welcome so in this video we're going to be continuing our conversation on sigfigs in our first video part one we talked about how to look at a measurement and figure out which digits are significant and which digits are not significant now in this part two video we're going to use that information and see how many sigfigs are in our final answer we're going to do that for addition and subtraction and multiplication and division so let's let's start with addition and subtraction first all right so addition and subtraction followed the same set of rules when we're looking at a problem involving addition or subtraction we're going to be looking for where our smallest common SigFig is we're going to be looking for the smallest SigFig that they both h h both have in common and then we're going to be cutting off our sigfigs there so I like to approach these problems visually so you'll see me lining up digit you'll see me using marks and underlines so that I know where to stop my Sig fix when it comes to addition and subtraction I don't um personally I don't really worry about counting a number of sig figs I don't find that very useful that is a skill we'll use later for addition and or excuse me multiplication and division but I don't count sigfigs when it comes to adding or subtracting all right so let's say we got a problem like this just written like this 103.6 G Plus 48.3 G the way I like to set up this problem is very uh visually so I'm going to be lining up my digits all right so I go one and I give about a equal spacing for my hundred's place my 10's Place ones place decimal point um my 10th Place my hund's place so on so forth and I'm going to stick with that same lining for the 48.3 I'm going to stick my four under the zero and my eight decimal point and then the three all right now these types of problems are quite common where you might have some gaps like over here and or over here and that's totally okay we're going to deal with that all right now for the Gap here that's really not a big deal at all when we add these two numbers they're going to be in the ballpark of like 150 about and uh yeah it it doesn't really matter that I don't have any information here for the the second measurement that doesn't mess with like the Precision or anything where you have to start worrying about Precision or sigfigs is on the smaller end of the problem so for us that would be here when we're thinking about like the 10th place and the 100th place so before we do the math I just want you to kind of think about why sigfigs matter so here we have oh forgot my uh units all right so here we have a measurement 103.6 5 that was measured more precisely than our second inform uh second measurement we have Precision information in the uh hundredths place whereas the second measurement does not what this means is if we add these two together because we don't have any sort of idea what's really here we're going to lose our precision in the second spot okay so what I like to do is I like to draw a line here right where that Precision ends okay some people like to underline I do this sometimes I like to underline all my significant digits this is definitely optional and then I'm going to do the same here and then visually I can see that my shared common Precision the smallest spot where it still happens is the hundreds place but afterwards I lose that shared Precision so in other words another way to think of it is because I have a blank here I have to draw that line there or another way you can think of it is because my numbers uh my digits more precisely my significant digits ended at this spot that's where I draw my line okay uh this line is not a decimal point that's our decimal point this is line is just telling me uh that everything to the left was will still be sigfigs everything to the right will not okay that doesn't mean we don't care about this five we're going to still type this into our calculator or do the calculation completely as normal okay type type everything so uh 103.6 5 plus 48.3 equals and we get um uh 150 might have to get a new marker 1519 and then I'll put the five here grams Okay 15195 g so again do the math in your calculator as normal okay the only difference is when you're considering SigFig is how you're going to round uh the number so um five and up at a chem 101 level uh this is a I teach chemistry and I realize not everyone watching this may be in chemstry but at a chem 101 level we just go five and up uh rounds up um there are more advanced rounding rules involving fives I think for this video we ain't going to worry about that all right don't worry about it so uh round up uh we go one five um oh this is this is one um that's going to round up to a zero and then we're going to have to carry a one 102.0 g is how I would report the answer okay so long story short uh we had common Precision in the T's Place and so we have to round accordingly um in the 10th place you're going to have to make a determination whether to round or or not okay here five and up we rounded so I have 152.021 intro course I would be okay with this answer this is good okay and maybe we can use it for future calculations if you want to um minimize rounding error down the road one thing you could do is you could write out something like 15195 and then you could underline those four digits to tell yourself or whoever is reading your work those are significant but that one's not I'm just carrying that around to minimize future uh rounding error another thing I see sometimes is [Music] 151.90 just carrying that to minimize rounding errors for future calculations down the road but it's actually not significant or you can just even combine uh the techniques okay uh going forward for uh the purposes of this video I'm not going to be worrying too much about this and I'm just going to go for rounding the answer to the correct number of Sig fix right so I'll just write that out correct answer when [Music] rounding to correct number of sigfigs okay all right got a few more addition and subtraction examples all right to save time I already WR them wrote most of them out um in this uh Neer format okay pause if you want to work on these on yourself this last one is similar to this one but I did something a little bit different especially with um that second part all right assuming you had a chance to pause the video and uh work on these I'll be using uh green so that uh for my an so it doesn't look like the problem itself okay so common sigfigs here and right there okay [Music] 27.9 + 3865 3 8 9 2 9 L all right so to report our final answer uh nine rounds up so we would write 3 8 9 3 lers as our final answer for this problem all right let's do the next one here we have a subtraction but the same uh rules apply okay um oh I said I was going to use green oh whatever U I'm going to um so I'm going to draw my line here okay maybe some of you were thinking about a zero and maybe you were thinking of drawing it there but remember um any zeros any trailing zeros that you have in a measurement where a decimal point is present it is significant okay so that that was measured that's significant all right here's my underlying Sig sigfigs to make sure we're on the same page all right so we have 54290 uh Kelvin uh Kelvin minus 68147 Kelvin okay 47 . 4753 all right four and Below we don't round up uh some people call it some people call that you don't round other people say round down um I don't know what to actually call it but this is what I'm going to write I don't round up okay and I get that answer 47.4 75 Kelvin all right so this next problem let's see if I trip up anyone so here we have a bunch of trailing zeros but there's no decimal point present in either me measurement which means for this first measurement our sigfigs will end there where that one is and the second measurement only has one SigFig which means our sigfigs are going to [Music] end in this place all right now I didn't draw a bunch of commas yet because I didn't want that to clutter a page but now after do this math I'll put them in 2 4 five 1 0 0 0us 8 5 zeros going to double check that yep and we get 1 6 5 1 0 0 0 milligrams okay so this is going to round up to a 17 0 0 0 z0 milligrams now I can worry about commas if I want that so uh 1.7 million in other words would be our answer all right last problem again said it'd be similar to this one let me go ahead and write it out so 2451 * 10 six power and then we have this one now I could type 8.00 * 10 5th into my calculator but then I'll get that all right so one thing I want you to take to heart and keep in mind is that calculators they're not going to uh think in sigfigs here I typed in a three SigFig value but it's not thinking in sigfigs here so it's us humans we have to think in sigfigs calculators don't thinking sig figs so let me show you what I mean so this is going to have one two three four five zeros notice how careful I am how much I take my time in writing out these problems all right and there's our problem so this might present a little bit of trickiness to you you're like well if I multiply this out I don't have a way of keeping those three sigfigs in this um measurement you're like 880,000 or 800,000 as written does not allow for me to keep track of those three sigfigs okay yeah that that can happen um so the the deal with scientific notation is scientific notation so powerful because it allows you to keep track of sigfigs in any situation whereas something written out as a standard number you may not be able to write down those sigfigs uh in a proper way so in other words scientific notation allows you write down any measurement whereas standard numbers don't always allow for that okay I'm not saying don't set it up this way I'm just saying there's a reason why this problem had to be written that way okay someone actually measured right here okay so the purposes of doing the math I'm just going to underline those three so I know to remember those are Sig uh sigfigs all right and then here I'll just underline those Okay so so those four sigfigs those three sigfigs and now we have a way of doing this math all right I'll just type this into the calculator so same calculator answer output as last last time that's where comma is going to go another comma but this time because we had shared Precision going down to this a thousand's place we can actually write that out as our final answer all right so that's the answer I don't like put a period there because that would imply that you suddenly had Precision there and we never uh did so that's just a way you would write that or you can just keep things in scientific notation depending on what you need to do all right cool that was addition and subtraction let's move on to multiplication and division all right multiplication and division we're not looking for a common shared SigFig actually we're just looking for the lowest number of sigfigs in our mathematical operation okay so in other words multiplication division lowest number of sigfigs in is equal to your number of sigfigs out all right so my tips just make sure you count the number of sigfigs in each item in your problem and um check your units at the end all right now this middle tip is uh this is where I see the most problems when people are doing sigfigs problems okay they'll understand the rules but then when they type things into the calculator uh they they forget the part that calculators don't automatically do sigfigs they don't they don't think in sigfigs sometimes by coincidence by luck it'll just have an answer that's the correct number of six figs but it wasn't trying to do that so be careful with the calculator you have to do SI things the calculator doesn't all right so what I did with these examples is I chose these examples with you know zero you'll see what happens okay I won't spoil my surprise so all right I snuck in um a division here for for its example all right so let's count a sigfigs in each thing all right 2.70 that's going to be four sigfigs and this uh second one has three sigfigs all right so even before we do the math uh I like to write the number of sigfigs in answer immediately okay uh we're going to need a three SigFig answer all right now this we can do in our head and um we we'd get the answer but let me let me show you let's just pretend we weren't doing this in our head and we were just typing it into the calculator and relying on the calculator see what happens 2.70 divided 1.00 it equals and it just says 2.7 so here's the Temptation you're going to write 2 7 and you remembered your units we have G over milliliters that's a density value all right that was the density of this aluminum block from our previous video okay um not this particular block but this is aluminum um but yeah as as written from the calculator it only gave a two sign two signicant answer but we need three so it's going to be on you to add a zero there okay we're not changing the value we are just simply showing the Precision correctly okay so it's on you to add that zero if the calculator doesn't do that okay so that would be the correct answer there all right Next Up 2 * 7 we get 14 14 M squared but here's the thing oh I moved a little fast so let me show you one SigFig one SigFig that means our answer also has to have one SigFig it's about the lowest number okay above we had four time or four divid by three so we need a three just the lowest number 1 1 we need a one signific answer okay you have to start with uh the biggest digit you have available so this has to round down or not round whatever you call it it's it's going to be 10 m squared okay that probably doesn't feel great to you knowing that it should in reality be in the ballpark it should be about uh 14 the ballpark of 14 but we we were forced to write 10 just due to our lack of good Precision in these measurements all right so yeah that's kind of what you're stuck doing all right so you say we got to do better let's increase the Precision on one of the measurements so here 2.0 this is uh two sigfigs and 7 m well that's just still one SigFig which means we still just need a one SigFig answer so that didn't solve anything for us and we're stuck with that same thing here one SigFig two sigfigs the lowest number so we're going to have a two SigFig answer here okay I really don't like writing this it it it hurts my heart that this was precise but that wasn't that wasn't and so I'm I'm forced to write something like that all right so how about here this is starting to look a little better here we have two sigfigs here we also have two sigfigs so finally we can have a two SigFig answer oh I should have wrote a one here okay but this problem here will have two sigic so now I can finally write um 14 but I'm not going to write 14.0 okay so let me demonstrate on the calculator again 2.0 * 7.0 it says 14 me squar that is all we're going to want okay you may have a Temptation you see a0 Z over there and a0 Z over there the Temptation might be to write 14.0 M squared but hopefully you got recognize that this has three sigfigs whereas the two parts going in had only two sigfigs each so we wouldn't be able to write that answer here next problem 36 sig figs three sigfigs we're going to have a three SigFig answer calculator still says 14 as it has every time but since we need three sigfigs we need to write that zero to maintain the Precision that we had going in all right let's do this last problem so this measurement here has 1 two 3 four five 6 seven eight sigfigs okay someone measured that perfectly oh I'm missing a unit I meant for this to be meters but this measurement here was not done very precisely so once again we would have one SigFig in our answer even if the first one was done very precisely all right I hope this helps if you encounter any examples uh that you feel weren't fully addressed here I'd be happy to tackle them in a separate video feel free to send them over to me but again hope this helps thanks and take care