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# Addition and subtraction with significant figures

## Video transcript

we saw in the last video that when you multiply or you divide numbers or I guess we should say when we divide multiply or divide measurements your result result can only have as many significant digits as the as the thing with the smallest significant digits that you ended up multiplying and dividing so just as a quick example if I have two point two point zero zero times I don't know let me three point five my answer over here can only have two significant digits this has two significant digits this has three two times three point five is seven and we can get to one level one we can go 1-0 to the right of the decimal because we can have two significant digits this was three this is two we only limit it to two because that was the smallest number of significant digits we had in all of the things that we were taking the product of when we do addition and subtraction it's a little bit different and I'll do an example first I'll just do kind of a numerical example first then we'll think of a little bit more of a real-world example and obviously even my real-world examples aren't really real world in the last video I talked about laying down carpet and someone rightfully pointed out hey if you're laying down carpet you always want to round up just because you don't want to you know it's easier to cut carpet away than to have to somehow glue carpet there but that's a particular to carpet I was just saying a general way to think about precision and significant figures but that was particularly to maybe carpets or tiles but when you add when you add or subtract when you add or subtract now the significant digits or the significant figures don't matter as much as the actual precision of the things that you're adding how many decimal places do you go so for example if I were to add one point two six and I were to add it to let's say I were to add it to two point three if you just add these two numbers up and let's say these are measurements so when you make it this is clearly three significant digits are we were able to measure to the nearest hundredth here this is two significant digits so three significant digits this is two significant digits we're able to measure to the nearest tenth let me label this this is the hundred hundredth the is the tenth when you add or subtract numbers your answer so if we just do this if we just add these two numbers I get what I get two point or I get three point five six the sum or the difference whatever you take you don't count significant figures so you don't say hey this can only have two significant figures what you say is this can only be as precise as the least precise thing that I ever had over here the least precise thing I had over here was 2.3 it only went to the tenths place so on our answer we can only go to the tenths place so we needed to we need to round this we need to round this guy up because we have a six right here so you round up so if you care about significant figures this is going to become this is going to become a three point seven and I want to be clear this time it worked out because this also has two significant figures this has two significant figures but this could have been let me do another situation you could have one point two six plus one hundred and two point three and then you would get obviously you would get one hundred and three point five six and in this situation this obviously over here has four significant figures this over here has three significant figures but on our answer we don't want to have three significant figures we want to have be only as precise as the least precise thing that we added up the least precise thing we only go one digit behind the decimal over here so we can only go to the tents only one digit behind the decimal there so what's going to be round up to 103 point six and to see why that makes sense let's do a little bit of an example here with actually measuring something so let's say I have a block here let's say that I have a block let me draw that block a little bit neater and let's say we have a pretty good meter stick and we're able to measure it to the nearest centimeter we get it it is two point zero nine two point zero nine meters and let's say we have another block let's say we have another block and this is the other block right over there and we are we have a let's say we have even a more precise a meter stick that can measure to the nearest millimeter and we get this to be one point nine zero one meter so we're measuring to the nearest milimeter and let's say that those measurements were done a long time ago and we don't have access to measure them anymore but someone says how tall is it if I were to stack the blue block on top of the red block or the orange block or whatever color that is so how high would this height be well if you didn't care about significant figures or precision you would literally just add them up you would add the one point nine zero one plus the two point zero nine so let me add those up so if you take one point nine zero one and add that to two point zero nine you get you get one plus nothing is 1 0 plus nine is nine nine plus zero is nine you get the decimal point one plus two is three so you would get three point nine nine one and the problem with this the reason why this is a little bit it's kind of misrepresenting how precise your measurement is is that you don't know what if I told you that the tower is three point nine nine one meters tall I'm implying that I somehow was able to measure the entire tower to the nearest millimeter the reality is I was only able to measure part of the tower to the nearest millimeter this part of the tower I was only able to measure the nearest centimeter so to make it clear that our measurement is only good to the nearest centimeter because there's more error here then you know it might it might overwhelm whatever precision we had on the middle of meters there to make that clear we need to make this only as precise as the least precise thing that we're adding up so over here the least precise thing was we went to the hundredths so over here we have to round to the hundredths and since one is less than five we're going to round down and so we can only legitimately say if we want to represent what we did properly that the tower is three point nine nine meters and I also want to make it clear that this doesn't just apply to when there is a decimal point if I were to tell you that let's say that I were to measure let's say that I were to measure let's say I were to measure a building and I was only able to measure the building to the nearest 10 feet so I tell you that that building is three hundred and fifty feet tall so this is the building this is a building building and let's say that there's a manufacturer of radio antennas so or radio towers and the manufacturer has measured their tower to the nearest foot and they say that their tower is 8 is 8 feet tall so notice here they measure to the nearest 10 feet here they measure to the nearest foot and actually to make it clear because it once again we as I said this is ambiguous it's not 100% clear how many significant figures there are maybe it was exactly 350 feet or maybe they just rounded to the nearest 10 feet so a better way to represent this they would be to say instead of writing at 350 a better way to write it would be 3.5 times 10 to the 2nd feet tall and when you write it in scientific notation it makes it very clear that there was only 3 there's only 2 significant digits here you only measure to the nearest 10 feet other ways to represent it you could write 350 this notation is done less but sometimes the last significant digit has a line on top of it or the last significant digit has a line below it so either of those are ways to specify this is probably the least ambiguous but assuming that they only measure to the nearest 10 feet how would you out and someone would ask you how tall is the building plus the tower well your first reaction which is let's just add the 350 plus 8 you would get 358 you would get 358 feet so this is the building plus the tower is 358 feet but once again we're misrepresenting it we're making it look like we were able to measure the combination to the nearest foot well we were only able to measure the tower to the nearest foot so in order to represent our measurement at the level of precision that we really did we really have to round this to the nearest 10 feet because that was our least precise measurement so we would really have to round this up 8 is greater than or equal to 5 so we'll round this up to 300 360 feet so once again whatever is and and just to make make it clear and what's it even this is ambiguous maybe we put a line over it to show that that is our level precision but then we only have two significant digits or we could write this as 3 point 6 times 10 to the 2nd which is just 100 three point six times ten to the second feet and scientific notation and this makes it very clear that we we have two significant digits here