it in scientific notation all right finally um let's just summarize a little bit about what we've learned so significant figures indicate the number of known values plus one place that is estimated so the number of known values plus one an estimated number and we have rules for determining which numbers in a quantity are significant and which aren't okay and usually the trickiest part of that the zeroes okay in calculations involving addition and subtraction we're going to limit to the lowest number of decimal places and in calculations involving multiplication and division we're going to limit to the least number of sig figs in all the data values