We saw in the last video that when you multiply or you divide numbers, or I guess we should say when we multiply or divide measurements, your result can only have as many significant digits as the thing with the smallest significant digits that you ended up multiplying and dividing. So just as a quick example, if I have 2.00 times, I don't know, let me, 3.00. 3.5, my answer over here can only have two significant digits. This has two significant digits, this has three. Two times 3.5 is seven, and we can get to one level, we can go one zero 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 numeric example first, and 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 particular to carpet. I was just saying a general way to think about precision and significant figures. But that was...
particular only to maybe carpets or tiles. But 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 1.26 and I were to add it to, let's say I were to add it to 2.3, 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.
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 hundredth. This 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 2 point, or I get 3.56.
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 had over here. The least precise thing I had over here was 2.3.
It only went to the tenths place. So in our answer, we can only go to the tenths place. So we need to round this guy up.
Because we have a 6 right here. So you round up. So if you care about significant figures, this is going to become a 3.7. 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 1.26 plus 102.3. And then you would get, obviously, you would get 103.56.
And in this situation, this obviously over here has four significant figures. This over here has three significant figures. But in our answer, we don't want to have three significant figures.
We want to 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 tenths, only one digit behind the decimal there.
So once again, we round up to 103.6. And to see why that makes sense, let's do a little bit of a. 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. It is 2.09 meters.
And let's say we have another block. This is the other block right over there. And 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 1.901 meters. So we're measuring to the nearest millimeter. 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 1.901 plus the 2.09.
So let me add those up. So if you take 1.901 and add that to, 2.09, you get 1 plus nothing is 1, 0 plus 9 is 9, 9 plus 0 is 9, you get the decimal point, 1 plus 2 is 3. So you would get 3.991. 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.
If I told you that the tower is 3.991 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 to the nearest centimeter.
So to make it clear that our measurement is only good to the nearest centimeter, because there's more error here than, you know, it might overwhelm whatever precision we had on the millimeters 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 hundreds, so over here we have to round to the hundreds. And since 1 is less than 5, 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 3.99 meters. And I also want to make it clear that this doesn't just apply to when there's a decimal point.
If I were to tell you that, let's say that 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 350 feet tall.
So this is the building. This is a 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 feet tall.
So notice, here they measured to the nearest 10 feet. Here they measured to the nearest foot. And actually, to make it clear, because once again, 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 would be to say, instead of writing it 350, a better way to write it would be 3.5 times 10 to the second feet tall. And when you write it in scientific notation, it makes it very clear that there's only two significant digits here. You only measured 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 it. This is probably the least ambiguous. But assuming that they only measured to the nearest 10 feet, how would you?
And someone would ask you, how tall is the building plus the tower? Well, your first reaction would just be, well, let's just add the 350 plus 8. 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 360 feet. So once again, whatever is...
And just to make it clear, even this is ambiguous, maybe we put a line over it to show that that is our level of precision, that we only have two significant digits, or... we could write this as 3.6 times 10 to the second, which is just 100, 3.6 times 10 to the second feet in scientific notation. And this makes it very clear that we only have two significant digits here.