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We saw in the last video that when you multiply or you divide

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numbers, or (I guess I should say when you multiply or

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divide measurements) your result can only have as many

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significant digits as the thing with the smallest significant

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digits you ended up multiplying and dividing.

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So just as a quick example, if I have 2.00 times (I don't know)

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3.5 my answer over here can only have 2 significant digits

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This has 2 significant digits, this has 3. 2 times 3.5

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is 7, and we can get to 1 zero to the right of the decimal.

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Because we can have 2 significant digits.

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This was 3, this is 2.

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We only limited it to 2, because that was the smallest

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number of significant digits we had in all of the things

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that we were taking the product of.

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When we do addition and subtraction, it's a little bit different.

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And I'll do an example first.

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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.

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And obviously even my real world examples aren't really real world.

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In my last video, I talked about laying down carpet

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and someone rightfully pointed out,"Hey, if you are laying down

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carpet, you always want to round up. Just because you don't wanna

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it's easier to cut carpet away, then somehow glue carpet there.

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But that's particular to carpet. I was just saying a general way

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to think about precision in significant figures.

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That was only particular to carpets or tiles.

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But when you add, when you add, or subtract,

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now these significant digits or these significant figures don't matter as much

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as the actual precision of the things that you are adding.

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How many decimal places do you go? For example,

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if I were to add 1.26, and I were to add it to - let's say - to 2.3.

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If you just add these two numbers up, and let's say these are measurements,

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so when you make it (these are clearly 3

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significant digits) we're able to measure to the nearest hundreth.

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Here this is two significant digits so three significant digits

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this is two significant digits, we are able to measure to the nearest tenth.

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Let me label this. This is the hundredth

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and this is the tenth. When you add or subtract numbers,

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your answer, so if you just do this, if we just add these

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two numbers, I get - what? - 3.56.

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The sum, or the difference whatever you take, you don't count significant figures

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You don't say,"Hey, this can only have two significant figures."

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What you can say is, "This can only be as precise as the least precise thing that I had over here.

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The least precise thing I had over here is 2.3.

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It only went to the tenths place, so in our answer

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we can only go to the tenths place. So we need to round

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this guy up. Cause we have a six right here, so we round up

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so if you care about significant figures, this is going to become a 3.7.

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And I want to be clear. This time it worked out,

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cause this also has 2 significant figures,

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this also has two significant figures. But this could have been...

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(let me do another situation) you could have 1.26 plus

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102.3, and you would get obviously 103.56.

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Then, in this situation - this obviously over here has 4 significant figures,

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this over here has 3 significant figures. But in our answer

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we don't want to have 3 significant figures. We wanna have the...

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only as precise as the least precise thing that we added up.

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The least precise thing we only go one digit behind the decimal over here,

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so we can only go to the tenth, only one digit over the decimal there.

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So once again, we round it up to 103.6.

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And to see why that makes sense, let's do a little bit of an example here

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with actually measuring something.

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So let's say we have a block here,

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let's say that I have a block, we draw that block a little bit neater,

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and let's say we have a pretty good meter stick,

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and we're able to measure to the nearest centimeter,

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we get, it is 2.09 meters.

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Let's say we have another block, and this is the other block right over there.

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We have a, let's say we have an even more precise meter stick,

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which can measure to the nearest millimeter.

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And we get this to be 1.901 meters.

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So measuring to the nearest millimeter.

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And let's say those measurements were done a long time ago,

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and we don't have access to measure them any more,

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but someone says 'How tall is it if I were stack the blue block

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on the top of the red block - or the orange block, or whatever that color that is?"

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So how high would this height be?

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Well, if you didn't care about significant figures or precision,

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you would just add them up.

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You'd add the 1.901 plus the 2.09.

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So let me add those up:

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so if you take 1.901 and add that to 2.09,

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you get 1 plus nothing is 1,

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0 plus 9 is 9,

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9 plus 0 is 9,

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you get the decimal point,

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1 plus 2 is 3. So you get 3.991.

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And the problem with this, the reason why this is a little bit...

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it's kind of misrepresenting how precise you measurement is.

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You don't know, if I told you that the tower is 3.991 meters tall,

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I'm implying that I somehow was able to measure

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the entire tower to the nearest millimeter.

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The reality is that I was only be able to

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measure the part of the tower to the millimeter.

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This part of the tower I was able to measure to the nearest centimeter.

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So to make it clear the our measurement is only good

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to the nearest centimeter,

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because there is more error here, then...

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it might overwhelm or whatever the precision we had on the millimeters there.

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To make that clear, we have to make this only as precise

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as the least precise thing that we are adding up.

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So over here, the least precise thing was, we went to the hundredths,

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so over here we have to round to the hundredths.

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So, since 1 is less then 5, we are going to round down,

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and so we can only legitimately say,

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if we want to represent what we did properly

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that the tower is 3.99 meters.

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And I also want to make it clear that this doesn't just apply

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to when there is a decimal point.

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If I were tell you that... Let's say that I were to measure...

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I want to measure a building.

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I was only able to measure the building to the nearest 10 feet.

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So I tell you that that building is 350 feet tall.

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So this is the building.

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This is a building.

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And let's say there is a manufacturer of radio antennas, so...

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or radio towers.

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And the manufacturers has measured their tower to the nearest foot.

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And they say, their tower is 8 feet tall.

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So notice: here they measure to the nearest 10 feet,

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here they measure to the nearest foot.

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And actually to make it clear, because once again,

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as I said, this is ambiguous,

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it's not 100% clear how many significant figures there are.

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Maybe it was exactly 350 feet or maybe they just rounded it to the nearest 10 feet.

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So a better way to represent this,

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they... would be to say instead of writing it 350,

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a better way to write it would be 3.5 times 10 to the second feet tall.

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And when you are writing in scientific notation,

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that makes it very clear that there is only 2 significant digits here,

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you are only measuring to the nearest 10 feet.

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Other way to represent it:

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you could write 350 this notation has done less,

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but sometimes the last significant digit has a line on the top of it,

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or the last significant digit has a line below it.

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Either of those are ways to specify it,

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this is probably the least ambiguous,

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but assuming that they only make measure to the nearest 10 feet,

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If someone were ask you: "How tall is the building plus the tower?"

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Well, your first reaction were, let's just add

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the 350 plus 8, you get 358.

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You'd get 358 feet. So this is the building plus the tower. 358 feet.

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For once again, we are misrepresenting it.

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We are making it look like we were able

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to measure the combination to the nearest foot.

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But we were able to measure only the tower to the nearest foot.

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So in order to represent our measurement at the level of precision at we really did,

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we really have to round this to the nearest 10 feet.

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Because that was our least precise measurement.

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So we would really have to round this up to,

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8 is greater-than-or-equal to 5,

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so we round this up to 360 feet.

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So once again, whatever is...

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Just to make it clear, even this ambiguous,

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maybe we put a line over to show, that is our level of precision,

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that we have 2 significant digits.

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Or we could write this as 3.6 times 10 to the second.

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Which is times 100.

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3.6 times 10 to the second feet in scientific notation.

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And this makes it very clear that we only have 2 significant digits here.