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## Arithmetic (all content)

# Addition and subtraction with significant figures

Learn to add and subtract with significant figures. Created by Sal Khan.

## Video transcript

We saw in the last video that when you multiply or you divide numbers, or (I guess I should say when you multiply or divide measurements) your result can only have as many significant digits as the thing with the smallest significant digits you ended up multiplying and dividing. So just as a quick example, if I have 2.00 times (I don't know) 3.5 my answer over here can only have 2 significant digits This has 2 significant digits, this has 3. 2 times 3.5 is 7, and we can get to 1 zero to the right of the decimal. Because we can have 2 significant digits. This was 3, this is 2. We only limited it to 2, 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 just do a kind of a numerical example first, and then I'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 my last video, I talked about laying down carpet and someone rightfully pointed out,"Hey, if you are laying down carpet, you always want to round up. Just because you don't wanna it's easier to cut carpet away, then somehow glue carpet there. But that's particular to carpet. I was just saying a general way to think about precision in significant figures. That was only particular to carpets or tiles. But when you add, when you add, or subtract, now these significant digits or these significant figures don't matter as much as the actual precision of the things that you are adding. How many decimal places do you go? For example, if I were to add 1.26, and I were to add it to - let's say - to 2.3. If you just add these two numbers up, and let's say these are measurements, so when you make it (these are clearly 3 significant digits) we're able to measure to the nearest hundreth. Here this is two significant digits so three significant digits this is two significant digits, we are able to measure to the nearest tenth. Let me label this. This is the hundredth and this is the tenth. When you add or subtract numbers, your answer, so if you just do this, if we just add these two numbers, I get - what? - 3.56. The sum, or the difference whatever you take, you don't count significant figures You don't say,"Hey, this can only have two significant figures." What you can 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 is 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. Cause we have a six right here, so we 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, cause this also has 2 significant figures, this also has two significant figures. But this could have been... (let me do another situation) you could have 1.26 plus 102.3, and you would get obviously 103.56. Then, in this situation - this obviously over here has 4 significant figures, this over here has 3 significant figures. But in our answer we don't want to have 3 significant figures. We wanna have the... 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 tenth, only one digit over the decimal there. So once again, we round it up to 103.6. 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 we have a block here, let's say that I have a block, we draw that block a little bit neater, and let's say we have a pretty good meter stick, and we're able to measure to the nearest centimeter, we get, it is 2.09 meters. Let's say we have another block, and this is the other block right over there. We have a, let's say we have an even more precise meter stick, which can measure to the nearest millimeter. And we get this to be 1.901 meters. So measuring to the nearest millimeter. And let's say those measurements were done a long time ago, and we don't have access to measure them any more, but someone says 'How tall is it if I were stack the blue block on the top of the red block - or the orange block, or whatever that color that is?" So how high would this height be? Well, if you didn't care about significant figures or precision, you would just add them up. You'd 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 get 3.991. And the problem with this, the reason why this is a little bit... it's kind of misrepresenting how precise you measurement is. 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 that I was only be able to measure the part of the tower to the millimeter. This part of the tower I was able to measure to the nearest centimeter. So to make it clear the our measurement is only good to the nearest centimeter, because there is more error here, then... it might overwhelm or whatever the precision we had on the millimeters there. To make that clear, we have to make this only as precise as the least precise thing that we are 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. So, since 1 is less then 5, we are 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 is a decimal point. If I were tell you that... Let's say that I were to measure... I want to measure a building. 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 there is a manufacturer of radio antennas, so... or radio towers. And the manufacturers has measured their tower to the nearest foot. And they say, their tower 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 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 it to the nearest 10 feet. So a better way to represent this, they... 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 are writing in scientific notation, that makes it very clear that there is only 2 significant digits here, you are only measuring to the nearest 10 feet. Other way to represent it: you could write 350 this notation has done less, but sometimes the last significant digit has a line on the top of it, or the last significant digit has a line below it. Either of those are ways to specify it, this is probably the least ambiguous, but assuming that they only make measure to the nearest 10 feet, If someone were ask you: "How tall is the building plus the tower?" Well, your first reaction were, let's just add the 350 plus 8, you get 358. You'd get 358 feet. So this is the building plus the tower. 358 feet. For once again, we are misrepresenting it. We are making it look like we were able to measure the combination to the nearest foot. But we were able to measure only the tower to the nearest foot. So in order to represent our measurement at the level of precision at 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 to, 8 is greater-than-or-equal to 5, so we round this up to 360 feet. So once again, whatever is... Just to make it clear, even this ambiguous, maybe we put a line over to show, that is our level of precision, that we have 2 significant digits. Or we could write this as 3.6 times 10 to the second. Which is times 100. 3.6 times 10 to the second feet in scientific notation. And this makes it very clear that we only have 2 significant digits here.