Let's look at how sig figs are propagated in calculations. Now I also want to put a big note here to tell you that I will tell you when you need to do sig figs in a calculation. If it does not say report your answer with the correct number of significant figures, then you do not have to explicitly track your sig figs through the calculation steps. I mean, please don't report a crazy number of sig figs or a crazy number of digits or a crazy low number of digits.
Go for somewhere between three and five sig like digits if you are not tracking sig figs. And if you have really big or really small numbers, put them in scientific notation. And your calculator will do this for you.
And this is something that I can show you when we are meeting in person. Okay. So propagating sig figs through calculations follows a set of rules based on the operation that you're doing. So multiplication and division, those two operations are related and the rule that we use to propagate sig figs is the same for multiplication and division.
And the rule that we use is that the answer has the same number of sig figs as the factor with the fewest number. of sig figs. So in this example, I have 1.052 times 12.504 times 0.53. Now, if I'm doing this calculation, I'm going to plug all of those values into my calculator, get the big number, everything.
I'm not going to cut it off. I'm just going to write it out. And if you do that, you get 6.7208.
And then I'm going to go back and check the number of sig figs. that each digit has and use that to determine the number of sig figs in my answer. So 1.052, I have an interior zero, right, it's trapped between the numbers that are definitely significant. This number has four sig figs. 12.504, same kind of case, five sig figs.
And then 0.53, the zero that's written here, It's a leading zero that is just kind of a placeholder. We didn't actually need it for the value, but when we write numbers that are less than one, we'll always put the zero there just so that like the decimal point stands out. So that zero is not significant, but the five and the three are. So this only has two sig figs and four versus five versus two sig figs.
Two is the least number of sig figs. So my answer will need to be reported with. two sig figs.
So I have 6.7208. If I want to cut this off at two sig figs, I would cut it off at 6.7. Let's do a division example.
If I take 2.0035, which has five sig figs, and divide by 3.20, that trailing zero after a decimal place is significant. So this has three sig figs. My full answer, without cutting it off, would be 0.626094, and the least number of sig figs I have is 3, so I would cut this off at 3 sig figs, which would be 0.626.
The second rule is for addition and subtraction. This rule, these operations are related. They have the same rule for propagating sig figs.
It is that the answer has the same number of decimal places as the quantity with the fewest decimal places. Now, this rule uses the term decimal places, but it means like placeholder. So like the least number of decimal places, you could have something where the last significant digit in that value cuts off in the text.
pens place, right? That's not something we would normally consider a decimal place, but we're looking at the value of the place where that last significant digit is. And for that reason, you will see me write my addition and subtraction sig fig propagation things stacked. So it will look, right, it'll look like I'm doing second grade arithmetic again, but It is the way that makes the most sense to me when I go to set this up. So I'm going to do 2.345 plus 0.07.
plus 2.9975. If I just add all of these numbers together, I get 5.4125. Now the values that I have that I've added up, I have 2.345. Now this is four significant figures, but the full number of significant figures doesn't matter in addition and subtraction.
It would matter if I was doing multiplication division. So what I care about is the fact that this goes to three decimal places. And so that's why I have this kind of in parentheses after. 0.07 goes to two decimal places, but only has one significant digit, right?
Because the zeros are leading zeros and placeholders. And then 2.9975 goes to four decimal places. Now the least number of decimal places is two decimal places.
from the point zero seven. So my answer will need to go to two decimal places. So 5.41 is how I would report my answer. Now what I do to make this even easier is if you find the spot at which it will be cut off, when it's written stacked like this you can just draw a great big vertical line after the digit that is determining the cutoff point.
And then if you write your answer in those placeholder spaces, it will show you exactly where your answer needs to cut off. So let's do this with a subtraction. The rule is the same.
So if we have 5.9, which goes to one decimal place, minus 0.221, which goes to three decimal places, the least number of decimal places is one decimal place. And so that's where I have my vertical line drawn. And my full answer would have been 5.679, but I'm going to cut this off at one decimal place. And I think this is actually the first case where I'm cutting it off, but I'm not going to report 5.6 because the digit after the six is a seven. So I'm going to round up based on that digit and I'm going to report 5.7, which takes us perfectly into talking about rounding.
So the rounding rules that we use are pretty much the same rules that you have always encountered when rounding with one exception. So when you go to round something, if the dropped digit, so the digit after the one that you're keeping, is between zero and four, you'll round down. If it's between six and nine, you'll round up. And if it is exactly five, the short version of this rule is to make the answer even.
Now, for how much time we will spend talking about rounding with fives, you will not encounter it very often. So when you round with something that has a five, it is an exact. 5. So it means that it is a 5 and then absolutely nothing else after that value. So if you have something that ends in 501, that's greater than 5 and you should round up. And if it's 5-0-0-0-1, that's still greater than 5 and you should round up.
So the only time that we use this make the answer even, which what that means is that if you had 92.85 exactly, your answer would be 92.8, which is even. If you had 92.75 exactly, your answer would be 92.8, which is even. So that's the way that that rule goes.
But again, it's only when you have exactly five at the end. And the last thing to consider when you're doing multi-step calculations, do not round any of the intermediate values. So when you're doing the calculation, you should mark.
where the sig figs would be cut off or just go back and go through the entire calculation again to count your significant digits. But if you do a calculation and then round and then do the next calculation and then round, it's very possible and it happens almost every time where you can accidentally round your way away from the correct answer. So it's possible. to round up and then round up again and just get so far from the correct answer that even though you did everything right, you don't have the correct answer because you rounded away from it.
So carry through extra digits when you do the calculations and then don't actually round anything until the end. You'll still need to track those sig figs, but you won't round them until you get to the very end. of the calculation.