we are going to talk about adding and subtracting significant figures the rule for adding and subtracting significant figures is different for than that for multiplication and division in multiplication and division you take the least number of significant figures for addition and subtraction you have to get the decimal point in the same position for all your numbers and then look for the least place value they have in common and that is going to stop or denote the position of your last SigFig on the right hand side for example if we want to add together these three numbers we'll notice that their decimal point is all in the same spot because they're all to the same power of 10 next what is the last place value they all have in common well this only goes to the hundred's place so that's the last place value they all have in common this number right here doesn't have any thousands place right so the hundreds is the last place value they have in common at this point you would proceed to add as you normally would having trouble getting my numbers lined up mentally here maybe I should have used the calculator okay there we go I think I did my addition successfully if I didn't that's not the point of it anyway this is the point of it since the last place the least place value all these numbers have in common is the hundreds that's going to be the place value of our last significant figure so that's going to be our last significant figure we're going to round on the basis of the number next to it or nine so our final answer is going to be 3.97 or three sigfigs in our answer now you're saying Dr Kelly we went with the number of significant figures in our last and our least number of significant figures of those numbers we added so how is this different than the multiplication and division rule here's how it's different the addition and subtraction rule can actually change your number of significant figures here's why and here's the basis for it and we'll look at some examples where it does before you do that why don't you write down Hawkman everybody write down Hawkman the addition subtraction rule is clearly linked to the measuring tools that you use and here's the logic to it if I have a ruler for example as my measuring tool can I use that measure that ruler to measure the width of a piece of hair the width not the length the width no right the the piece of hair is way too thin to be measured using a common household ruler so there is a limit as to how small an object a tool can measure there's a small limit on how small a tool can measure if I take that same ruler however and I lay that ruler end on end on end on end I could use that 12in ruler to measure the distance between cities so I can measure using my measuring tool infinitely large things by adding the results of repeated measurements together but I can't measure the width of a hair with it so I'm stopped by the capabilities of my ruler on the so I'm stopped by the capabilities of my ruler on the right hand side but on the leftand side I can get infinitely large so there's an example of how we can change our number of sigfigs let's take a look at this second example here real quickly if we look at these two numbers the last place value they have in common is the hundreds so in our answer the least place value we're going to report is the hundreds so I add these numbers together and because we're back stopped this direction but we can expand in this direction for addition and subtraction our answer is 10.47 that we would report and we're going to report an answer that has actually going to have four significant figures despite the fact we had a three SigFig measurement added to a another three SigFig measurement so we can increase our number of significant figures by adding and subtracting we can also decrease our number of sigfigs for example and this one's much more easier for people to buy the last place value that they have in common for these two measurements is the 10 spot the tenths so when we subtract 5 4 3 1 zero that's our last place value we round on the basis of that three and our answer we would have to report is 0.1 so we wind up reporting an answer with only one SigFig despite the fact we had a number with two and a number with five sigfigs involved in our operation so addition and subtraction can totally change the number of significant figures you have how bizarre everybody write down Bizarro another thing that will cause cause a little bit of trouble with addition and subtraction is scientific notation scientific notation creates the illusion when you line up two numbers that they have the same place value when in reality they have much different place values let's take a look here we have a number with the fifth power and the number with the third power so even though it looks like you could line up their decimals right here the problem is their decimals not in the same spot because their power of 10's different so before I can add these two numbers or subtract them I have to get their decimal in the same location so we're going to have to do exactly what the preacher tells us on Sunday and we're going to have to go to the higher power go to the higher power raise your small power to your large power there's a two power difference between three and five so I'm going to move my decimal two spots to the left one two put in my decimal there so I have 1. 235 * 10 5th plus 0.0 924 time 10 5th so now my decimal is in the same spot and the last place value they have in common is now right there so I can add my numbers now four 7 2 3 1 * 10 the 5th that's the last place value they have in common going to round on the basis of that number so the answer I would report would be 1 372 I think I said the wrong thing but I wrote the correct thing chemistry is easy life is hard let's keep going um work one more example here again it looks like these two numbers have their decimals in the same spot but they don't 1es to the 7th 1es to the 3 so there is a four power difference so I'm going to move my decimal four spots to the left one 2 3 four fill in my loops with zeros so I have 1. 235 * 10 7 0.0000 24 time 10 to the 7th add them together the least place value they have in common is right there so let's add them together that's the last place value they have in common so I would report as my answer that number should look familiar to you does that number look familiar to anybody where have we seen that number before that was one of the numbers we started out with wait a second did we just perform an addition Rea an addition and wind up with the exact same number we started out with despite adding something to it whoa mind blown that's exactly what happened because the quantity that we added is so much smaller than the quantity we added it to that it didn't significantly change the value it's sort of like if Bill Gates bent down and picked up a penny would it really change his net worth no right we use this principle in chemistry of some numbers being so much smaller than another that they're insignificant all the time classic example is when we talk about the mass of an atom when we talk about the mass of an atom we say only that only protons and neutrons contribute to the mass of an atom despite the fact that electrons also have mass it's just the mass of the electrons is so much less than the mass of the protons and neutrons that it doesn't significantly change the mass of an atom so we use this concept of not changing a number by adding or subtracting things all the time and it's something we'll talk about again um real quickly let me look at an example using negative signs simply because they can throw people let's zoom in on this reaction right here 7.3 * 10 9+ 3.23 * 10 to the -6 we have to get our decimal spot in the same place however what's our highest power remember negative numbers work backwards so our higher Powers actually are -6 and there's three differ so we're going to move our decimal 1 2 3 so 0. 73 * 10 6 + 3.23 * 10 to the -6 last place value they have in common is right there so we add those together we round on the basis of that seven and that seven alone so we get 2 3.24 * 10 -6 G let me show you one other way of writing an answer just in case we haven't seen it before often you're doing one calculation in order to use the answer of that calculation in another calculation if you round every single time you do a calculation you'll cause a phenomenon known as propagation of error and when you get to um calculus based physics you will have lots of fun doing the math behind propagation of error but for our purposes right now we're wanting to minimize it so you can minimize propagation of error by carrying forward in your calculations one insignificant number in other words delaying your rounding to the end to help us keep track of our significant figures as we're carrying a number forward in calculations we use a special notation instead of writing 3.24 * 10 6 for my answer for this problem I could write it 3.23 7 as a subscript time 10 to the6 G writing this number as a subscript tells me that that is an insignificant number that I wish to carry forward for purposes of calculation and for purposes of avoiding propagation of error this is notation that you will see used all the time by chemists and physicists um I will often lapse into it by default you'll see a lot of the examples in this class I will write my answers like this as opposed to like this where I'm showing you I have three Sig whoops where I'm showing you that I have three sigfigs but I'm also carrying forward an insignificant digit in my calculation by writing it as a subscript and that's considered good form it's notation you're going to see um quite frequently everybody write down Ahsoka I'm really enjoying that TV show and that's it for this segment