Addition and subtraction with significant figures Addition and Subtraction with Significant Figures
Addition and subtraction with significant figures
- 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.
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